Optimal. Leaf size=53 \[ -2 \tan ^{-1}\left (\frac {\left (a x^5+b x\right )^{3/4}}{a x^4+b}\right )-2 \tanh ^{-1}\left (\frac {\left (a x^5+b x\right )^{3/4}}{a x^4+b}\right ) \]
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Rubi [F] time = 2.29, 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 {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \int \frac {-3 b+a x^4}{\sqrt [4]{x} \sqrt [4]{b+a x^4} \left (b-x^3+a x^4\right )} \, dx}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3 b+a x^{16}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{b+a x^{16}}}+\frac {x^2 \left (-4 b+x^{12}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-4 b+x^{12}\right )}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 b x^2}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}+\frac {x^{14}}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+\frac {a x^4}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+\frac {a x^{16}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a x^4}{b}} \, _2F_1\left (\frac {3}{16},\frac {1}{4};\frac {19}{16};-\frac {a x^4}{b}\right )}{3 \sqrt [4]{b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}-\frac {\left (16 b \sqrt [4]{x} \sqrt [4]{b+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{16}} \left (b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^5}}\\ \end {align*}
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Mathematica [F] time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.16, size = 53, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right )-2 \tanh ^{-1}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\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^{4} - 3 \, b}{{\left (a x^{5} + b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} + b\right )}}\,{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 \[\int \frac {a \,x^{4}-3 b}{\left (a \,x^{4}-x^{3}+b \right ) \left (a \,x^{5}+b x \right )^{\frac {1}{4}}}\, dx\]
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 {a x^{4} - 3 \, b}{{\left (a x^{5} + b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} + 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 = -0.02 \begin {gather*} \int -\frac {3\,b-a\,x^4}{{\left (a\,x^5+b\,x\right )}^{1/4}\,\left (a\,x^4-x^3+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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