Optimal. Leaf size=192 \[ \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^4 x^{12}-4 a p^3 q x^{10}+24 a p^3 q x^9-16 a p^2 q^2 x^8-8 a p^2 q^2 x^7+36 a p^2 q^2 x^6-4 a p q^3 x^4+24 a p q^3 x^3+6 a q^4+15 b p x^9+15 b q x^6\right )}{30 x^{10}}-b p q \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+2 b p q \log (x) \]
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Rubi [F] time = 1.33, 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx &=\int \left (-\frac {2 a q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}}-\frac {5 a p q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8}-\frac {q \left (2 b+3 a p^2 q\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {p \left (b+a p^2 q\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+a p^4 x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \, dx\\ &=\left (a p^4\right ) \int x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-\left (5 a p q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8} \, dx-\left (2 a q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}} \, dx+\left (p \left (b+a p^2 q\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx-\left (q \left (2 b+3 a p^2 q\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.65, 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.58, size = 192, normalized size = 1.00 \begin {gather*} \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^4 x^{12}-4 a p^3 q x^{10}+24 a p^3 q x^9-16 a p^2 q^2 x^8-8 a p^2 q^2 x^7+36 a p^2 q^2 x^6-4 a p q^3 x^4+24 a p q^3 x^3+6 a q^4+15 b p x^9+15 b q x^6\right )}{30 x^{10}}-b p q \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+2 b p q \log (x) \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 (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, 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 (b \,x^{6}+a \left (p \,x^{3}+q \right )^{3}\right )}{x^{11}}\, 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 (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}}\,{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 (a\,{\left (p\,x^3+q\right )}^3+b\,x^6\right )\,\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^{11}} \,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^{3} x^{9} + 3 a p^{2} q x^{6} + 3 a p q^{2} x^{3} + a q^{3} + b x^{6}\right )}{x^{11}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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