Optimal. Leaf size=112 \[ \frac {2 \sqrt {b} \sqrt {a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {a-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 a^2}+\frac {(2 b-a) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )}{3 a^2}+\frac {\sqrt {x^4-x} x}{3 a} \]
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Rubi [A] time = 0.26, antiderivative size = 149, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2042, 466, 465, 478, 523, 217, 206, 377, 205} \begin {gather*} \frac {2 \sqrt {b} \sqrt {x^4-x} \sqrt {a-b} \tan ^{-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} (a-2 b) \tanh ^{-1}\left (\frac {x^{3/2}}{\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 205
Rule 206
Rule 217
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 (a-b) 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 {\left ((a-2 b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) 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 {2 \sqrt {a-b} \sqrt {b} \sqrt {-x+x^4} \tan ^{-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}}-\frac {(a-2 b) \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.54, size = 136, normalized size = 1.21 \begin {gather*} \frac {x^2 \left (\frac {\sqrt {1-x^3} x^3 (a-2 b) F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};x^3,\frac {a x^3}{b}\right )}{b}+\frac {3 \sqrt {1-x^3} \sin ^{-1}\left (\frac {\sqrt {\frac {x^3 (b-a)}{b}}}{\sqrt {1-\frac {a x^3}{b}}}\right )}{\sqrt {\frac {x^3 (b-a)}{b}}}+3 \left (x^3-1\right )\right )}{9 a \sqrt {x \left (x^3-1\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.58, size = 112, 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 {(2 b-a) \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 = 247, normalized size = 2.21 \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 = 0.56, size = 100, normalized size = 0.89 \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 \, \sqrt {a b - b^{2}} \arctan \left (\frac {b \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {a b - b^{2}}}\right )}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 946, normalized size = 8.45
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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