Optimal. Leaf size=56 \[ \frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \]
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Rubi [F]
time = 1.55, 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^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\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^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {q+p x^6}}{a x^3}+\frac {x \left (2 b+3 a p x^2\right ) \sqrt {q+p x^6}}{a \left (a q+b x^4+a p x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (2 b+3 a p x^2\right ) \sqrt {q+p x^6}}{a q+b x^4+a p x^6} \, dx}{a}-\frac {2 \int \frac {\sqrt {q+p x^6}}{x^3} \, dx}{a}\\ &=\frac {\text {Subst}\left (\int \frac {(2 b+3 a p x) \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{2 a}-\frac {\text {Subst}\left (\int \frac {\sqrt {q+p x^3}}{x^2} \, dx,x,x^2\right )}{a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\text {Subst}\left (\int \left (\frac {2 b \sqrt {q+p x^3}}{a q+b x^2+a p x^3}+\frac {3 a p x \sqrt {q+p x^3}}{a q+b x^2+a p x^3}\right ) \, dx,x,x^2\right )}{2 a}-\frac {(3 p) \text {Subst}\left (\int \frac {x}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {b \text {Subst}\left (\int \frac {\sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{a}-\frac {\left (3 p^{2/3}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{2 a}+\frac {1}{2} (3 p) \text {Subst}\left (\int \frac {x \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )-\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} p^{2/3} \sqrt [3]{q}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}-\frac {3 \sqrt [3]{p} \sqrt {q+p x^6}}{a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x^2+p^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}\right )|-7-4 \sqrt {3}\right )}{2 a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} \sqrt {q+p x^6}}-\frac {\sqrt {2} 3^{3/4} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x^2+p^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}\right )|-7-4 \sqrt {3}\right )}{a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} \sqrt {q+p x^6}}+\frac {b \text {Subst}\left (\int \frac {\sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{a}+\frac {1}{2} (3 p) \text {Subst}\left (\int \frac {x \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )\\ \end {align*}
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Mathematica [A]
time = 8.20, size = 56, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{6}-2 q \right ) \sqrt {p \,x^{6}+q}}{x^{3} \left (a p \,x^{6}+b \,x^{4}+a q \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int -\frac {\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{x^3\,\left (a\,p\,x^6+b\,x^4+a\,q\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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