Optimal. Leaf size=112 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^5+b x^2}}{\sqrt {a x^5+b x^2}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b x^2}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b x^2}}\right ) \]
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Rubi [F] time = 1.62, 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 {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^3}\right ) \int \frac {-2 b+a x^3}{\sqrt {x} \sqrt [4]{b+a x^3} \left (b+x^2+a x^3\right )} \, dx}{\sqrt [4]{b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {-2 b+a x^6}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{b+a x^6}}-\frac {3 b+x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {3 b+x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}+\frac {x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1+\frac {a x^3}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {a x^6}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}\\ &=\frac {2 x \sqrt [4]{1+\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-\frac {a x^3}{b}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^6} \left (b+x^4+a x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^5}}\\ \end {align*}
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Mathematica [F] time = 0.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 b+a x^3}{\left (b+x^2+a x^3\right ) \sqrt [4]{b x^2+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.92, size = 112, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^5+b x^2}}{\sqrt {a x^5+b x^2}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b x^2}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5+b 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 {a x^{3} - 2 \, b}{{\left (a x^{5} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{3} + x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{3}-2 b}{\left (a \,x^{3}+x^{2}+b \right ) \left (a \,x^{5}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx \end {gather*}
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 {a x^{3} - 2 \, b}{{\left (a x^{5} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{3} + x^{2} + b\right )}}\,{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 {2\,b-a\,x^3}{{\left (a\,x^5+b\,x^2\right )}^{1/4}\,\left (a\,x^3+x^2+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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