3.5.72 \(\int \frac {(a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4)}{x^{11}} \, dx\) [472]

Optimal. Leaf size=764 \[ -\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {b^2 (b d-6 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (b c-4 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{7/3} \left (7 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c-4 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{2240 a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-1/2520*(252*c/x^10+280*d/x^9+315*e/x^8+360*f/x^7+420*g/x^6)*(b*x^3+a)^(3/2)+1/24*b^2*(-6*a*g+b*d)*arctanh((b*
x^3+a)^(1/2)/a^(1/2))/a^(3/2)-1/1680*b*(108*c/x^7+140*d/x^6+189*e/x^5+270*f/x^4+420*g/x^3)*(b*x^3+a)^(1/2)-27/
1120*b^2*c*(b*x^3+a)^(1/2)/a/x^4-1/24*b^2*d*(b*x^3+a)^(1/2)/a/x^3-27/320*b^2*e*(b*x^3+a)^(1/2)/a/x^2+27/448*b^
2*(-4*a*f+b*c)*(b*x^3+a)^(1/2)/a^2/x-27/448*b^(7/3)*(-4*a*f+b*c)*(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*f+b*c)*(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)-9/2240*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*a^(2/3)*b^(1/3)*e-5*(-4*a*f+b*c)*(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^(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)

________________________________________________________________________________________

Rubi [A]
time = 0.90, antiderivative size = 764, normalized size of antiderivative = 1.00, number of steps used = 13, 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 {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{7/3} \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 (\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 ) \left (7 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c-4 a f)\right )}{2240 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 {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \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}} (b c-4 a f) 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 )}{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 {b^2 (b d-6 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{3/2}}-\frac {27 b^{7/3} \sqrt {a+b x^3} (b c-4 a f)}{448 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {27 b^2 \sqrt {a+b x^3} (b c-4 a f)}{448 a^2 x}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}-\frac {b \sqrt {a+b x^3} \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right )}{1680}-\frac {\left (a+b x^3\right )^{3/2} \left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right )}{2520} \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^11,x]

[Out]

-1/1680*(b*((108*c)/x^7 + (140*d)/x^6 + (189*e)/x^5 + (270*f)/x^4 + (420*g)/x^3)*Sqrt[a + b*x^3]) - (27*b^2*c*
Sqrt[a + b*x^3])/(1120*a*x^4) - (b^2*d*Sqrt[a + b*x^3])/(24*a*x^3) - (27*b^2*e*Sqrt[a + b*x^3])/(320*a*x^2) +
(27*b^2*(b*c - 4*a*f)*Sqrt[a + b*x^3])/(448*a^2*x) - (27*b^(7/3)*(b*c - 4*a*f)*Sqrt[a + b*x^3])/(448*a^2*((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((252*c)/x^10 + (280*d)/x^9 + (315*e)/x^8 + (360*f)/x^7 + (420*g)/x^6)*(a +
b*x^3)^(3/2))/2520 + (b^2*(b*d - 6*a*g)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(24*a^(3/2)) + (27*3^(1/4)*Sqrt[2 -
Sqrt[3]]*b^(7/3)*(b*c - 4*a*f)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + S
qrt[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*a^(2/3)*b^(1/3)*e - 5*(1 - Sqrt[3])*(b*c -
4*a*f))*(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 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqr
t[3]])/(2240*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])

