Optimal. Leaf size=84 \[ \frac {\sqrt {a+b x^3} (2 a B+A b)}{3 a}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 84, 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, 50, 63, 208} \[ \frac {\sqrt {a+b x^3} (2 a B+A b)}{3 a}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x^2} \, dx,x,x^3\right )\\ &=-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3}+\frac {(A b+2 a B) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^3\right )}{6 a}\\ &=\frac {(A b+2 a B) \sqrt {a+b x^3}}{3 a}-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3}+\frac {1}{6} (A b+2 a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {(A b+2 a B) \sqrt {a+b x^3}}{3 a}-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3}+\frac {(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 b}\\ &=\frac {(A b+2 a B) \sqrt {a+b x^3}}{3 a}-\frac {A \left (a+b x^3\right )^{3/2}}{3 a x^3}-\frac {(A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.75 \[ \frac {1}{3} \left (\frac {\sqrt {a+b x^3} \left (2 B x^3-A\right )}{x^3}-\frac {(2 a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 143, normalized size = 1.70 \[ \left [\frac {{\left (2 \, B a + A b\right )} \sqrt {a} x^{3} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (2 \, B a x^{3} - A a\right )} \sqrt {b x^{3} + a}}{6 \, a x^{3}}, \frac {{\left (2 \, B a + A b\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, B a x^{3} - A a\right )} \sqrt {b x^{3} + a}}{3 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 68, normalized size = 0.81 \[ \frac {2 \, \sqrt {b x^{3} + a} B b + \frac {{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b x^{3} + a} A b}{x^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 0.86 \[ \left (-\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {\sqrt {b \,x^{3}+a}}{3 x^{3}}\right ) A +\left (-\frac {2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {2 \sqrt {b \,x^{3}+a}}{3}\right ) B \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 107, normalized size = 1.27 \[ \frac {1}{6} \, {\left (\frac {b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, \sqrt {b x^{3} + a}}{x^{3}}\right )} A + \frac {1}{3} \, {\left (\sqrt {a} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x^{3} + a}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.93, size = 76, normalized size = 0.90 \[ \frac {2\,B\,\sqrt {b\,x^3+a}}{3}-\frac {A\,\sqrt {b\,x^3+a}}{3\,x^3}+\frac {\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )\,\left (\frac {A\,b}{2}+B\,a\right )}{3\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.66, size = 134, normalized size = 1.60 \[ - \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} - \frac {2 B \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {2 B a}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 B \sqrt {b} x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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