Integrand size = 22, antiderivative size = 299 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\frac {(A b-16 a B) \sqrt {a+b x^3}}{80 x^5}+\frac {13 b (A b-16 a B) \sqrt {a+b x^3}}{320 a x^2}-\frac {A \left (a+b x^3\right )^{5/2}}{8 a x^8}-\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} (A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{320 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/80*(A*b-16*B*a)*(b*x^3+a)^(1/2)/x^5+13/320*b*(A*b-16*B*a)*(b*x^3+a)^(1/2 )/a/x^2-1/8*A*(b*x^3+a)^(5/2)/a/x^8-9/320*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2) )*b^(5/3)*(A*b-16*B*a)*(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)/a/(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 = 10.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\frac {\sqrt {a+b x^3} \left (-\frac {5 A \left (a+b x^3\right )^2}{a}+\frac {\left (\frac {A b}{2}-8 a B\right ) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {3}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{40 x^8} \] Input:
Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^9,x]
Output:
(Sqrt[a + b*x^3]*((-5*A*(a + b*x^3)^2)/a + (((A*b)/2 - 8*a*B)*x^3*Hypergeo metric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(40*x^8)
Time = 0.54 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 809, 809, 759}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(A b-16 a B) \int \frac {\left (b x^3+a\right )^{3/2}}{x^6}dx}{16 a}-\frac {A \left (a+b x^3\right )^{5/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(A b-16 a B) \left (\frac {9}{10} b \int \frac {\sqrt {b x^3+a}}{x^3}dx-\frac {\left (a+b x^3\right )^{3/2}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{5/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(A b-16 a B) \left (\frac {9}{10} b \left (\frac {3}{4} b \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {\sqrt {a+b x^3}}{2 x^2}\right )-\frac {\left (a+b x^3\right )^{3/2}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{5/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(A b-16 a B) \left (\frac {9}{10} b \left (\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 x^2}\right )-\frac {\left (a+b x^3\right )^{3/2}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{5/2}}{8 a x^8}\) |
Input:
Int[((a + b*x^3)^(3/2)*(A + B*x^3))/x^9,x]
Output:
-1/8*(A*(a + b*x^3)^(5/2))/(a*x^8) - ((A*b - 16*a*B)*(-1/5*(a + b*x^3)^(3/ 2)/x^5 + (9*b*(-1/2*Sqrt[a + b*x^3]/x^2 + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/ 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]*EllipticF[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]])/( 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])))/10))/(16*a)
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 + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 2.06 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (27 A \,b^{2} x^{6}+208 B a b \,x^{6}+76 a A b \,x^{3}+64 B \,a^{2} x^{3}+40 a^{2} A \right )}{320 x^{8} a}+\frac {9 i b \left (A b -16 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{320 a \sqrt {b \,x^{3}+a}}\) | \(351\) |
| elliptic | \(-\frac {A a \sqrt {b \,x^{3}+a}}{8 x^{8}}-\frac {\left (\frac {19 A b}{16}+B a \right ) \sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {b \left (27 A b +208 B a \right ) \sqrt {b \,x^{3}+a}}{320 a \,x^{2}}-\frac {2 i \left (B \,b^{2}-\frac {b^{2} \left (27 A b +208 B a \right )}{640 a}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(372\) |
| default | \(A \left (-\frac {a \sqrt {b \,x^{3}+a}}{8 x^{8}}-\frac {19 b \sqrt {b \,x^{3}+a}}{80 x^{5}}-\frac {27 b^{2} \sqrt {b \,x^{3}+a}}{320 a \,x^{2}}+\frac {9 i b^{2} \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}}}}\, \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 )}{320 a \sqrt {b \,x^{3}+a}}\right )+B \left (-\frac {a \sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {13 b \sqrt {b \,x^{3}+a}}{20 x^{2}}-\frac {9 i b \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}}}}\, \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 )}{20 \sqrt {b \,x^{3}+a}}\right )\) | \(653\) |
Input:
int((b*x^3+a)^(3/2)*(B*x^3+A)/x^9,x,method=_RETURNVERBOSE)
Output:
-1/320*(b*x^3+a)^(1/2)*(27*A*b^2*x^6+208*B*a*b*x^6+76*A*a*b*x^3+64*B*a^2*x ^3+40*A*a^2)/x^8/a+9/320*I*b*(A*b-16*B*a)/a*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1 /2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/ 3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*( -a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1 /2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/( -a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/ 2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\frac {27 \, {\left (16 \, B a b - A b^{2}\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left ({\left (208 \, B a b + 27 \, A b^{2}\right )} x^{6} + 4 \, {\left (16 \, B a^{2} + 19 \, A a b\right )} x^{3} + 40 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{320 \, a x^{8}} \] Input:
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^9,x, algorithm="fricas")
Output:
1/320*(27*(16*B*a*b - A*b^2)*sqrt(b)*x^8*weierstrassPInverse(0, -4*a/b, x) - ((208*B*a*b + 27*A*b^2)*x^6 + 4*(16*B*a^2 + 19*A*a*b)*x^3 + 40*A*a^2)*s qrt(b*x^3 + a))/(a*x^8)
Time = 3.00 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {A \sqrt {a} b \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {B \sqrt {a} b \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \] Input:
integrate((b*x**3+a)**(3/2)*(B*x**3+A)/x**9,x)
Output:
A*a**(3/2)*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*sqrt(a)*b*gamma(-5/3)*hyper((-5/3, -1/2), (-2 /3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + B*a**(3/2)*gamma(-5 /3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(- 2/3)) + B*sqrt(a)*b*gamma(-2/3)*hyper((-2/3, -1/2), (1/3,), b*x**3*exp_pol ar(I*pi)/a)/(3*x**2*gamma(1/3))
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{9}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^9,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^9, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\int { \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{x^{9}} \,d x } \] Input:
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^9,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^9, x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\int \frac {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2}}{x^9} \,d x \] Input:
int(((A + B*x^3)*(a + b*x^3)^(3/2))/x^9,x)
Output:
int(((A + B*x^3)*(a + b*x^3)^(3/2))/x^9, x)
\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^9} \, dx=\frac {-62 \sqrt {b \,x^{3}+a}\, a^{2}+26 \sqrt {b \,x^{3}+a}\, a b \,x^{3}-182 \sqrt {b \,x^{3}+a}\, b^{2} x^{6}-405 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{12}+a \,x^{9}}d x \right ) a^{3} x^{8}}{91 x^{8}} \] Input:
int((b*x^3+a)^(3/2)*(B*x^3+A)/x^9,x)
Output:
( - 62*sqrt(a + b*x**3)*a**2 + 26*sqrt(a + b*x**3)*a*b*x**3 - 182*sqrt(a + b*x**3)*b**2*x**6 - 405*int(sqrt(a + b*x**3)/(a*x**9 + b*x**12),x)*a**3*x **8)/(91*x**8)