Optimal. Leaf size=64 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {b}{a^2 \sqrt {a+b x^3}}-\frac {1}{3 a x^3 \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac {\sqrt {a+b x^3}}{a^2 x^3}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{3 a x^3 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2}{3 a x^3 \sqrt {a+b x^3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^3\right )}{a}\\ &=\frac {2}{3 a x^3 \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{a^2 x^3}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{2 a^2}\\ &=\frac {2}{3 a x^3 \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{a^2 x^3}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{a^2}\\ &=\frac {2}{3 a x^3 \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{a^2 x^3}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.58 \[ -\frac {2 b \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x^3}{a}+1\right )}{3 a^2 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 173, normalized size = 2.70 \[ \left [\frac {3 \, {\left (b^{2} x^{6} + a b x^{3}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, {\left (3 \, a b x^{3} + a^{2}\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, -\frac {3 \, {\left (b^{2} x^{6} + a b x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x^{3} + a^{2}\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 72, normalized size = 1.12 \[ -\frac {b \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {3 \, {\left (b x^{3} + a\right )} b - 2 \, a b}{3 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{3} + a} a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 0.89 \[ \frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}-\frac {2 b}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{2}}-\frac {\sqrt {b \,x^{3}+a}}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 86, normalized size = 1.34 \[ -\frac {3 \, {\left (b x^{3} + a\right )} b - 2 \, a b}{3 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x^{3} + a} a^{3}\right )}} - \frac {b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 74, normalized size = 1.16 \[ \frac {b\,\ln \left (\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}^3}{x^6}\right )}{2\,a^{5/2}}-\frac {2\,b}{3\,a^2\,\sqrt {b\,x^3+a}}-\frac {\sqrt {b\,x^3+a}}{3\,a^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.18, size = 75, normalized size = 1.17 \[ - \frac {1}{3 a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b}}{a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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