Optimal. Leaf size=49 \[ \frac {1}{3} (a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {\sqrt {x^4+x} \left (a x^3+2 b\right )}{3 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2038, 2004, 2029, 206} \begin {gather*} \frac {1}{3} x \sqrt {x^4+x} (a-2 b)+\frac {1}{3} (a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {2 b \left (x^4+x\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2004
Rule 2029
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^3\right ) \sqrt {x+x^4}}{x^3} \, dx &=\frac {2 b \left (x+x^4\right )^{3/2}}{3 x^3}-(-a+2 b) \int \sqrt {x+x^4} \, dx\\ &=\frac {1}{3} (a-2 b) x \sqrt {x+x^4}+\frac {2 b \left (x+x^4\right )^{3/2}}{3 x^3}-\frac {1}{2} (-a+2 b) \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{3} (a-2 b) x \sqrt {x+x^4}+\frac {2 b \left (x+x^4\right )^{3/2}}{3 x^3}-\frac {1}{3} (-a+2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=\frac {1}{3} (a-2 b) x \sqrt {x+x^4}+\frac {2 b \left (x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{3} (a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 1.27 \begin {gather*} \frac {\sqrt {x^4+x} \left (x^{3/2} (a-2 b) \sinh ^{-1}\left (x^{3/2}\right )+\sqrt {x^3+1} \left (a x^3+2 b\right )\right )}{3 x^2 \sqrt {x^3+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 49, normalized size = 1.00 \begin {gather*} \frac {\left (2 b+a x^3\right ) \sqrt {x+x^4}}{3 x^2}+\frac {1}{3} (a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 51, normalized size = 1.04 \begin {gather*} -\frac {{\left (a - 2 \, b\right )} x^{2} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x + 1\right ) - 2 \, {\left (a x^{3} + 2 \, b\right )} \sqrt {x^{4} + x}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 57, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} + x} a x + \frac {1}{6} \, {\left (a - 2 \, b\right )} \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, {\left (a - 2 \, b\right )} \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) + \frac {2}{3} \, b \sqrt {\frac {1}{x^{3}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 48, normalized size = 0.98
method | result | size |
trager | \(\frac {\left (a \,x^{3}+2 b \right ) \sqrt {x^{4}+x}}{3 x^{2}}+\frac {\left (a -2 b \right ) \ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{6}\) | \(48\) |
meijerg | \(-\frac {a \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}+\frac {b \left (\frac {4 \sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{\frac {3}{2}}}-4 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}\) | \(64\) |
elliptic | \(\frac {2 b \sqrt {x^{4}+x}}{3 x^{2}}+\frac {a x \sqrt {x^{4}+x}}{3}-\frac {2 \left (\frac {a}{2}-b \right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(322\) |
risch | \(\frac {\left (x^{3}+1\right ) \left (a \,x^{3}+2 b \right )}{3 x \sqrt {x \left (x^{3}+1\right )}}-\frac {2 \left (\frac {a}{2}-b \right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(326\) |
default | \(a \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-b \left (-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}+\frac {2 \sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, -\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\) | \(606\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - b\right )} \sqrt {x^{4} + x}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {\left (b-a\,x^3\right )\,\sqrt {x^4+x}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{3} - b\right )}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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