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^{11}} \, dx &=-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{10}-\frac {d x}{9}-\frac {e x^2}{8}-\frac {f x^3}{7}-\frac {g x^4}{6}\right )}{x^8} \, dx\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {1}{4} \left (27 b^2\right ) \int \frac {\frac {c}{70}+\frac {d x}{54}+\frac {e x^2}{40}+\frac {f x^3}{28}+\frac {g x^4}{18}}{x^5 \sqrt {a+b x^3}} \, dx\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (27 b^2\right ) \int \frac {-\frac {4 a d}{27}-\frac {a e x}{5}+\frac {1}{14} (b c-4 a f) x^2-\frac {4}{9} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx}{32 a}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {\left (9 b^2\right ) \int \frac {\frac {6 a^2 e}{5}-\frac {3}{7} a (b c-4 a f) x-\frac {4}{9} a (b d-6 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{64 a^2}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (9 b^2\right ) \int \frac {\frac {12}{7} a^2 (b c-4 a f)+\frac {16}{9} a^2 (b d-6 a g) x+\frac {6}{5} a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{256 a^3}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {\left (9 b^2\right ) \int \frac {-\frac {32}{9} a^3 (b d-6 a g)-\frac {12}{5} a^3 b e x-\frac {12}{7} a^2 b (b c-4 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{512 a^4}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {\left (9 b^2\right ) \int \frac {-\frac {12}{5} a^3 b e-\frac {12}{7} a^2 b (b c-4 a f) x}{\sqrt {a+b x^3}} \, dx}{512 a^4}-\frac {\left (b^2 (b d-6 a g)\right ) \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{16 a}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {\left (\frac {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac {\left (27 b^{8/3} (b c-4 a f)\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 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c-4 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{4480 a^{5/3}}-\frac {\left (b^2 (b d-6 a g)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{48 a}\\ &=-\frac {b \left (\frac {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b c-4 a f) \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 {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (b c-4 a f) \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 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c-4 a f)\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 )}{2240 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 {(b (b d-6 a g)) \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 {108 c}{x^7}+\frac {140 d}{x^6}+\frac {189 e}{x^5}+\frac {270 f}{x^4}+\frac {420 g}{x^3}\right ) \sqrt {a+b x^3}}{1680}-\frac {27 b^2 c \sqrt {a+b x^3}}{1120 a x^4}-\frac {b^2 d \sqrt {a+b x^3}}{24 a x^3}-\frac {27 b^2 e \sqrt {a+b x^3}}{320 a x^2}+\frac {27 b^2 (b c-4 a f) \sqrt {a+b x^3}}{448 a^2 x}-\frac {27 b^{7/3} (b c-4 a f) \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 {252 c}{x^{10}}+\frac {280 d}{x^9}+\frac {315 e}{x^8}+\frac {360 f}{x^7}+\frac {420 g}{x^6}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac {b^2 (b d-6 a g) \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 c-4 a f) \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 a^{2/3} \sqrt [3]{b} e-5 \left (1-\sqrt {3}\right ) (b c-4 a f)\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 )}{2240 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}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.70, size = 930, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (-1215 b^3 c x^9+8 a^3 \left (252 c+5 x \left (56 d+63 e x+72 f x^2+84 g x^3\right )\right )+3 a b^2 x^6 (162 c+x (280 d+81 x (7 e+20 f x)))+4 a^2 b x^3 \left (828 c+x \left (980 d+3 x \left (399 e+510 f x+700 g x^2\right )\right )\right )\right )}{20160 a^2 x^{10}}+\frac {b^2 \left (280 \sqrt {a} b d \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 )-1680 a^{3/2} g \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 )+567 a b^{2/3} e \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 )-405 \sqrt {2} \sqrt [3]{a} b^{4/3} c \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 )+1620 \sqrt {2} a^{4/3} \sqrt [3]{b} f \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 )}{6720 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^11,x]

[Out]

-1/20160*(Sqrt[a + b*x^3]*(-1215*b^3*c*x^9 + 8*a^3*(252*c + 5*x*(56*d + 63*e*x + 72*f*x^2 + 84*g*x^3)) + 3*a*b
^2*x^6*(162*c + x*(280*d + 81*x*(7*e + 20*f*x))) + 4*a^2*b*x^3*(828*c + x*(980*d + 3*x*(399*e + 510*f*x + 700*
g*x^2)))))/(a^2*x^10) + (b^2*(280*Sqrt[a]*b*d*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]] - 1680*a^(3/2)*g*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]] + 567*a*b^(2/3)*e*((-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)] - 405*Sqrt[2]*a^(1/3)*b^(4/3)*c*((-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))]
) + 1620*Sqrt[2]*a^(4/3)*b^(1/3)*f*((-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))])))/(6720*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. 1469 vs. \(2 (598 ) = 1196\).
time = 0.44, size = 1470, normalized size = 1.92

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(976\)
risch	\(\text {Expression too large to display}\)	\(1343\)
default	\(\text {Expression too large to display}\)	\(1470\)

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^11,x,method=_RETURNVERBOSE)

[Out]

