Integrand size = 22, antiderivative size = 334 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}-\frac {19 A b-10 a B}{45 a^2 x^2 \left (a+b x^3\right )^{3/2}}-\frac {13 (19 A b-10 a B)}{135 a^3 x^2 \sqrt {a+b x^3}}+\frac {91 (19 A b-10 a B) \sqrt {a+b x^3}}{540 a^4 x^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{2/3} (19 A b-10 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{540 \sqrt [4]{3} a^4 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-1/5*A/a/x^5/(b*x^3+a)^(3/2)-1/45*(19*A*b-10*B*a)/a^2/x^2/(b*x^3+a)^(3/2)- 13/135*(19*A*b-10*B*a)/a^3/x^2/(b*x^3+a)^(1/2)+91/540*(19*A*b-10*B*a)*(b*x ^3+a)^(1/2)/a^4/x^2+91/1620*(1/2*6^(1/2)+1/2*2^(1/2))*b^(2/3)*(19*A*b-10*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)*3^(3/4)/a^4/(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 = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.25 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\frac {-2 a^2 A+\left (\frac {19 A b}{2}-5 a B\right ) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{2},\frac {1}{3},-\frac {b x^3}{a}\right )}{10 a^3 x^5 \left (a+b x^3\right )^{3/2}} \] Input:
Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^(5/2)),x]
Output:
(-2*a^2*A + ((19*A*b)/2 - 5*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hyper geometric2F1[-2/3, 5/2, 1/3, -((b*x^3)/a)])/(10*a^3*x^5*(a + b*x^3)^(3/2))
Time = 0.62 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 819, 819, 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.
| \(\displaystyle \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(19 A b-10 a B) \int \frac {1}{x^3 \left (b x^3+a\right )^{5/2}}dx}{10 a}-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(19 A b-10 a B) \left (\frac {13 \int \frac {1}{x^3 \left (b x^3+a\right )^{3/2}}dx}{9 a}+\frac {2}{9 a x^2 \left (a+b x^3\right )^{3/2}}\right )}{10 a}-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(19 A b-10 a B) \left (\frac {13 \left (\frac {7 \int \frac {1}{x^3 \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x^2 \left (a+b x^3\right )^{3/2}}\right )}{10 a}-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(19 A b-10 a B) \left (\frac {13 \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a}}dx}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x^2 \left (a+b x^3\right )^{3/2}}\right )}{10 a}-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(19 A b-10 a B) \left (\frac {13 \left (\frac {7 \left (-\frac {\sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 a x^2}\right )}{3 a}+\frac {2}{3 a x^2 \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x^2 \left (a+b x^3\right )^{3/2}}\right )}{10 a}-\frac {A}{5 a x^5 \left (a+b x^3\right )^{3/2}}\) |
Input:
Int[(A + B*x^3)/(x^6*(a + b*x^3)^(5/2)),x]
Output:
-1/5*A/(a*x^5*(a + b*x^3)^(3/2)) - ((19*A*b - 10*a*B)*(2/(9*a*x^2*(a + b*x ^3)^(3/2)) + (13*(2/(3*a*x^2*Sqrt[a + b*x^3]) + (7*(-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]*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]])/(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])))/(3*a)))/ (9*a)))/(10*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 + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[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 = 2.85 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{5 a^{3} x^{5}}+\frac {\left (27 A b -10 B a \right ) \sqrt {b \,x^{3}+a}}{20 a^{4} x^{2}}+\frac {2 x \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 a^{3} b \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2 b x \left (25 A b -16 B a \right )}{27 a^{4} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 i \left (\frac {b \left (27 A b -10 B a \right )}{40 a^{4}}+\frac {b \left (25 A b -16 B a \right )}{27 a^{4}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(425\) |
| default | \(A \left (-\frac {\sqrt {b \,x^{3}+a}}{5 a^{3} x^{5}}+\frac {27 b \sqrt {b \,x^{3}+a}}{20 a^{4} x^{2}}+\frac {2 x \sqrt {b \,x^{3}+a}}{9 a^{3} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {50 b^{2} x}{27 a^{4} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {1729 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 )}{1620 a^{4} \sqrt {b \,x^{3}+a}}\right )+B \left (-\frac {\sqrt {b \,x^{3}+a}}{2 a^{3} x^{2}}-\frac {2 x \sqrt {b \,x^{3}+a}}{9 a^{2} b \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {32 b x}{27 a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {91 i \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 )}{162 a^{3} \sqrt {b \,x^{3}+a}}\right )\) | \(722\) |
| risch | \(\text {Expression too large to display}\) | \(1273\) |
Input:
int((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/5/a^3*A*(b*x^3+a)^(1/2)/x^5+1/20/a^4*(27*A*b-10*B*a)*(b*x^3+a)^(1/2)/x^ 2+2/9*x/a^3/b*(A*b-B*a)*(b*x^3+a)^(1/2)/(x^3+a/b)^2+2/27*b*x/a^4*(25*A*b-1 6*B*a)/((x^3+a/b)*b)^(1/2)-2/3*I*(1/40*b*(27*A*b-10*B*a)/a^4+1/27*b/a^4*(2 5*A*b-16*B*a))*3^(1/2)/b*(-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 = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.53 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=-\frac {91 \, {\left ({\left (10 \, B a b^{2} - 19 \, A b^{3}\right )} x^{11} + 2 \, {\left (10 \, B a^{2} b - 19 \, A a b^{2}\right )} x^{8} + {\left (10 \, B a^{3} - 19 \, A a^{2} b\right )} x^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (91 \, {\left (10 \, B a b^{2} - 19 \, A b^{3}\right )} x^{9} + 130 \, {\left (10 \, B a^{2} b - 19 \, A a b^{2}\right )} x^{6} + 108 \, A a^{3} + 27 \, {\left (10 \, B a^{3} - 19 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{540 \, {\left (a^{4} b^{2} x^{11} + 2 \, a^{5} b x^{8} + a^{6} x^{5}\right )}} \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
-1/540*(91*((10*B*a*b^2 - 19*A*b^3)*x^11 + 2*(10*B*a^2*b - 19*A*a*b^2)*x^8 + (10*B*a^3 - 19*A*a^2*b)*x^5)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (91*(10*B*a*b^2 - 19*A*b^3)*x^9 + 130*(10*B*a^2*b - 19*A*a*b^2)*x^6 + 10 8*A*a^3 + 27*(10*B*a^3 - 19*A*a^2*b)*x^3)*sqrt(b*x^3 + a))/(a^4*b^2*x^11 + 2*a^5*b*x^8 + a^6*x^5)
Timed out. \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x**3+A)/x**6/(b*x**3+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} x^{6}} \,d x } \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^6), x)
\[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} x^{6}} \,d x } \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^6), x)
Timed out. \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\int \frac {B\,x^3+A}{x^6\,{\left (b\,x^3+a\right )}^{5/2}} \,d x \] Input:
int((A + B*x^3)/(x^6*(a + b*x^3)^(5/2)),x)
Output:
int((A + B*x^3)/(x^6*(a + b*x^3)^(5/2)), x)
\[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{12}+2 a b \,x^{9}+a^{2} x^{6}}d x \] Input:
int((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x)
Output:
int(sqrt(a + b*x**3)/(a**2*x**6 + 2*a*b*x**9 + b**2*x**12),x)