Optimal. Leaf size=178 \[ \frac {1}{2} \left (-a p^2 q^2-2 b p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\log (x) \left (a p^2 q^2+2 b p q\right )+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p^3 x^9-a p^2 q x^7+3 a p^2 q x^6-a p q^2 x^4+3 a p q^2 x^3+a q^3+2 b p x^7+2 b q x^4\right )}{4 x^8} \]
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Rubi [F] time = 1.61, 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} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, 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} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx &=\int \left (a p^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}-\frac {2 a q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9}-\frac {3 a p q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6}-\frac {2 b q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {b p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}\right ) \, dx\\ &=(b p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx+\left (a p^3\right ) \int \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-(2 b q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx-\left (3 a p q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6} \, dx-\left (2 a q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9} \, dx\\ \end {align*}
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Mathematica [F] time = 0.78, 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} \left (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.40, size = 178, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^3+3 a p q^2 x^3+2 b q x^4-a p q^2 x^4+3 a p^2 q x^6+2 b p x^7-a p^2 q x^7+a p^3 x^9\right )}{4 x^8}+\left (2 b p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 b p q-a p^2 q^2\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\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 {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{9}}\,{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 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (a \,p^{2} x^{6}+2 a p q \,x^{3}+b \,x^{4}+a \,q^{2}\right )}{x^{9}}\, dx\]
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 {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{9}}\,{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}\,\left (a\,p^2\,x^6+2\,a\,p\,q\,x^3+a\,q^2+b\,x^4\right )}{x^9} \,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}} \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}\right )}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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