Optimal. Leaf size=56 \[ \frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {p x^5+q}}\right )}{a^{3/2}}+\frac {2 \sqrt {p x^5+q}}{a x} \]
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Rubi [F] time = 1.41, 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 {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{x^2 \left (a q+b x^2+a p x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{x^2 \left (a q+b x^2+a p x^5\right )} \, dx &=\int \left (-\frac {2 \sqrt {q+p x^5}}{a x^2}+\frac {\left (2 b+5 a p x^3\right ) \sqrt {q+p x^5}}{a \left (a q+b x^2+a p x^5\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (2 b+5 a p x^3\right ) \sqrt {q+p x^5}}{a q+b x^2+a p x^5} \, dx}{a}-\frac {2 \int \frac {\sqrt {q+p x^5}}{x^2} \, dx}{a}\\ &=\frac {2 \sqrt {q+p x^5}}{a x}+\frac {\int \left (\frac {2 b \sqrt {q+p x^5}}{a q+b x^2+a p x^5}+\frac {5 a p x^3 \sqrt {q+p x^5}}{a q+b x^2+a p x^5}\right ) \, dx}{a}-\frac {(5 p) \int \frac {x^3}{\sqrt {q+p x^5}} \, dx}{a}\\ &=\frac {2 \sqrt {q+p x^5}}{a x}+\frac {(2 b) \int \frac {\sqrt {q+p x^5}}{a q+b x^2+a p x^5} \, dx}{a}+(5 p) \int \frac {x^3 \sqrt {q+p x^5}}{a q+b x^2+a p x^5} \, dx-\frac {\left (5 p \sqrt {1+\frac {p x^5}{q}}\right ) \int \frac {x^3}{\sqrt {1+\frac {p x^5}{q}}} \, dx}{a \sqrt {q+p x^5}}\\ &=\frac {2 \sqrt {q+p x^5}}{a x}-\frac {5 p x^4 \sqrt {1+\frac {p x^5}{q}} \, _2F_1\left (\frac {1}{2},\frac {4}{5};\frac {9}{5};-\frac {p x^5}{q}\right )}{4 a \sqrt {q+p x^5}}+\frac {(2 b) \int \frac {\sqrt {q+p x^5}}{a q+b x^2+a p x^5} \, dx}{a}+(5 p) \int \frac {x^3 \sqrt {q+p x^5}}{a q+b x^2+a p x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{x^2 \left (a q+b x^2+a p x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.78, size = 56, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {q+p x^5}}{a x}+\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {q+p x^5}}\right )}{a^{3/2}} \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 (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{{\left (a p x^{5} + b x^{2} + a q\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{x^{2} \left (a p \,x^{5}+b \,x^{2}+a q \right )}\, dx\]
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 {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{{\left (a p x^{5} + b x^{2} + a q\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.40, size = 102, normalized size = 1.82 \begin {gather*} \frac {2\,\sqrt {p\,x^5+q}}{a\,x}+\frac {\sqrt {b}\,\ln \left (\frac {a^5\,b\,p^4\,x^2-a^6\,p^4\,\left (p\,x^5+q\right )+a^{11/2}\,\sqrt {b}\,p^4\,x\,\sqrt {p\,x^5+q}\,2{}\mathrm {i}}{4\,b^2\,q\,x^2+4\,a\,b\,q\,\left (p\,x^5+q\right )}\right )\,1{}\mathrm {i}}{a^{3/2}} \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 {\sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{x^{2} \left (a p x^{5} + a q + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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