Optimal. Leaf size=251 \[ \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a c p x^3+a c q-2 a d x^2+2 b c x^2\right )}{2 c^2 x^4}+\frac {\log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right ) \left (-a c^2 p q+a d^2-b c d\right )}{c^3}-\frac {2 (a d-b c) \sqrt {2 c^2 p q-d^2} \tan ^{-1}\left (\frac {x^2 \sqrt {2 c^2 p q-d^2}}{c \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+c p x^3+c q+d x^2}\right )}{c^3}+\frac {2 \log (x) \left (a c^2 p q-a d^2+b c d\right )}{c^3} \]
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Rubi [F] time = 13.94, 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 ) \left (a q+b x^2+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left (c q+d x^2+c p x^3\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 ) \left (a q+b x^2+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left (c q+d x^2+c p x^3\right )} \, dx &=\int \left (-\frac {2 a q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c x^5}-\frac {2 (b c-a d) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c^2 x^3}+\frac {a p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c x^2}+\frac {2 d (b c-a d) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c^3 q x}+\frac {(b c-a d) \left (3 c^2 p q-2 d^2 x-2 c d p x^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c^3 q \left (c q+d x^2+c p x^3\right )}\right ) \, dx\\ &=-\frac {(2 (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3} \, dx}{c^2}+\frac {(a p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx}{c}+\frac {(b c-a d) \int \frac {\left (3 c^2 p q-2 d^2 x-2 c d p x^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3} \, dx}{c^3 q}+\frac {(2 d (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{c^3 q}-\frac {(2 a q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx}{c}\\ &=-\frac {(2 (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3} \, dx}{c^2}+\frac {(a p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx}{c}+\frac {(b c-a d) \int \left (\frac {3 c^2 p q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3}-\frac {2 d^2 x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3}-\frac {2 c d p x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3}\right ) \, dx}{c^3 q}+\frac {(2 d (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{c^3 q}-\frac {(2 a q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx}{c}\\ &=-\frac {(2 (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^3} \, dx}{c^2}+\frac {(a p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx}{c}+\frac {(3 (b c-a d) p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3} \, dx}{c}+\frac {(2 d (b c-a d)) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx}{c^3 q}-\frac {\left (2 d^2 (b c-a d)\right ) \int \frac {x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3} \, dx}{c^3 q}-\frac {(2 d (b c-a d) p) \int \frac {x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{c q+d x^2+c p x^3} \, dx}{c^2 q}-\frac {(2 a q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx}{c}\\ \end {align*}
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Mathematica [F] time = 4.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 q+p x^3\right ) \left (a q+b x^2+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left (c q+d x^2+c p x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.22, size = 251, normalized size = 1.00 \begin {gather*} \frac {\left (a c q+2 b c x^2-2 a d x^2+a c p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{2 c^2 x^4}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x^2}{c q+d x^2+c p x^3+c \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right )}{c^3}+\frac {2 \left (b c d-a d^2+a c^2 p q\right ) \log (x)}{c^3}+\frac {\left (-b c d+a d^2-a c^2 p q\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 )}{c^3} \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(-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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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \left (a p \,x^{3}+b \,x^{2}+a q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}}{x^{5} \left (c p \,x^{3}+d \,x^{2}+c q \right )}\, dx\]
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^{4} + 2 \, p q x^{3} + q^{2}} {\left (a p x^{3} + b x^{2} + a q\right )} {\left (p x^{3} - 2 \, q\right )}}{{\left (c p x^{3} + d x^{2} + c q\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 = -0.00 \begin {gather*} \int -\frac {\left (2\,q-p\,x^3\right )\,\left (a\,p\,x^3+b\,x^2+a\,q\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}}{x^5\,\left (c\,p\,x^3+d\,x^2+c\,q\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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