Optimal. Leaf size=159 \[ \frac {\sqrt {2 a p q+b} \tanh ^{-1}\left (\frac {\sqrt {b} x^2 \sqrt {2 a p q+b}}{\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (a p x^3+a q\right )+a p^2 x^6+2 a p q x^3+a q^2+b x^2}\right )}{a \sqrt {b}}+\frac {\log \left (\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+p x^3+q\right )}{a}-\frac {\log (x)}{a} \]
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Rubi [F] time = 5.50, 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 (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+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 (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+a \left (q+p x^3\right )^2\right )} \, dx &=\int \left (-\frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q x}+\frac {x \left (b+4 a p q x+a p^2 x^4\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q \left (a q^2+b x^2+2 a p q x^3+a p^2 x^6\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x} \, dx}{a q}+\frac {\int \frac {x \left (b+4 a p q x+a p^2 x^4\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6} \, dx}{a q}\\ &=-\frac {\int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x} \, dx}{a q}+\frac {\int \left (\frac {b x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6}+\frac {4 a p q x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6}+\frac {a p^2 x^5 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6}\right ) \, dx}{a q}\\ &=(4 p) \int \frac {x^2 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6} \, dx-\frac {\int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x} \, dx}{a q}+\frac {b \int \frac {x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6} \, dx}{a q}+\frac {p^2 \int \frac {x^5 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{a q^2+b x^2+2 a p q x^3+a p^2 x^6} \, dx}{q}\\ \end {align*}
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Mathematica [F] time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x \left (b x^2+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 = 0.74, size = 159, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b+2 a p q} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^2}{a q^2+b x^2+2 a p q x^3+a p^2 x^6+\left (a q+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right )}{a \sqrt {b}}-\frac {\log (x)}{a}+\frac {\log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right )}{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^{3} - 2 \, p q x^{2} + q^{2}} {\left (2 \, p x^{3} - q\right )}}{{\left ({\left (p x^{3} + q\right )}^{2} a + b x^{2}\right )} x}\,{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 \[\int \frac {\left (2 p \,x^{3}-q \right ) \sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}}{x \left (b \,x^{2}+a \left (p \,x^{3}+q \right )^{2}\right )}\, dx\]
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 {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left (2 \, p x^{3} - q\right )}}{{\left ({\left (p x^{3} + q\right )}^{2} a + b x^{2}\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 (q-2\,p\,x^3\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x\,\left (a\,{\left (p\,x^3+q\right )}^2+b\,x^2\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 (2 p x^{3} - q\right ) \sqrt {p^{2} x^{6} + 2 p q x^{3} - 2 p q x^{2} + q^{2}}}{x \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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