Integrand size = 22, antiderivative size = 305 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\frac {(7 A b-16 a B) \sqrt {a+b x^3}}{80 a x^5}+\frac {3 b (7 A b-16 a B) \sqrt {a+b x^3}}{320 a^2 x^2}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{320 a^2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/80*(7*A*b-16*B*a)*(b*x^3+a)^(1/2)/a/x^5+3/320*b*(7*A*b-16*B*a)*(b*x^3+a) ^(1/2)/a^2/x^2-1/8*A*(b*x^3+a)^(3/2)/a/x^8+1/320*3^(3/4)*(1/2*6^(1/2)+1/2* 2^(1/2))*b^(5/3)*(7*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^2/(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 = 80, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\frac {\sqrt {a+b x^3} \left (-5 A \left (a+b x^3\right )+\frac {\left (\frac {7 A b}{2}-8 a B\right ) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},-\frac {1}{2},-\frac {2}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{40 a x^8} \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^9,x]
Output:
(Sqrt[a + b*x^3]*(-5*A*(a + b*x^3) + (((7*A*b)/2 - 8*a*B)*x^3*Hypergeometr ic2F1[-5/3, -1/2, -2/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(40*a*x^8)
Time = 0.55 (sec) , antiderivative size = 297, 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, 847, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(7 A b-16 a B) \int \frac {\sqrt {b x^3+a}}{x^6}dx}{16 a}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(7 A b-16 a B) \left (\frac {3}{10} b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx-\frac {\sqrt {a+b x^3}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(7 A b-16 a B) \left (\frac {3}{10} b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a}}dx}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )-\frac {\sqrt {a+b x^3}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(7 A b-16 a B) \left (\frac {3}{10} b \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )-\frac {\sqrt {a+b x^3}}{5 x^5}\right )}{16 a}-\frac {A \left (a+b x^3\right )^{3/2}}{8 a x^8}\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^9,x]
Output:
-1/8*(A*(a + b*x^3)^(3/2))/(a*x^8) - ((7*A*b - 16*a*B)*(-1/5*Sqrt[a + b*x^ 3]/x^5 + (3*b*(-1/2*Sqrt[a + b*x^3]/(a*x^2) - (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*3^ (1/4)*a*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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[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 = 352, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-21 A \,b^{2} x^{6}+48 B a b \,x^{6}+12 a A b \,x^{3}+64 B \,a^{2} x^{3}+40 a^{2} A \right )}{320 x^{8} a^{2}}-\frac {i b \left (7 A b -16 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{320 a^{2} \sqrt {b \,x^{3}+a}}\) | \(352\) |
| elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{8 x^{8}}-\frac {\left (3 A b +16 B a \right ) \sqrt {b \,x^{3}+a}}{80 a \,x^{5}}+\frac {3 b \left (7 A b -16 B a \right ) \sqrt {b \,x^{3}+a}}{320 a^{2} x^{2}}-\frac {i b \left (7 A b -16 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{320 a^{2} \sqrt {b \,x^{3}+a}}\) | \(362\) |
| default | \(A \left (-\frac {\sqrt {b \,x^{3}+a}}{8 x^{8}}-\frac {3 b \sqrt {b \,x^{3}+a}}{80 a \,x^{5}}+\frac {21 b^{2} \sqrt {b \,x^{3}+a}}{320 a^{2} x^{2}}-\frac {7 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^{2} \sqrt {b \,x^{3}+a}}\right )+B \left (-\frac {\sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {3 b \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}+\frac {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 a \sqrt {b \,x^{3}+a}}\right )\) | \(660\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^9,x,method=_RETURNVERBOSE)
Output:
-1/320*(b*x^3+a)^(1/2)*(-21*A*b^2*x^6+48*B*a*b*x^6+12*A*a*b*x^3+64*B*a^2*x ^3+40*A*a^2)/x^8/a^2-1/320*I*b*(7*A*b-16*B*a)/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)*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 = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=-\frac {3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} x^{6} + 4 \, {\left (16 \, B a^{2} + 3 \, A a b\right )} x^{3} + 40 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{320 \, a^{2} x^{8}} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^9,x, algorithm="fricas")
Output:
-1/320*(3*(16*B*a*b - 7*A*b^2)*sqrt(b)*x^8*weierstrassPInverse(0, -4*a/b, x) + (3*(16*B*a*b - 7*A*b^2)*x^6 + 4*(16*B*a^2 + 3*A*a*b)*x^3 + 40*A*a^2)* sqrt(b*x^3 + a))/(a^2*x^8)
Time = 1.64 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{8} \Gamma \left (- \frac {5}{3}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x**9,x)
Output:
A*sqrt(a)*gamma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi)/ a)/(3*x**8*gamma(-5/3)) + B*sqrt(a)*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3, ), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{9}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^9,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^9, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{9}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^9,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^9, x)
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^9} \,d x \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^9,x)
Output:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^9, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx=\frac {-8 \sqrt {b \,x^{3}+a}\, a -26 \sqrt {b \,x^{3}+a}\, b \,x^{3}+27 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{12}+a \,x^{9}}d x \right ) a^{2} x^{8}}{91 x^{8}} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^9,x)
Output:
( - 8*sqrt(a + b*x**3)*a - 26*sqrt(a + b*x**3)*b*x**3 + 27*int(sqrt(a + b* x**3)/(a*x**9 + b*x**12),x)*a**2*x**8)/(91*x**8)