Optimal. Leaf size=159 \[ \frac {\sqrt {2 a p q+b} \tanh ^{-1}\left (\frac {\sqrt {b} x^4 \sqrt {2 a p q+b}}{\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p x^3+a q\right )+a p^2 x^6+2 a p q x^3+a q^2+b x^4}\right )}{a \sqrt {b}}+\frac {\log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )}{a}-\frac {2 \log (x)}{a} \]
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Rubi [F] time = 6.06, 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 {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x \left (b x^4+a \left (q+p x^3\right )^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x \left (b x^4+a \left (q+p x^3\right )^2\right )} \, dx &=\int \left (-\frac {2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q x}+\frac {x^2 \left (5 a p q+2 b x+2 a p^2 x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {x^2 \left (5 a p q+2 b x+2 a p^2 x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6} \, dx}{a q}-\frac {2 \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{a q}\\ &=\frac {\int \left (\frac {5 a p q x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6}+\frac {2 b x^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6}+\frac {2 a p^2 x^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6}\right ) \, dx}{a q}-\frac {2 \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{a q}\\ &=(5 p) \int \frac {x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6} \, dx-\frac {2 \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{a q}+\frac {(2 b) \int \frac {x^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6} \, dx}{a q}+\frac {\left (2 p^2\right ) \int \frac {x^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a q^2+2 a p q x^3+b x^4+a p^2 x^6} \, dx}{q}\\ \end {align*}
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Mathematica [F] time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x \left (b x^4+a \left (q+p x^3\right )^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.78, size = 159, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2 a p q+b} \tanh ^{-1}\left (\frac {\sqrt {b} x^4 \sqrt {2 a p q+b}}{\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p x^3+a q\right )+a p^2 x^6+2 a p q x^3+a q^2+b x^4}\right )}{a \sqrt {b}}+\frac {\log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )}{a}-\frac {2 \log (x)}{a} \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 {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{{\left (b x^{4} + {\left (p x^{3} + q\right )}^{2} a\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}}{x \left (b \,x^{4}+a \left (p \,x^{3}+q \right )^{2}\right )}\, 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 {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{{\left (b x^{4} + {\left (p x^{3} + q\right )}^{2} a\right )} x}\,{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 {\left (2\,q-p\,x^3\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}}{x\,\left (a\,{\left (p\,x^3+q\right )}^2+b\,x^4\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 {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}}}{x \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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