Optimal. Leaf size=129 \[ -\frac {(5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {a B-5 A b}{4 a^3 b x^{3/2}}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 290, 325, 329, 275, 205} \begin {gather*} \frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}-\frac {5 A b-a B}{4 a^3 b x^{3/2}}-\frac {(5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 275
Rule 290
Rule 325
Rule 329
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^{5/2} \left (a+b x^3\right )^3} \, dx &=\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {\left (\frac {15 A b}{2}-\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}+\frac {(3 (5 A b-a B)) \int \frac {1}{x^{5/2} \left (a+b x^3\right )} \, dx}{8 a^2 b}\\ &=-\frac {5 A b-a B}{4 a^3 b x^{3/2}}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}-\frac {(3 (5 A b-a B)) \int \frac {\sqrt {x}}{a+b x^3} \, dx}{8 a^3}\\ &=-\frac {5 A b-a B}{4 a^3 b x^{3/2}}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}-\frac {(3 (5 A b-a B)) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^6} \, dx,x,\sqrt {x}\right )}{4 a^3}\\ &=-\frac {5 A b-a B}{4 a^3 b x^{3/2}}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}-\frac {(5 A b-a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^{3/2}\right )}{4 a^3}\\ &=-\frac {5 A b-a B}{4 a^3 b x^{3/2}}+\frac {A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2}+\frac {5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}-\frac {(5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 102, normalized size = 0.79 \begin {gather*} \frac {(a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {a^2 \left (5 B x^3-8 A\right )+a \left (3 b B x^6-25 A b x^3\right )-15 A b^2 x^6}{12 a^3 x^{3/2} \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.17, size = 102, normalized size = 0.79 \begin {gather*} \frac {(a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {-8 a^2 A+5 a^2 B x^3-25 a A b x^3+3 a b B x^6-15 A b^2 x^6}{12 a^3 x^{3/2} \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 347, normalized size = 2.69 \begin {gather*} \left [\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{3} + 2 \, \sqrt {-a b} x^{\frac {3}{2}} - a}{b x^{3} + a}\right ) + 2 \, {\left (3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 8 \, A a^{3} b + 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{3} x^{8} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{2}\right )}}, \frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {3}{2}}}{a}\right ) + {\left (3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 8 \, A a^{3} b + 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{3} x^{8} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 88, normalized size = 0.68 \begin {gather*} \frac {{\left (B a - 5 \, A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} - \frac {2 \, A}{3 \, a^{3} x^{\frac {3}{2}}} + \frac {3 \, B a b x^{\frac {9}{2}} - 7 \, A b^{2} x^{\frac {9}{2}} + 5 \, B a^{2} x^{\frac {3}{2}} - 9 \, A a b x^{\frac {3}{2}}}{12 \, {\left (b x^{3} + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 133, normalized size = 1.03 \begin {gather*} -\frac {7 A \,b^{2} x^{\frac {9}{2}}}{12 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {B b \,x^{\frac {9}{2}}}{4 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {3 A b \,x^{\frac {3}{2}}}{4 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {5 B \,x^{\frac {3}{2}}}{12 \left (b \,x^{3}+a \right )^{2} a}-\frac {5 A b \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{3}}+\frac {B \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{2}}-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 100, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (B a b - 5 \, A b^{2}\right )} x^{6} + 5 \, {\left (B a^{2} - 5 \, A a b\right )} x^{3} - 8 \, A a^{2}}{12 \, {\left (a^{3} b^{2} x^{\frac {15}{2}} + 2 \, a^{4} b x^{\frac {9}{2}} + a^{5} x^{\frac {3}{2}}\right )}} + \frac {{\left (B a - 5 \, A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.73, size = 163, normalized size = 1.26 \begin {gather*} -\frac {\frac {2\,A}{3\,a}+\frac {5\,x^3\,\left (5\,A\,b-B\,a\right )}{12\,a^2}+\frac {b\,x^6\,\left (5\,A\,b-B\,a\right )}{4\,a^3}}{a^2\,x^{3/2}+b^2\,x^{15/2}+2\,a\,b\,x^{9/2}}-\frac {\mathrm {atan}\left (\frac {8\,a^{7/2}\,\sqrt {b}\,x^{3/2}\,\left (86400\,A^2\,a^9\,b^5-34560\,A\,B\,a^{10}\,b^4+3456\,B^2\,a^{11}\,b^3\right )}{\left (5\,A\,b-B\,a\right )\,\left (138240\,A\,a^{13}\,b^4-27648\,B\,a^{14}\,b^3\right )}\right )\,\left (5\,A\,b-B\,a\right )}{4\,a^{7/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________