Optimal. Leaf size=86 \[ \frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}+\frac {2 a B-3 A b}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ -\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^4 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {A}{3 a x^3 \sqrt {a+b x^3}}+\frac {\left (-\frac {3 A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}-\frac {(3 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}-\frac {(3 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a^2 b}\\ &=-\frac {3 A b-2 a B}{3 a^2 \sqrt {a+b x^3}}-\frac {A}{3 a x^3 \sqrt {a+b x^3}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.66 \[ \frac {x^3 (2 a B-3 A b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^3}{a}+1\right )-a A}{3 a^2 x^3 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 233, normalized size = 2.71 \[ \left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, {\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 99, normalized size = 1.15 \[ \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{2}} + \frac {2 \, {\left (b x^{3} + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x^{3} + a\right )} A b + 2 \, A a b}{3 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{3} + a} a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 100, normalized size = 1.16 \[ \left (\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}-\frac {2 b}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{2}}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{2} x^{3}}\right ) A +\left (-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a}\right ) B \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 144, normalized size = 1.67 \[ -\frac {1}{6} \, A {\left (\frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} b - 2 \, a b\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x^{3} + a} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} + \frac {1}{3} \, B {\left (\frac {\log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {b x^{3} + a} a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.93, size = 131, normalized size = 1.52 \[ \frac {\ln \left (\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}^3}{x^6}\right )\,\left (3\,A\,b-2\,B\,a\right )}{6\,a^{5/2}}-\frac {\frac {2\,B\,a^2-3\,A\,a\,b}{2\,a^3}-\frac {a\,\left (\frac {A\,b^2}{3\,a^3}+\frac {5\,b\,\left (2\,B\,a^2-3\,A\,a\,b\right )}{6\,a^4}\right )}{b}}{\sqrt {b\,x^3+a}}-\frac {A\,\sqrt {b\,x^3+a}}{3\,a^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 79.79, size = 264, normalized size = 3.07 \[ A \left (- \frac {1}{3 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b}}{a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{2} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{2} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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