Optimal. Leaf size=68 \[ -\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {b \sqrt {a+b x^3}}{4 x^3}-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, 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, 47, 63, 208} \[ -\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {b \sqrt {a+b x^3}}{4 x^3}-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{3/2}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {b \sqrt {a+b x^3}}{4 x^3}-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=-\frac {b \sqrt {a+b x^3}}{4 x^3}-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6}+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=-\frac {b \sqrt {a+b x^3}}{4 x^3}-\frac {\left (a+b x^3\right )^{3/2}}{6 x^6}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 76, normalized size = 1.12 \[ -\frac {2 a^2+3 b^2 x^6 \sqrt {\frac {b x^3}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )+7 a b x^3+5 b^2 x^6}{12 x^6 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 138, normalized size = 2.03 \[ \left [\frac {3 \, \sqrt {a} b^{2} x^{6} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, {\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{24 \, a x^{6}}, \frac {3 \, \sqrt {-a} b^{2} x^{6} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) - {\left (5 \, a b x^{3} + 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{12 \, a x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 70, normalized size = 1.03 \[ \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{3} - 3 \, \sqrt {b x^{3} + a} a b^{3}}{b^{2} x^{6}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 0.79 \[ -\frac {b^{2} \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {5 \sqrt {b \,x^{3}+a}\, b}{12 x^{3}}-\frac {\sqrt {b \,x^{3}+a}\, a}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 98, normalized size = 1.44 \[ \frac {b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{8 \, \sqrt {a}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {b x^{3} + a} a b^{2}}{12 \, {\left ({\left (b x^{3} + a\right )}^{2} - 2 \, {\left (b x^{3} + a\right )} a + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 74, normalized size = 1.09 \[ \frac {b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{8\,\sqrt {a}}-\frac {5\,b\,\sqrt {b\,x^3+a}}{12\,x^3}-\frac {a\,\sqrt {b\,x^3+a}}{6\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.54, size = 78, normalized size = 1.15 \[ - \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{6 x^{\frac {9}{2}}} - \frac {5 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}}{12 x^{\frac {3}{2}}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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