Optimal. Leaf size=173 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2 \sqrt {q+p x^6}}{\sqrt {a} q-\sqrt {b} x^4+\sqrt {a} p x^6}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^4}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^6}{\sqrt {2} \sqrt [4]{b}}}{x^2 \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \]
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Rubi [A]
time = 0.38, antiderivative size = 255, normalized size of antiderivative = 1.47, number of steps
used = 10, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6846, 217,
1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {p x^6+q}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {p x^6+q}}+\sqrt {a}+\frac {\sqrt {b} x^4}{p x^6+q}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6846
Rubi steps
\begin {align*} \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx &=-\text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}}-\frac {\text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{2 \sqrt {a}}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {a} \sqrt {b}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a} \sqrt {q+p x^6}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} x^4}{q+p x^6}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2}{\sqrt {q+p x^6}}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [F]
time = 10.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{b x^8+a \left (q+p x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (p \,x^{6}-2 q \right ) \sqrt {p \,x^{6}+q}}{b \,x^{8}+a \left (p \,x^{6}+q \right )^{2}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs.
\(2 (124) = 248\).
time = 0.83, size = 361, normalized size = 2.09 \begin {gather*} \frac {1}{2} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} b x^{2} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}}{\sqrt {p x^{6} + q}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} + 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{12} + 2 \, a p q x^{6} - b x^{8} + a q^{2} - 2 \, {\left (a b x^{6} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{8} + a^{3} b q x^{2}\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{6} + q} - 2 \, {\left (a^{2} b p x^{10} + a^{2} b q x^{4}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{12} + 2 \, a p q x^{6} + b x^{8} + a q^{2}}\right ) \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 {x \left (p x^{6} - 2 q\right ) \sqrt {p x^{6} + q}}{a p^{2} x^{12} + 2 a p q x^{6} + a q^{2} + b x^{8}}\, dx \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 = -0.01 \begin {gather*} \int -\frac {x\,\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{a\,{\left (p\,x^6+q\right )}^2+b\,x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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