Optimal. Leaf size=52 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^3-b x}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x \left (a x^3-b x\right )^{3/4}}{a x^2-b}\right ) \]
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Rubi [F] time = 2.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx &=\int \frac {x \left (-7 b+5 a x^2\right )}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{3/4} \left (-7 b+5 a x^2\right )}{\sqrt [4]{-b+a x^2} \left (b-a x^2+x^7\right )} \, dx}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-7 b+5 a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {5 a x^{14}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8-x^{28}\right )}-\frac {7 b x^6}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=-\frac {\left (20 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8-x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (28 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [4]{-b+a x^8} \left (b-a x^8+x^{28}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ \end {align*}
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Mathematica [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-7 b x+5 a x^3}{\sqrt [4]{-b x+a x^3} \left (b-a x^2+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.24, size = 52, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{-b x+a x^3}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x \left (-b x+a x^3\right )^{3/4}}{-b+a x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {5 a \,x^{3}-7 b x}{\left (a \,x^{3}-b x \right )^{\frac {1}{4}} \left (x^{7}-a \,x^{2}+b \right )}\, dx\]
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 {5 \, a x^{3} - 7 \, b x}{{\left (x^{7} - a x^{2} + b\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{4}}}\,{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 {7\,b\,x-5\,a\,x^3}{{\left (a\,x^3-b\,x\right )}^{1/4}\,\left (x^7-a\,x^2+b\right )} \,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 \left (5 a x^{2} - 7 b\right )}{\sqrt [4]{x \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{7}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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