Optimal. Leaf size=58 \[ \frac {2 (A b-a B)}{3 a b \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {446, 78, 63, 208} \[ \frac {2 (A b-a B)}{3 a b \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2 (A b-a B)}{3 a b \sqrt {a+b x^3}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {2 (A b-a B)}{3 a b \sqrt {a+b x^3}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a b}\\ &=\frac {2 (A b-a B)}{3 a b \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 58, normalized size = 1.00 \[ \frac {1}{3} \left (\frac {2 (A b-a B)}{a b \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 170, normalized size = 2.93 \[ \left [\frac {{\left (A b^{2} x^{3} + A a b\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, \sqrt {b x^{3} + a} {\left (B a^{2} - A a b\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {2 \, {\left ({\left (A b^{2} x^{3} + A a b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) - \sqrt {b x^{3} + a} {\left (B a^{2} - A a b\right )}\right )}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 0.91 \[ \frac {2 \, A \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a} - \frac {2 \, {\left (B a - A b\right )}}{3 \, \sqrt {b x^{3} + a} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 57, normalized size = 0.98 \[ \left (-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}+\frac {2}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a}\right ) A -\frac {2 B}{3 \sqrt {b \,x^{3}+a}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 70, normalized size = 1.21 \[ \frac {1}{3} \, A {\left (\frac {\log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {b x^{3} + a} a}\right )} - \frac {2 \, B}{3 \, \sqrt {b x^{3} + a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.77, size = 65, normalized size = 1.12 \[ \frac {\frac {2\,A}{3\,a}-\frac {2\,B}{3\,b}}{\sqrt {b\,x^3+a}}+\frac {A\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.82, size = 56, normalized size = 0.97 \[ \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {a + b x^{3}}}{\sqrt {- a}} \right )}}{3 a \sqrt {- a}} - \frac {2 \left (- A b + B a\right )}{3 a b \sqrt {a + b x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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