3.5.73 \(\int \frac {(a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4)}{x^{12}} \, dx\) [473]

Optimal. Leaf size=796 \[ -\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {b^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{7/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{49280 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-1/27720*(2520*c/x^11+2772*d/x^10+3080*e/x^9+3465*f/x^8+3960*g/x^7)*(b*x^3+a)^(3/2)+1/24*b^3*e*arctanh((b*x^3+
a)^(1/2)/a^(1/2))/a^(3/2)-1/18480*b*(945*c/x^8+1188*d/x^7+1540*e/x^6+2079*f/x^5+2970*g/x^4)*(b*x^3+a)^(1/2)-27
/1760*b^2*c*(b*x^3+a)^(1/2)/a/x^5-27/1120*b^2*d*(b*x^3+a)^(1/2)/a/x^4-1/24*b^2*e*(b*x^3+a)^(1/2)/a/x^3+27/7040
*b^2*(-22*a*f+7*b*c)*(b*x^3+a)^(1/2)/a^2/x^2+27/448*b^2*(-4*a*g+b*d)*(b*x^3+a)^(1/2)/a^2/x-27/448*b^(7/3)*(-4*
a*g+b*d)*(b*x^3+a)^(1/2)/a^2/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+27/896*3^(1/4)*b^(7/3)*(-4*a*g+b*d)*(a^(1/3)+b^(1
/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1
/2*2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/a^(5/3)/(b*x^3+a
)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)+9/49280*3^(3/4)*b^(7/3)*(a^(1/3)
+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(7*b^(1/3
)*(-22*a*f+7*b*c)+110*a^(1/3)*(-4*a*g+b*d)*(1-3^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+
b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/a^2/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3
)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.62, antiderivative size = 796, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 1839, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \begin {gather*} \frac {e \tanh ^{-1}\left (\frac {\sqrt {b x^3+a}}{\sqrt {a}}\right ) b^3}{24 a^{3/2}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b d-4 a g) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{7/3}}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {b x^3+a}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) b^{7/3}}{49280 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {b x^3+a}}-\frac {27 (b d-4 a g) \sqrt {b x^3+a} b^{7/3}}{448 a^2 \left (\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}+\frac {27 (b d-4 a g) \sqrt {b x^3+a} b^2}{448 a^2 x}+\frac {27 (7 b c-22 a f) \sqrt {b x^3+a} b^2}{7040 a^2 x^2}-\frac {e \sqrt {b x^3+a} b^2}{24 a x^3}-\frac {27 d \sqrt {b x^3+a} b^2}{1120 a x^4}-\frac {27 c \sqrt {b x^3+a} b^2}{1760 a x^5}-\frac {\left (\frac {945 c}{x^8}+\frac {2970 g}{x^4}+\frac {2079 f}{x^5}+\frac {1540 e}{x^6}+\frac {1188 d}{x^7}\right ) \sqrt {b x^3+a} b}{18480}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {3960 g}{x^7}+\frac {3465 f}{x^8}+\frac {3080 e}{x^9}+\frac {2772 d}{x^{10}}\right ) \left (b x^3+a\right )^{3/2}}{27720} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x]

[Out]

-1/18480*(b*((945*c)/x^8 + (1188*d)/x^7 + (1540*e)/x^6 + (2079*f)/x^5 + (2970*g)/x^4)*Sqrt[a + b*x^3]) - (27*b
^2*c*Sqrt[a + b*x^3])/(1760*a*x^5) - (27*b^2*d*Sqrt[a + b*x^3])/(1120*a*x^4) - (b^2*e*Sqrt[a + b*x^3])/(24*a*x
^3) + (27*b^2*(7*b*c - 22*a*f)*Sqrt[a + b*x^3])/(7040*a^2*x^2) + (27*b^2*(b*d - 4*a*g)*Sqrt[a + b*x^3])/(448*a
^2*x) - (27*b^(7/3)*(b*d - 4*a*g)*Sqrt[a + b*x^3])/(448*a^2*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((2520*c)/
x^11 + (2772*d)/x^10 + (3080*e)/x^9 + (3465*f)/x^8 + (3960*g)/x^7)*(a + b*x^3)^(3/2))/27720 + (b^3*e*ArcTanh[S
qrt[a + b*x^3]/Sqrt[a]])/(24*a^(3/2)) + (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/3)*(b*d - 4*a*g)*(a^(1/3) + b^(1/3)
*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[(
(1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(896*a^(5/3)*Sqrt[(a
^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]) + (9*3^(3/4)*Sqrt[2 + Sq
rt[3]]*b^(7/3)*(7*b^(1/3)*(7*b*c - 22*a*f) + 110*(1 - Sqrt[3])*a^(1/3)*(b*d - 4*a*g))*(a^(1/3) + b^(1/3)*x)*Sq
rt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - S
qrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(49280*a^2*Sqrt[(a^(1/3)*(
a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1839

