Integrand size = 15, antiderivative size = 556 \[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=-\frac {108 a^3 x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {54 a^2 x^5 \sqrt {a+b x^3}}{6175 b}+\frac {18}{475} a x^8 \sqrt {a+b x^3}+\frac {432 a^4 \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}-\frac {216 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {144 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-108/8645*a^3*x^2*(b*x^3+a)^(1/2)/b^2+54/6175*a^2*x^5*(b*x^3+a)^(1/2)/b+18 /475*a*x^8*(b*x^3+a)^(1/2)+432/8645*a^4*(b*x^3+a)^(1/2)/b^(8/3)/((1+3^(1/2 ))*a^(1/3)+b^(1/3)*x)+2/25*x^8*(b*x^3+a)^(3/2)-216/8645*3^(1/4)*(1/2*6^(1/ 2)-1/2*2^(1/2))*a^(13/3)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b ^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2) )*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)/b^(8/3 )/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b *x^3+a)^(1/2)+144/8645*2^(1/2)*3^(3/4)*a^(13/3)*(a^(1/3)+b^(1/3)*x)*((a^(2 /3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2 )*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x ),I*3^(1/2)+2*I)/b^(8/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3) +b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.57 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.15 \[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\frac {2 x^2 \sqrt {a+b x^3} \left (-\left (\left (10 a-19 b x^3\right ) \left (a+b x^3\right )^2\right )+\frac {10 a^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{475 b^2} \] Input:
Integrate[x^7*(a + b*x^3)^(3/2),x]
Output:
(2*x^2*Sqrt[a + b*x^3]*(-((10*a - 19*b*x^3)*(a + b*x^3)^2) + (10*a^3*Hyper geometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(475*b^2 )
Time = 0.96 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {811, 811, 843, 843, 832, 759, 2416}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^7 \left (a+b x^3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {9}{25} a \int x^7 \sqrt {b x^3+a}dx+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \int \frac {x^7}{\sqrt {b x^3+a}}dx+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \left (\frac {2 x^5 \sqrt {a+b x^3}}{13 b}-\frac {10 a \int \frac {x^4}{\sqrt {b x^3+a}}dx}{13 b}\right )+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \left (\frac {2 x^5 \sqrt {a+b x^3}}{13 b}-\frac {10 a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \int \frac {x}{\sqrt {b x^3+a}}dx}{7 b}\right )}{13 b}\right )+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \left (\frac {2 x^5 \sqrt {a+b x^3}}{13 b}-\frac {10 a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{7 b}\right )}{13 b}\right )+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \left (\frac {2 x^5 \sqrt {a+b x^3}}{13 b}-\frac {10 a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{7 b}\right )}{13 b}\right )+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {9}{25} a \left (\frac {3}{19} a \left (\frac {2 x^5 \sqrt {a+b x^3}}{13 b}-\frac {10 a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{7 b}\right )}{13 b}\right )+\frac {2}{19} x^8 \sqrt {a+b x^3}\right )+\frac {2}{25} x^8 \left (a+b x^3\right )^{3/2}\) |
Input:
Int[x^7*(a + b*x^3)^(3/2),x]
Output:
(2*x^8*(a + b*x^3)^(3/2))/25 + (9*a*((2*x^8*Sqrt[a + b*x^3])/19 + (3*a*((2 *x^5*Sqrt[a + b*x^3])/(13*b) - (10*a*((2*x^2*Sqrt[a + b*x^3])/(7*b) - (4*a *(((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^ (1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(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]*Ellipt icE[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]])/(b^(1/3)*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]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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]*El lipticF[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]])/(3^(1/4)*b^(2/3)*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])))/(7*b )))/(13*b)))/19))/25
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]*Sqrt[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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[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] - S imp[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] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 0.56 (sec) , antiderivative size = 488, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-\frac {2 x^{2} \left (-1729 b^{3} x^{9}-2548 a \,b^{2} x^{6}-189 a^{2} b \,x^{3}+270 a^{3}\right ) \sqrt {b \,x^{3}+a}}{43225 b^{2}}-\frac {144 i a^{4} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{8645 b^{3} \sqrt {b \,x^{3}+a}}\) | \(488\) |
| default | \(\frac {2 b \,x^{11} \sqrt {b \,x^{3}+a}}{25}+\frac {56 a \,x^{8} \sqrt {b \,x^{3}+a}}{475}+\frac {54 a^{2} x^{5} \sqrt {b \,x^{3}+a}}{6175 b}-\frac {108 a^{3} x^{2} \sqrt {b \,x^{3}+a}}{8645 b^{2}}-\frac {144 i a^{4} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{8645 b^{3} \sqrt {b \,x^{3}+a}}\) | \(509\) |
| elliptic | \(\frac {2 b \,x^{11} \sqrt {b \,x^{3}+a}}{25}+\frac {56 a \,x^{8} \sqrt {b \,x^{3}+a}}{475}+\frac {54 a^{2} x^{5} \sqrt {b \,x^{3}+a}}{6175 b}-\frac {108 a^{3} x^{2} \sqrt {b \,x^{3}+a}}{8645 b^{2}}-\frac {144 i a^{4} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{8645 b^{3} \sqrt {b \,x^{3}+a}}\) | \(509\) |
Input:
int(x^7*(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/43225*x^2*(-1729*b^3*x^9-2548*a*b^2*x^6-189*a^2*b*x^3+270*a^3)/b^2*(b*x ^3+a)^(1/2)-144/8645*I/b^3*a^4*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)))
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.14 \[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (1080 \, a^{4} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (1729 \, b^{4} x^{11} + 2548 \, a b^{3} x^{8} + 189 \, a^{2} b^{2} x^{5} - 270 \, a^{3} b x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{43225 \, b^{3}} \] Input:
integrate(x^7*(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
-2/43225*(1080*a^4*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse( 0, -4*a/b, x)) - (1729*b^4*x^11 + 2548*a*b^3*x^8 + 189*a^2*b^2*x^5 - 270*a ^3*b*x^2)*sqrt(b*x^3 + a))/b^3
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \] Input:
integrate(x**7*(b*x**3+a)**(3/2),x)
Output:
a**(3/2)*x**8*gamma(8/3)*hyper((-3/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi )/a)/(3*gamma(11/3))
\[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{7} \,d x } \] Input:
integrate(x^7*(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(3/2)*x^7, x)
\[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{7} \,d x } \] Input:
integrate(x^7*(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(3/2)*x^7, x)
Timed out. \[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\int x^7\,{\left (b\,x^3+a\right )}^{3/2} \,d x \] Input:
int(x^7*(a + b*x^3)^(3/2),x)
Output:
int(x^7*(a + b*x^3)^(3/2), x)
\[ \int x^7 \left (a+b x^3\right )^{3/2} \, dx=\frac {-\frac {108 \sqrt {b \,x^{3}+a}\, a^{3} x^{2}}{8645}+\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} b \,x^{5}}{6175}+\frac {56 \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{8}}{475}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} x^{11}}{25}+\frac {216 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{4}}{8645}}{b^{2}} \] Input:
int(x^7*(b*x^3+a)^(3/2),x)
Output:
(2*( - 270*sqrt(a + b*x**3)*a**3*x**2 + 189*sqrt(a + b*x**3)*a**2*b*x**5 + 2548*sqrt(a + b*x**3)*a*b**2*x**8 + 1729*sqrt(a + b*x**3)*b**3*x**11 + 54 0*int((sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a**4))/(43225*b**2)