Optimal. Leaf size=95 \[ -\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {5 b^2}{4 a^3 \sqrt {a+b x^3}}+\frac {5 b}{12 a^2 x^3 \sqrt {a+b x^3}}-\frac {1}{6 a x^6 \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {5 b \sqrt {a+b x^3}}{4 a^3 x^3}-\frac {5 \sqrt {a+b x^3}}{6 a^2 x^6}+\frac {2}{3 a x^6 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2}{3 a x^6 \sqrt {a+b x^3}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {2}{3 a x^6 \sqrt {a+b x^3}}-\frac {5 \sqrt {a+b x^3}}{6 a^2 x^6}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^3\right )}{4 a^2}\\ &=\frac {2}{3 a x^6 \sqrt {a+b x^3}}-\frac {5 \sqrt {a+b x^3}}{6 a^2 x^6}+\frac {5 b \sqrt {a+b x^3}}{4 a^3 x^3}+\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{8 a^3}\\ &=\frac {2}{3 a x^6 \sqrt {a+b x^3}}-\frac {5 \sqrt {a+b x^3}}{6 a^2 x^6}+\frac {5 b \sqrt {a+b x^3}}{4 a^3 x^3}+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{4 a^3}\\ &=\frac {2}{3 a x^6 \sqrt {a+b x^3}}-\frac {5 \sqrt {a+b x^3}}{6 a^2 x^6}+\frac {5 b \sqrt {a+b x^3}}{4 a^3 x^3}-\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 39, normalized size = 0.41 \[ \frac {2 b^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {b x^3}{a}+1\right )}{3 a^3 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 203, normalized size = 2.14 \[ \left [\frac {15 \, {\left (b^{3} x^{9} + a b^{2} x^{6}\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (15 \, a b^{2} x^{6} + 5 \, a^{2} b x^{3} - 2 \, a^{3}\right )} \sqrt {b x^{3} + a}}{24 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}}, \frac {15 \, {\left (b^{3} x^{9} + a b^{2} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{2} x^{6} + 5 \, a^{2} b x^{3} - 2 \, a^{3}\right )} \sqrt {b x^{3} + a}}{12 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 0.93 \[ \frac {5 \, b^{2} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3}} + \frac {2 \, b^{2}}{3 \, \sqrt {b x^{3} + a} a^{3}} + \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 9 \, \sqrt {b x^{3} + a} a b^{2}}{12 \, a^{3} b^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 80, normalized size = 0.84 \[ -\frac {5 b^{2} \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {7}{2}}}+\frac {2 b^{2}}{3 \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}\, a^{3}}+\frac {7 \sqrt {b \,x^{3}+a}\, b}{12 a^{3} x^{3}}-\frac {\sqrt {b \,x^{3}+a}}{6 a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 122, normalized size = 1.28 \[ \frac {15 \, {\left (b x^{3} + a\right )}^{2} b^{2} - 25 \, {\left (b x^{3} + a\right )} a b^{2} + 8 \, a^{2} b^{2}}{12 \, {\left ({\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3} - 2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4} + \sqrt {b x^{3} + a} a^{5}\right )}} + \frac {5 \, b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{8 \, a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 96, normalized size = 1.01 \[ \frac {2\,b^2}{3\,a^3\,\sqrt {b\,x^3+a}}-\frac {\sqrt {b\,x^3+a}}{6\,a^2\,x^6}+\frac {5\,b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{8\,a^{7/2}}+\frac {7\,b\,\sqrt {b\,x^3+a}}{12\,a^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.24, size = 112, normalized size = 1.18 \[ - \frac {1}{6 a \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {5 \sqrt {b}}{12 a^{2} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {5 b^{\frac {3}{2}}}{4 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {5 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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