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a +
b*x^n)^p, x] - Dist[b*n*p, Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a
, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1, 0]

Rule 1846

Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[Coeff[Pq, x, 0], Int[1/(x*Sqrt[a + b*x^n]), x
], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] &
& IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]

Rule 1849

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0
*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[2*a*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 1892

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{12}} \, dx &=-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{11}-\frac {d x}{10}-\frac {e x^2}{9}-\frac {f x^3}{8}-\frac {g x^4}{7}\right )}{x^9} \, dx\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {1}{4} \left (27 b^2\right ) \int \frac {\frac {c}{88}+\frac {d x}{70}+\frac {e x^2}{54}+\frac {f x^3}{40}+\frac {g x^4}{28}}{x^6 \sqrt {a+b x^3}} \, dx\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {\left (27 b^2\right ) \int \frac {-\frac {a d}{7}-\frac {5 a e x}{27}+\frac {1}{88} (7 b c-22 a f) x^2-\frac {5}{14} a g x^3}{x^5 \sqrt {a+b x^3}} \, dx}{40 a}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {\left (27 b^2\right ) \int \frac {\frac {40 a^2 e}{27}-\frac {1}{11} a (7 b c-22 a f) x-\frac {5}{7} a (b d-4 a g) x^2}{x^4 \sqrt {a+b x^3}} \, dx}{320 a^2}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {\left (9 b^2\right ) \int \frac {\frac {6}{11} a^2 (7 b c-22 a f)+\frac {30}{7} a^2 (b d-4 a g) x+\frac {40}{9} a^2 b e x^2}{x^3 \sqrt {a+b x^3}} \, dx}{640 a^3}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {\left (9 b^2\right ) \int \frac {-\frac {120}{7} a^3 (b d-4 a g)-\frac {160}{9} a^3 b e x+\frac {6}{11} a^2 b (7 b c-22 a f) x^2}{x^2 \sqrt {a+b x^3}} \, dx}{2560 a^4}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {\left (9 b^2\right ) \int \frac {\frac {320}{9} a^4 b e-\frac {12}{11} a^3 b (7 b c-22 a f) x+\frac {120}{7} a^3 b (b d-4 a g) x^2}{x \sqrt {a+b x^3}} \, dx}{5120 a^5}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {\left (9 b^2\right ) \int \frac {-\frac {12}{11} a^3 b (7 b c-22 a f)+\frac {120}{7} a^3 b (b d-4 a g) x}{\sqrt {a+b x^3}} \, dx}{5120 a^5}-\frac {\left (b^3 e\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}-\frac {\left (b^3 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{48 a}-\frac {\left (27 b^{8/3} (b d-4 a g)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{896 a^2}+\frac {\left (27 b^{8/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d-4 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{98560 a^2}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{7/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{49280 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\left (b^2 e\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{24 a}\\ &=-\frac {b \left (\frac {945 c}{x^8}+\frac {1188 d}{x^7}+\frac {1540 e}{x^6}+\frac {2079 f}{x^5}+\frac {2970 g}{x^4}\right ) \sqrt {a+b x^3}}{18480}-\frac {27 b^2 c \sqrt {a+b x^3}}{1760 a x^5}-\frac {27 b^2 d \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 e \sqrt {a+b x^3}}{24 a x^3}+\frac {27 b^2 (7 b c-22 a f) \sqrt {a+b x^3}}{7040 a^2 x^2}+\frac {27 b^2 (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b d-4 a g) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {2520 c}{x^{11}}+\frac {2772 d}{x^{10}}+\frac {3080 e}{x^9}+\frac {3465 f}{x^8}+\frac {3960 g}{x^7}\right ) \left (a+b x^3\right )^{3/2}}{27720}+\frac {b^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{7/3} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{49280 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.80, size = 1017, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (-243 b^3 x^9 (49 c+110 d x)+16 a^3 (2520 c+11 x (252 d+5 x (56 e+9 x (7 f+8 g x))))+6 a b^2 x^6 (1134 c+11 x (162 d+x (280 e+81 x (7 f+20 g x))))+8 a^2 b x^3 (7875 c+11 x (828 d+x (980 e+9 x (133 f+170 g x))))\right )}{443520 a^2 x^{11}}+\frac {b^{7/3} \left (6160 \sqrt {a} b^{2/3} e \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-3969 b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )+12474 a \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )-8910 \sqrt {2} \sqrt [3]{a} b d \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )+35640 \sqrt {2} a^{4/3} g \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )\right )}{147840 a^2 \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x]

[Out]

