Optimal. Leaf size=110 \[ \frac {2 \sqrt {b} \sqrt {-a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-a-b}}{\sqrt {b} (x+1) \left (x^2-x+1\right )}\right )}{3 a^2}+\frac {(a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 a^2}+\frac {\sqrt {x^4+x} x}{3 a} \]
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Rubi [A] time = 0.26, antiderivative size = 129, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2042, 466, 465, 478, 523, 215, 377, 208} \begin {gather*} \frac {\sqrt {x^4+x} (a+2 b) \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x^3+1} \sqrt {x}}-\frac {2 \sqrt {b} \sqrt {x^4+x} \sqrt {a+b} \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {a+b}}{\sqrt {b} \sqrt {x^3+1}}\right )}{3 a^2 \sqrt {x^3+1} \sqrt {x}}+\frac {\sqrt {x^4+x} x}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 215
Rule 377
Rule 465
Rule 466
Rule 478
Rule 523
Rule 2042
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {x+x^4}}{-b+a x^3} \, dx &=\frac {\sqrt {x+x^4} \int \frac {x^{7/2} \sqrt {1+x^3}}{-b+a x^3} \, dx}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt {1+x^6}}{-b+a x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^2}}{-b+a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}-\frac {\sqrt {x+x^4} \operatorname {Subst}\left (\int \frac {-b+(-a-2 b) x^2}{\sqrt {1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}-\frac {\left ((-a-2 b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 b (a+b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {(a+2 b) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 b (a+b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(-a-b) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {(a+2 b) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {2 \sqrt {b} \sqrt {a+b} \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b} x^{3/2}}{\sqrt {b} \sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 156, normalized size = 1.42 \begin {gather*} -\frac {x \sqrt {x^4+x} \left (x^3 (a+2 b) \sqrt {-\frac {x^3 (a+b)}{b}} F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};-x^3,\frac {a x^3}{b}\right )-3 b \left (\sqrt {x^3+1} \sqrt {-\frac {x^3 (a+b)}{b}}-\sin ^{-1}\left (\frac {\sqrt {-\frac {x^3 (a+b)}{b}}}{\sqrt {1-\frac {a x^3}{b}}}\right )\right )\right )}{9 a b \sqrt {x^3+1} \sqrt {-\frac {x^3 (a+b)}{b}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.66, size = 110, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b} \sqrt {-a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-a-b}}{\sqrt {b} (x+1) \left (x^2-x+1\right )}\right )}{3 a^2}+\frac {(a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 a^2}+\frac {\sqrt {x^4+x} x}{3 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.72, size = 233, normalized size = 2.12 \begin {gather*} \left [\frac {2 \, \sqrt {x^{4} + x} a x + {\left (a + 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \sqrt {a b + b^{2}} \log \left (-\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} x^{3} - 4 \, {\left ({\left (a + 2 \, b\right )} x^{4} + b x\right )} \sqrt {x^{4} + x} \sqrt {a b + b^{2}} + b^{2}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right )}{6 \, a^{2}}, \frac {2 \, \sqrt {x^{4} + x} a x + {\left (a + 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + 2 \, \sqrt {-a b - b^{2}} \arctan \left (\frac {2 \, \sqrt {x^{4} + x} \sqrt {-a b - b^{2}} x}{{\left (a + 2 \, b\right )} x^{3} + b}\right )}{6 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 101, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x^{4} + x} x}{3 \, a} + \frac {{\left (a + 2 \, b\right )} \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a^{2}} - \frac {{\left (a + 2 \, b\right )} \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a^{2}} + \frac {2 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {b \sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 963, normalized size = 8.75
result too large to display
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 {\sqrt {x^{4} + x} x^{3}}{a x^{3} - b}\,{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 = -0.01 \begin {gather*} -\int \frac {x^3\,\sqrt {x^4+x}}{b-a\,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 {x^{3} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{a x^{3} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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