Optimal. Leaf size=181 \[ \frac {1}{2} \left (-a p^2 q^2-2 b p q\right ) \log \left (\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+p x^3+q\right )+\frac {1}{2} \log (x) \left (a p^2 q^2+2 b p q\right )+\frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (a p^3 x^9+3 a p^2 q x^6-a p^2 q x^5+3 a p q^2 x^3-a p q^2 x^2+a q^3+2 b p x^5+2 b q x^2\right )}{4 x^4} \]
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Rubi [F] time = 1.26, 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} \left (b x^2+a \left (q+p x^3\right )^2\right )}{x^5} \, 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} \left (b x^2+a \left (q+p x^3\right )^2\right )}{x^5} \, dx &=\int \left (2 b p \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}-\frac {a q^3 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5}-\frac {b q \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3}+3 a p^2 q x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}+2 a p^3 x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \, dx\\ &=(2 b p) \int \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx+\left (2 a p^3\right ) \int x^4 \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-(b q) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3} \, dx+\left (3 a p^2 q\right ) \int x \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \, dx-\left (a q^3\right ) \int \frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.77, 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} \left (b x^2+a \left (q+p x^3\right )^2\right )}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.37, size = 181, normalized size = 1.00 \begin {gather*} \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^3-2 p q x^2+q^2}+p x^3+q\right )+\frac {1}{2} \log (x) \left (a p^2 q^2+2 b p q\right )+\frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (a p^3 x^9+3 a p^2 q x^6-a p^2 q x^5+3 a p q^2 x^3-a p q^2 x^2+a q^3+2 b p x^5+2 b q x^2\right )}{4 x^4} \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^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, 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}}\, \left (b \,x^{2}+a \left (p \,x^{3}+q \right )^{2}\right )}{x^{5}}\, 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^{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^{5}}\,{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 )\,\left (a\,{\left (p\,x^3+q\right )}^2+b\,x^2\right )\,\sqrt {p^2\,x^6+2\,p\,q\,x^3-2\,p\,q\,x^2+q^2}}{x^5} \,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}} \left (a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{2}\right )}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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