-1/443520*(Sqrt[a + b*x^3]*(-243*b^3*x^9*(49*c + 110*d*x) + 16*a^3*(2520*c + 11*x*(252*d + 5*x*(56*e + 9*x*(7*
f + 8*g*x)))) + 6*a*b^2*x^6*(1134*c + 11*x*(162*d + x*(280*e + 81*x*(7*f + 20*g*x)))) + 8*a^2*b*x^3*(7875*c +
11*x*(828*d + x*(980*e + 9*x*(133*f + 170*g*x))))))/(a^2*x^11) + (b^(7/3)*(6160*Sqrt[a]*b^(2/3)*e*Sqrt[(a^(1/3
) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]] - 3969*
b^(4/3)*c*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[((-1)^(
1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)
*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + 12474*a*b^(1/3)*f*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqr
t[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (
-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)
^(1/3)] - 8910*Sqrt[2]*a^(1/3)*b*d*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(
1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*
EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF
[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) + 35640*Sqrt[2]*a^(4
/3)*g*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^
(1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1
/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*
b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))])))/(147840*a^2*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*
x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1772 vs. \(2 (628 ) = 1256\).
time = 0.45, size = 1773, normalized size = 2.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1006\)
risch	\(\text {Expression too large to display}\)	\(1639\)
default	\(\text {Expression too large to display}\)	\(1773\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^12,x,method=_RETURNVERBOSE)

[Out]

c*(-1/11*a*(b*x^3+a)^(1/2)/x^11-25/176*b*(b*x^3+a)^(1/2)/x^8-27/1760*b^2/a*(b*x^3+a)^(1/2)/x^5+189/7040*b^3/a^
2*(b*x^3+a)^(1/2)/x^2-63/7040*I*b^3/a^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*
b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2
)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(
-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)
))+e*(-1/9*a*(b*x^3+a)^(1/2)/x^9-7/36*b*(b*x^3+a)^(1/2)/x^6-1/24*b^2/a*(b*x^3+a)^(1/2)/x^3+1/24*b^3/a^(3/2)*ar
ctanh((b*x^3+a)^(1/2)/a^(1/2)))+f*(-1/8*a*(b*x^3+a)^(1/2)/x^8-19/80*b*(b*x^3+a)^(1/2)/x^5-27/320*b^2/a*(b*x^3+
a)^(1/2)/x^2+9/320*I*b^2/a*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3
^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))
^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(
1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3)
)^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+g*(-1/7*a*
(b*x^3+a)^(1/2)/x^7-17/56*b*(b*x^3+a)^(1/2)/x^4-27/112*b^2/a*(b*x^3+a)^(1/2)/x-9/112*I*b^2/a*3^(1/2)*(-a*b^2)^
(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b
^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(
1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b
*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-
a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))
+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b
/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/
2))))+d*(-1/10*a*(b*x^3+a)^(1/2)/x^10-23/140*b*(b*x^3+a)^(1/2)/x^7-27/1120*b^2/a*(b*x^3+a)^(1/2)/x^4+27/448*b^
3/a^2*(b*x^3+a)^(1/2)/x+9/448*I*b^3/a^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*
b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2
)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-
a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2
/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b
*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-
3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^12,x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + x^2*e + d*x + c)*(b*x^3 + a)^(3/2)/x^12, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 606, normalized size = 0.76 \begin {gather*} \left [\frac {4620 \, \sqrt {a} b^{3} e x^{11} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} + 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} \sqrt {b} x^{11} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} \sqrt {b} x^{11} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (18480 \, a b^{2} e x^{8} - 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} x^{10} - 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} x^{9} + 86240 \, a^{2} b e x^{5} + 396 \, {\left (27 \, a b^{2} d + 340 \, a^{2} b g\right )} x^{7} + 252 \, {\left (27 \, a b^{2} c + 418 \, a^{2} b f\right )} x^{6} + 49280 \, a^{3} e x^{2} + 44352 \, a^{3} d x + 3168 \, {\left (23 \, a^{2} b d + 20 \, a^{3} g\right )} x^{4} + 40320 \, a^{3} c + 2520 \, {\left (25 \, a^{2} b c + 22 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{443520 \, a^{2} x^{11}}, -\frac {9240 \, \sqrt {-a} b^{3} e x^{11} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) - 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} \sqrt {b} x^{11} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} \sqrt {b} x^{11} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (18480 \, a b^{2} e x^{8} - 26730 \, {\left (b^{3} d - 4 \, a b^{2} g\right )} x^{10} - 1701 \, {\left (7 \, b^{3} c - 22 \, a b^{2} f\right )} x^{9} + 86240 \, a^{2} b e x^{5} + 396 \, {\left (27 \, a b^{2} d + 340 \, a^{2} b g\right )} x^{7} + 252 \, {\left (27 \, a b^{2} c + 418 \, a^{2} b f\right )} x^{6} + 49280 \, a^{3} e x^{2} + 44352 \, a^{3} d x + 3168 \, {\left (23 \, a^{2} b d + 20 \, a^{3} g\right )} x^{4} + 40320 \, a^{3} c + 2520 \, {\left (25 \, a^{2} b c + 22 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{443520 \, a^{2} x^{11}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^12,x, algorithm="fricas")

