Integrand size = 26, antiderivative size = 364 \[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {27 a^2 (22 A b-a B) \sqrt {e x} \sqrt {a+b x^3}}{1408 b e}+\frac {3 a (22 A b-a B) \sqrt {e x} \left (a+b x^3\right )^{3/2}}{352 b e}+\frac {(22 A b-a B) \sqrt {e x} \left (a+b x^3\right )^{5/2}}{176 b e}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}+\frac {27\ 3^{3/4} a^{8/3} (22 A b-a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2816 b e \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
3/352*a*(22*A*b-B*a)*(b*x^3+a)^(3/2)*(e*x)^(1/2)/b/e+1/176*(22*A*b-B*a)*(b *x^3+a)^(5/2)*(e*x)^(1/2)/b/e+1/11*B*(b*x^3+a)^(7/2)*(e*x)^(1/2)/b/e+27/14 08*a^2*(22*A*b-B*a)*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b/e+27/2816*3^(3/4)*a^(8/3 )*(22*A*b-B*a)*(a^(1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^( 1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1 /3)+b^(1/3)*x*(1+3^(1/2)))*EllipticF((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/ (a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(e*x)^(1 /2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2) ))^2)^(1/2)/b/e/(b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^ (1/3)*x*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {x \sqrt {a+b x^3} \left (B \left (a+b x^3\right )^3-\frac {a^2 (-22 A b+a B) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{6},\frac {7}{6},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{11 b \sqrt {e x}} \]
(x*Sqrt[a + b*x^3]*(B*(a + b*x^3)^3 - (a^2*(-22*A*b + a*B)*Hypergeometric2 F1[-5/2, 1/6, 7/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(11*b*Sqrt[e*x])
Time = 0.43 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {959, 811, 811, 811, 851, 766}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(22 A b-a B) \int \frac {\left (b x^3+a\right )^{5/2}}{\sqrt {e x}}dx}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(22 A b-a B) \left (\frac {15}{16} a \int \frac {\left (b x^3+a\right )^{3/2}}{\sqrt {e x}}dx+\frac {\sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 e}\right )}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(22 A b-a B) \left (\frac {15}{16} a \left (\frac {9}{10} a \int \frac {\sqrt {b x^3+a}}{\sqrt {e x}}dx+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 e}\right )}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(22 A b-a B) \left (\frac {15}{16} a \left (\frac {9}{10} a \left (\frac {3}{4} a \int \frac {1}{\sqrt {e x} \sqrt {b x^3+a}}dx+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 e}\right )}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(22 A b-a B) \left (\frac {15}{16} a \left (\frac {9}{10} a \left (\frac {3 a \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 e}+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 e}\right )}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {(22 A b-a B) \left (\frac {15}{16} a \left (\frac {9}{10} a \left (\frac {3^{3/4} a^{2/3} \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 e^2 \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}+\frac {\sqrt {e x} \sqrt {a+b x^3}}{2 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{3/2}}{5 e}\right )+\frac {\sqrt {e x} \left (a+b x^3\right )^{5/2}}{8 e}\right )}{22 b}+\frac {B \sqrt {e x} \left (a+b x^3\right )^{7/2}}{11 b e}\) |
(B*Sqrt[e*x]*(a + b*x^3)^(7/2))/(11*b*e) + ((22*A*b - a*B)*((Sqrt[e*x]*(a + b*x^3)^(5/2))/(8*e) + (15*a*((Sqrt[e*x]*(a + b*x^3)^(3/2))/(5*e) + (9*a* ((Sqrt[e*x]*Sqrt[a + b*x^3])/(2*e) + (3^(3/4)*a^(2/3)*Sqrt[e*x]*(a^(1/3)*e + b^(1/3)*e*x)*Sqrt[(a^(2/3)*e^2 - a^(1/3)*b^(1/3)*e^2*x + b^(2/3)*e^2*x^ 2)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*EllipticF[ArcCos[(a^(1/3)*e + (1 - Sqrt[3])*b^(1/3)*e*x)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)], (2 + Sqrt[3])/4])/(4*e^2*Sqrt[(b^(1/3)*e*x*(a^(1/3)*e + b^(1/3)*e*x))/(a^(1/3 )*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*Sqrt[a + b*x^3])))/10))/16))/(22*b)
3.6.39.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Result contains complex when optimal does not.
