Optimal. Leaf size=58 \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {A \sqrt {a+b x^3}}{3 a x^3} \]
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Rubi [A] time = 0.05, 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} \begin {gather*} \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {A \sqrt {a+b x^3}}{3 a x^3} \end {gather*}
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^4 \sqrt {a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=-\frac {A \sqrt {a+b x^3}}{3 a x^3}+\frac {\left (-\frac {A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac {A \sqrt {a+b x^3}}{3 a x^3}+\frac {\left (2 \left (-\frac {A b}{2}+a B\right )\right ) \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 {A \sqrt {a+b x^3}}{3 a x^3}+\frac {(A b-2 a B) \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 = 57, normalized size = 0.98 \begin {gather*} \frac {1}{3} \left (\frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A \sqrt {a+b x^3}}{a x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 58, normalized size = 1.00 \begin {gather*} \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {A \sqrt {a+b x^3}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.51, size = 126, normalized size = 2.17 \begin {gather*} \left [-\frac {{\left (2 \, B a - A b\right )} \sqrt {a} x^{3} \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} A a}{6 \, a^{2} x^{3}}, \frac {{\left (2 \, B a - A b\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) - \sqrt {b x^{3} + a} A a}{3 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 62, normalized size = 1.07 \begin {gather*} \frac {\frac {{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {\sqrt {b x^{3} + a} A b}{a x^{3}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.07 \begin {gather*} -\frac {2 B \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}+\left (\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{3}+a}}{3 a \,x^{3}}\right ) A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 109, normalized size = 1.88 \begin {gather*} -\frac {1}{6} \, A {\left (\frac {2 \, \sqrt {b x^{3} + a} b}{{\left (b x^{3} + a\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )} + \frac {B \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.89, size = 67, normalized size = 1.16 \begin {gather*} \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 (A\,b-2\,B\,a\right )}{6\,a^{3/2}}-\frac {A\,\sqrt {b\,x^3+a}}{3\,a\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.83, size = 80, normalized size = 1.38 \begin {gather*} - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{3 a x^{\frac {3}{2}}} + \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 a^{\frac {3}{2}}} - \frac {2 B \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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