[Out]

[1/443520*(4620*sqrt(a)*b^3*e*x^11*log((b^2*x^6 + 8*a*b*x^3 + 4*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(a) + 8*a^2)
/x^6) + 1701*(7*b^3*c - 22*a*b^2*f)*sqrt(b)*x^11*weierstrassPInverse(0, -4*a/b, x) + 26730*(b^3*d - 4*a*b^2*g)
*sqrt(b)*x^11*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (18480*a*b^2*e*x^8 - 26730*(b^3*
d - 4*a*b^2*g)*x^10 - 1701*(7*b^3*c - 22*a*b^2*f)*x^9 + 86240*a^2*b*e*x^5 + 396*(27*a*b^2*d + 340*a^2*b*g)*x^7
+ 252*(27*a*b^2*c + 418*a^2*b*f)*x^6 + 49280*a^3*e*x^2 + 44352*a^3*d*x + 3168*(23*a^2*b*d + 20*a^3*g)*x^4 + 4
0320*a^3*c + 2520*(25*a^2*b*c + 22*a^3*f)*x^3)*sqrt(b*x^3 + a))/(a^2*x^11), -1/443520*(9240*sqrt(-a)*b^3*e*x^1
1*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) - 1701*(7*b^3*c - 22*a*b^2*f)*sqrt(b)*x^1
1*weierstrassPInverse(0, -4*a/b, x) - 26730*(b^3*d - 4*a*b^2*g)*sqrt(b)*x^11*weierstrassZeta(0, -4*a/b, weiers
trassPInverse(0, -4*a/b, x)) + (18480*a*b^2*e*x^8 - 26730*(b^3*d - 4*a*b^2*g)*x^10 - 1701*(7*b^3*c - 22*a*b^2*
f)*x^9 + 86240*a^2*b*e*x^5 + 396*(27*a*b^2*d + 340*a^2*b*g)*x^7 + 252*(27*a*b^2*c + 418*a^2*b*f)*x^6 + 49280*a
^3*e*x^2 + 44352*a^3*d*x + 3168*(23*a^2*b*d + 20*a^3*g)*x^4 + 40320*a^3*c + 2520*(25*a^2*b*c + 22*a^3*f)*x^3)*
sqrt(b*x^3 + a))/(a^2*x^11)]

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Sympy [A]
time = 13.53, size = 541, normalized size = 0.68 \begin {gather*} \frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{3}, - \frac {1}{2} \\ - \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{11} \Gamma \left (- \frac {8}{3}\right )} + \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {10}{3}, - \frac {1}{2} \\ - \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{10} \Gamma \left (- \frac {7}{3}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} g \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {\sqrt {a} b g \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} - \frac {a^{2} e}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {11 a \sqrt {b} e}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {17 b^{\frac {3}{2}} e}{72 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {5}{2}} e}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{3} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{24 a^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**12,x)

[Out]

a**(3/2)*c*gamma(-11/3)*hyper((-11/3, -1/2), (-8/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**11*gamma(-8/3)) + a**(3/
2)*d*gamma(-10/3)*hyper((-10/3, -1/2), (-7/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**10*gamma(-7/3)) + a**(3/2)*f*g
amma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**8*gamma(-5/3)) + a**(3/2)*g*gamma(-7/3
)*hyper((-7/3, -1/2), (-4/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3)) + sqrt(a)*b*c*gamma(-8/3)*hyper(
(-8/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**8*gamma(-5/3)) + sqrt(a)*b*d*gamma(-7/3)*hyper((-7/3, -
1/2), (-4/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3)) + sqrt(a)*b*f*gamma(-5/3)*hyper((-5/3, -1/2), (-
2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt(a)*b*g*gamma(-4/3)*hyper((-4/3, -1/2), (-1/3,), b
*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) - a**2*e/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - 11*a*sqrt(
b)*e/(36*x**(15/2)*sqrt(a/(b*x**3) + 1)) - 17*b**(3/2)*e/(72*x**(9/2)*sqrt(a/(b*x**3) + 1)) - b**(5/2)*e/(24*a
*x**(3/2)*sqrt(a/(b*x**3) + 1)) + b**3*e*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(24*a**(3/2))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^12,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + x^2*e + d*x + c)*(b*x^3 + a)^(3/2)/x^12, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^{12}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x)

[Out]

int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12, x)

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