Optimal. Leaf size=52 \[ \frac {\sqrt {x^4+x} \left (3 a x^6+2 b x^3+2 b\right )}{9 x^5}+\frac {1}{3} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]
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Rubi [C] time = 0.30, antiderivative size = 163, normalized size of antiderivative = 3.13, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2048, 2052, 2014, 2020, 2011, 329, 225} \begin {gather*} \frac {a \left (x^4+x\right )^{3/2}}{3 x^3}-\frac {2 a \sqrt {x^4+x}}{5 x^3}+\frac {3^{3/4} a x (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^4+x}}+\frac {2 b \left (x^4+x\right )^{3/2}}{9 x^6} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 225
Rule 329
Rule 2011
Rule 2014
Rule 2020
Rule 2048
Rule 2052
Rubi steps
\begin {align*} \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx &=\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\int \frac {\left (-b+a x^2\right ) \sqrt {x+x^4}}{x^6} \, dx\\ &=\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\int \left (-\frac {b \sqrt {x+x^4}}{x^6}+\frac {a \sqrt {x+x^4}}{x^4}\right ) \, dx\\ &=\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+a \int \frac {\sqrt {x+x^4}}{x^4} \, dx-b \int \frac {\sqrt {x+x^4}}{x^6} \, dx\\ &=-\frac {2 a \sqrt {x+x^4}}{5 x^3}+\frac {2 b \left (x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{5} (3 a) \int \frac {1}{\sqrt {x+x^4}} \, dx\\ &=-\frac {2 a \sqrt {x+x^4}}{5 x^3}+\frac {2 b \left (x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\frac {\left (3 a \sqrt {x} \sqrt {1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx}{5 \sqrt {x+x^4}}\\ &=-\frac {2 a \sqrt {x+x^4}}{5 x^3}+\frac {2 b \left (x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\frac {\left (6 a \sqrt {x} \sqrt {1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x+x^4}}\\ &=-\frac {2 a \sqrt {x+x^4}}{5 x^3}+\frac {2 b \left (x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (x+x^4\right )^{3/2}}{3 x^3}+\frac {3^{3/4} a x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 65, normalized size = 1.25 \begin {gather*} \frac {\sqrt {x^4+x} \left (\sqrt {x^3+1} \left (3 a x^6+2 b \left (x^3+1\right )\right )+3 a x^{9/2} \sinh ^{-1}\left (x^{3/2}\right )\right )}{9 x^5 \sqrt {x^3+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 52, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x+x^4} \left (2 b+2 b x^3+3 a x^6\right )}{9 x^5}+\frac {1}{3} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 55, normalized size = 1.06 \begin {gather*} \frac {3 \, a x^{5} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + 2 \, {\left (3 \, a x^{6} + 2 \, b x^{3} + 2 \, b\right )} \sqrt {x^{4} + x}}{18 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 49, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} + x} a x + \frac {2}{9} \, b {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} + \frac {1}{6} \, a \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, a \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 46, normalized size = 0.88
method | result | size |
meijerg | \(-\frac {a \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}+\frac {2 b \left (x^{3}+1\right )^{\frac {3}{2}}}{9 x^{\frac {9}{2}}}\) | \(46\) |
trager | \(\frac {\sqrt {x^{4}+x}\, \left (3 a \,x^{6}+2 b \,x^{3}+2 b \right )}{9 x^{5}}+\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{6}\) | \(51\) |
risch | \(\frac {\left (x^{3}+1\right ) \left (3 a \,x^{6}+2 b \,x^{3}+2 b \right )}{9 x^{4} \sqrt {x \left (x^{3}+1\right )}}-\frac {a \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(327\) |
elliptic | \(\frac {2 b \sqrt {x^{4}+x}}{9 x^{5}}+\frac {2 b \sqrt {x^{4}+x}}{9 x^{2}}+\frac {a x \sqrt {x^{4}+x}}{3}-\frac {a \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(329\) |
default | \(a \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-b \left (-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}-\frac {2 \sqrt {x^{4}+x}}{9 x^{2}}\right )\) | \(332\) |
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^{6} - b\right )} \sqrt {x^{4} + x}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 47, normalized size = 0.90 \begin {gather*} \frac {2\,b\,\left (x^3+1\right )\,\sqrt {x^4+x}}{9\,x^5}+\frac {2\,a\,x\,\sqrt {x^4+x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ -x^3\right )}{3\,\sqrt {x^3+1}} \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^{6} - b\right )}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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