Time = 4.84 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.18
| method | result | size |
| risch | \(\frac {\left (128 b^{3} B \,x^{9}+176 x^{6} b^{3} A +376 B \,x^{6} a \,b^{2}+616 a A \,b^{2} x^{3}+356 B \,a^{2} b \,x^{3}+1034 a^{2} b A +81 a^{3} B \right ) x \sqrt {b \,x^{3}+a}}{1408 b \sqrt {e x}}+\frac {81 a^{3} \left (22 A b -B a \right ) \left (\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 {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, F\left (\sqrt {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\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 ) \left (\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 )}{\left (\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 ) \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 ) \sqrt {\left (b \,x^{3}+a \right ) e x}}{1408 \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 ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b e x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \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 ) \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 {e x}\, \sqrt {b \,x^{3}+a}}\) | \(792\) |
| elliptic | \(\text {Expression too large to display}\) | \(959\) |
| default | \(\text {Expression too large to display}\) | \(4617\) |
1/1408/b*(128*B*b^3*x^9+176*A*b^3*x^6+376*B*a*b^2*x^6+616*A*a*b^2*x^3+356* B*a^2*b*x^3+1034*A*a^2*b+81*B*a^3)*x*(b*x^3+a)^(1/2)/(e*x)^(1/2)+81/1408*a ^3*(22*A*b-B*a)*(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))*x/(-1/2/b*(-a*b^2)^(1/ 3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(x-1/b*(- a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/ b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/( x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1 /2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a* b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^( 1/2)/b*(-a*b^2)^(1/3))/(-a*b^2)^(1/3)/(b*e*x*(x-1/b*(-a*b^2)^(1/3))*(x+1/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)- 1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1 /2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(- a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3 ^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1 /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))*((b*x^3+a)*e*x)^(1/2)/(e*x) ^(1/2)/(b*x^3+a)^(1/2)
\[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{\sqrt {e x}} \,d x } \]
integral((B*b^2*x^9 + (2*B*a*b + A*b^2)*x^6 + (B*a^2 + 2*A*a*b)*x^3 + A*a^ 2)*sqrt(b*x^3 + a)*sqrt(e*x)/(e*x), x)
Result contains complex when optimal does not.
Time = 17.85 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\frac {A a^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {7}{6}\right )} + \frac {2 A a^{\frac {3}{2}} b x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {13}{6}\right )} + \frac {A \sqrt {a} b^{2} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {19}{6}\right )} + \frac {B a^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {13}{6}\right )} + \frac {2 B a^{\frac {3}{2}} b x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {19}{6}\right )} + \frac {B \sqrt {a} b^{2} x^{\frac {19}{2}} \Gamma \left (\frac {19}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{6} \\ \frac {25}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {e} \Gamma \left (\frac {25}{6}\right )} \]
A*a**(5/2)*sqrt(x)*gamma(1/6)*hyper((-1/2, 1/6), (7/6,), b*x**3*exp_polar( I*pi)/a)/(3*sqrt(e)*gamma(7/6)) + 2*A*a**(3/2)*b*x**(7/2)*gamma(7/6)*hyper ((-1/2, 7/6), (13/6,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(e)*gamma(13/6)) + A*sqrt(a)*b**2*x**(13/2)*gamma(13/6)*hyper((-1/2, 13/6), (19/6,), b*x**3* exp_polar(I*pi)/a)/(3*sqrt(e)*gamma(19/6)) + B*a**(5/2)*x**(7/2)*gamma(7/6 )*hyper((-1/2, 7/6), (13/6,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(e)*gamma(1 3/6)) + 2*B*a**(3/2)*b*x**(13/2)*gamma(13/6)*hyper((-1/2, 13/6), (19/6,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(e)*gamma(19/6)) + B*sqrt(a)*b**2*x**(19/ 2)*gamma(19/6)*hyper((-1/2, 19/6), (25/6,), b*x**3*exp_polar(I*pi)/a)/(3*s qrt(e)*gamma(25/6))
\[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{\sqrt {e x}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{\sqrt {e x}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{\sqrt {e x}} \, dx=\int \frac {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{5/2}}{\sqrt {e\,x}} \,d x \]