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

Optimal. Leaf size=796 \[ \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 (\sin ^{-1}\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 (\sin ^{-1}\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} \]

[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 = 1.53, 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, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \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 (\sin ^{-1}\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 (\sin ^{-1}\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} \]

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]

-(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])/18480 - (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[Sqr
t[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 + Sqrt
[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)*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]*EllipticF[ArcSin[((1 - Sqr
t[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 63

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 208

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

Rule 218

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

Rule 266

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 1825

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 1832

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 1835

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq
0*(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 1877

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]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 1878

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 ) \operatorname {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 ) \operatorname {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]  time = 0.47, size = 194, normalized size = 0.24 \[ \frac {\sqrt {a+b x^3} \left (11 x^3 \left (-105 a^5 f \, _2F_1\left (-\frac {8}{3},-\frac {3}{2};-\frac {5}{3};-\frac {b x^3}{a}\right )-120 a^5 g x \, _2F_1\left (-\frac {7}{3},-\frac {3}{2};-\frac {4}{3};-\frac {b x^3}{a}\right )+112 b^3 e x^8 \left (a+b x^3\right )^2 \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {b x^3}{a}+1\right )\right )-840 a^5 c \, _2F_1\left (-\frac {11}{3},-\frac {3}{2};-\frac {8}{3};-\frac {b x^3}{a}\right )-924 a^5 d x \, _2F_1\left (-\frac {10}{3},-\frac {3}{2};-\frac {7}{3};-\frac {b x^3}{a}\right )\right )}{9240 a^4 x^{11} \sqrt {\frac {b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^3]*(-840*a^5*c*Hypergeometric2F1[-11/3, -3/2, -8/3, -((b*x^3)/a)] - 924*a^5*d*x*Hypergeometric2F
1[-10/3, -3/2, -7/3, -((b*x^3)/a)] + 11*x^3*(-105*a^5*f*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] - 12
0*a^5*g*x*Hypergeometric2F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] + 112*b^3*e*x^8*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*
Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^3)/a])))/(9240*a^4*x^11*Sqrt[1 + (b*x^3)/a])

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b g x^{7} + b f x^{6} + b e x^{5} + {\left (b d + a g\right )} x^{4} + a e x^{2} + {\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a}}{x^{12}}, x\right ) \]

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]

integral((b*g*x^7 + b*f*x^6 + b*e*x^5 + (b*d + a*g)*x^4 + a*e*x^2 + (b*c + a*f)*x^3 + a*d*x + a*c)*sqrt(b*x^3
+ a)/x^12, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{12}}\,{d x} \]

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

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maple [B]  time = 0.10, size = 1773, normalized size = 2.23 \[ \text {result too large to display} \]

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)

[Out]

f*(-1/8*(b*x^3+a)^(1/2)*a/x^8-19/80*(b*x^3+a)^(1/2)*b/x^5-27/320*(b*x^3+a)^(1/2)/a*b^2/x^2+9/320*I/a*b^2*3^(1/
2)*(-a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*(
(x-(-a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b))^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b
+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1
/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/
(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2)))+d*(-1/10*(b*x^3+a)^(1/2)*a/x^10-23/140*(b*x^
3+a)^(1/2)*b/x^7-27/1120*(b*x^3+a)^(1/2)/a*b^2/x^4+27/448*(b*x^3+a)^(1/2)/a^2*b^3/x+9/448*I*b^3/a^2*3^(1/2)*(-
a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*((x-(-
a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b))^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b+1/2*
I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)^(1/2)*((-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1
/2)*(-a*b^2)^(1/3)/b)*EllipticE(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)
/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(
1/2))+(-a*b^2)^(1/3)/b*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2
)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^
(1/2))))+e*(-1/9*(b*x^3+a)^(1/2)*a/x^9-7/36*(b*x^3+a)^(1/2)*b/x^6-1/24*(b*x^3+a)^(1/2)/a*b^2/x^3+1/24/a^(3/2)*
b^3*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+g*(-1/7*(b*x^3+a)^(1/2)*a/x^7-17/56*(b*x^3+a)^(1/2)*b/x^4-27/112*(b*x^3+
a)^(1/2)/a*b^2/x-9/112*I/a*b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/
b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b
))^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)
^(1/2)*((-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*EllipticE(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/
b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/
3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2))+(-a*b^2)^(1/3)/b*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)
/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1
/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2))))+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*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/
3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b))^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*
(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/
3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^
(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{12}}\,{d x} \]

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

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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sympy [A]  time = 24.05, size = 541, normalized size = 0.68 \[ \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}}} \]

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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