d*(-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)*arcta
nh((b*x^3+a)^(1/2)/a^(1/2)))+g*(-1/6*a*(b*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2*arctanh((b*x^3+a
)^(1/2)/a^(1/2))/a^(1/2))+e*(-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)*E
llipticF(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)))+f*(-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))))+
c*(-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^11,x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + x^2*e + d*x + c)*(b*x^3 + a)^(3/2)/x^11, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 559, normalized size = 0.73 \begin {gather*} \left [-\frac {1701 \, a b^{\frac {5}{2}} e x^{10} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 210 \, {\left (b^{3} d - 6 \, a b^{2} g\right )} \sqrt {a} x^{10} \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 ) - 1215 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (1701 \, a b^{2} e x^{8} - 1215 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{9} + 4788 \, a^{2} b e x^{5} + 840 \, {\left (a b^{2} d + 10 \, a^{2} b g\right )} x^{7} + 18 \, {\left (27 \, a b^{2} c + 340 \, a^{2} b f\right )} x^{6} + 2520 \, a^{3} e x^{2} + 2240 \, a^{3} d x + 560 \, {\left (7 \, a^{2} b d + 6 \, a^{3} g\right )} x^{4} + 2016 \, a^{3} c + 144 \, {\left (23 \, a^{2} b c + 20 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{10}}, -\frac {1701 \, a b^{\frac {5}{2}} e x^{10} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 420 \, {\left (b^{3} d - 6 \, a b^{2} g\right )} \sqrt {-a} x^{10} \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 ) - 1215 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (1701 \, a b^{2} e x^{8} - 1215 \, {\left (b^{3} c - 4 \, a b^{2} f\right )} x^{9} + 4788 \, a^{2} b e x^{5} + 840 \, {\left (a b^{2} d + 10 \, a^{2} b g\right )} x^{7} + 18 \, {\left (27 \, a b^{2} c + 340 \, a^{2} b f\right )} x^{6} + 2520 \, a^{3} e x^{2} + 2240 \, a^{3} d x + 560 \, {\left (7 \, a^{2} b d + 6 \, a^{3} g\right )} x^{4} + 2016 \, a^{3} c + 144 \, {\left (23 \, a^{2} b c + 20 \, a^{3} f\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{20160 \, a^{2} x^{10}}\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^11,x, algorithm="fricas")

[Out]

[-1/20160*(1701*a*b^(5/2)*e*x^10*weierstrassPInverse(0, -4*a/b, x) + 210*(b^3*d - 6*a*b^2*g)*sqrt(a)*x^10*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) - 1215*(b^3*c - 4*a*b^2*f)*sqrt(b
)*x^10*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (1701*a*b^2*e*x^8 - 1215*(b^3*c - 4*a*b
^2*f)*x^9 + 4788*a^2*b*e*x^5 + 840*(a*b^2*d + 10*a^2*b*g)*x^7 + 18*(27*a*b^2*c + 340*a^2*b*f)*x^6 + 2520*a^3*e
*x^2 + 2240*a^3*d*x + 560*(7*a^2*b*d + 6*a^3*g)*x^4 + 2016*a^3*c + 144*(23*a^2*b*c + 20*a^3*f)*x^3)*sqrt(b*x^3
+ a))/(a^2*x^10), -1/20160*(1701*a*b^(5/2)*e*x^10*weierstrassPInverse(0, -4*a/b, x) + 420*(b^3*d - 6*a*b^2*g)
*sqrt(-a)*x^10*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) - 1215*(b^3*c - 4*a*b^2*f)*s
qrt(b)*x^10*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (1701*a*b^2*e*x^8 - 1215*(b^3*c -
4*a*b^2*f)*x^9 + 4788*a^2*b*e*x^5 + 840*(a*b^2*d + 10*a^2*b*g)*x^7 + 18*(27*a*b^2*c + 340*a^2*b*f)*x^6 + 2520*
a^3*e*x^2 + 2240*a^3*d*x + 560*(7*a^2*b*d + 6*a^3*g)*x^4 + 2016*a^3*c + 144*(23*a^2*b*c + 20*a^3*f)*x^3)*sqrt(
b*x^3 + a))/(a^2*x^10)]

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Sympy [A]
time = 15.16, size = 576, normalized size = 0.75 \begin {gather*} \frac {a^{\frac {3}{2}} c \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}} e \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}} f \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 {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 e \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 f \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} d}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a^{2} g}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {11 a \sqrt {b} d}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} g}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {17 b^{\frac {3}{2}} d}{72 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} g \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} g}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {5}{2}} d}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} + \frac {b^{3} d \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**11,x)

[Out]

a**(3/2)*c*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)*e*gamma(-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)*f*gamm
a(-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(-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*e*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*f*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*d/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)
) - a**2*g/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - 11*a*sqrt(b)*d/(36*x**(15/2)*sqrt(a/(b*x**3) + 1)) - a
*sqrt(b)*g/(4*x**(9/2)*sqrt(a/(b*x**3) + 1)) - 17*b**(3/2)*d/(72*x**(9/2)*sqrt(a/(b*x**3) + 1)) - b**(3/2)*g*s
qrt(a/(b*x**3) + 1)/(3*x**(3/2)) - b**(3/2)*g/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**(5/2)*d/(24*a*x**(3/2)*s
qrt(a/(b*x**3) + 1)) - b**2*g*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a)) + b**3*d*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^11,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + x^2*e + d*x + c)*(b*x^3 + a)^(3/2)/x^11, 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^{11}} \,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^11,x)

[Out]

int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^11, x)

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