Optimal. Leaf size=59 \[ \frac {\left (-3 a x^3-2 b\right ) \sqrt {a x^3+b}}{6 x^6}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{6 \sqrt {b}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 78, 47, 63, 208} \begin {gather*} -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{6 \sqrt {b}}-\frac {a \sqrt {a x^3+b}}{6 x^3}-\frac {\left (a x^3+b\right )^{3/2}}{3 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {b+a x^3} \left (2 b+a x^3\right )}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b+a x} (2 b+a x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (b+a x^3\right )^{3/2}}{3 x^6}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {\sqrt {b+a x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {a \sqrt {b+a x^3}}{6 x^3}-\frac {\left (b+a x^3\right )^{3/2}}{3 x^6}+\frac {1}{12} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right )\\ &=-\frac {a \sqrt {b+a x^3}}{6 x^3}-\frac {\left (b+a x^3\right )^{3/2}}{3 x^6}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right )\\ &=-\frac {a \sqrt {b+a x^3}}{6 x^3}-\frac {\left (b+a x^3\right )^{3/2}}{3 x^6}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{6 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 1.27 \begin {gather*} -\frac {a^2 x^6 \sqrt {\frac {a x^3}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {a x^3}{b}+1}\right )+3 a^2 x^6+5 a b x^3+2 b^2}{6 x^6 \sqrt {a x^3+b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 59, normalized size = 1.00 \begin {gather*} \frac {\left (-2 b-3 a x^3\right ) \sqrt {b+a x^3}}{6 x^6}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{6 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 136, normalized size = 2.31 \begin {gather*} \left [\frac {a^{2} \sqrt {b} x^{6} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) - 2 \, {\left (3 \, a b x^{3} + 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{12 \, b x^{6}}, \frac {a^{2} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right ) - {\left (3 \, a b x^{3} + 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{6 \, b x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 69, normalized size = 1.17 \begin {gather*} \frac {\frac {a^{3} \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{3} - \sqrt {a x^{3} + b} a^{3} b}{a^{2} x^{6}}}{6 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 48, normalized size = 0.81
method | result | size |
risch | \(-\frac {\sqrt {a \,x^{3}+b}\, \left (3 a \,x^{3}+2 b \right )}{6 x^{6}}-\frac {a^{2} \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{6 \sqrt {b}}\) | \(48\) |
elliptic | \(-\frac {b \sqrt {a \,x^{3}+b}}{3 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{2 x^{3}}-\frac {a^{2} \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{6 \sqrt {b}}\) | \(54\) |
default | \(a \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )+2 b \left (-\frac {\sqrt {a \,x^{3}+b}}{6 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{12 b \,x^{3}}+\frac {a^{2} \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{12 b^{\frac {3}{2}}}\right )\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 158, normalized size = 2.68 \begin {gather*} \frac {1}{6} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} a - \frac {1}{12} \, {\left (\frac {a^{2} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{2} + \sqrt {a x^{3} + b} a^{2} b\right )}}{{\left (a x^{3} + b\right )}^{2} b - 2 \, {\left (a x^{3} + b\right )} b^{2} + b^{3}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 74, normalized size = 1.25 \begin {gather*} \frac {a^2\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{12\,\sqrt {b}}-\frac {b\,\sqrt {a\,x^3+b}}{3\,x^6}-\frac {a\,\sqrt {a\,x^3+b}}{2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 68.22, size = 128, normalized size = 2.17 \begin {gather*} - \frac {a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} - \frac {a^{\frac {3}{2}}}{6 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {\sqrt {a} b}{2 x^{\frac {9}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{6 \sqrt {b}} - \frac {b^{2}}{3 \sqrt {a} x^{\frac {15}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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