Optimal. Leaf size=167 \[ \frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{x^2 \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.294319, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1251, 1646, 12, 724, 206} \[ \frac{d^2 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{x^2 \left (-a b e-2 a c d+b^2 d\right )+a (b d-2 a e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 1646
Rule 12
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{\operatorname{Subst}\left (\int -\frac{\left (b^2-4 a c\right ) d^2}{2 \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=-\frac{a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x^2}{\sqrt{a+b x^2+c x^4}}\right )}{c d^2-b d e+a e^2}\\ &=-\frac{a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x^2+c x^4}}+\frac{d^2 \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x^2}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x^2+c x^4}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.652703, size = 204, normalized size = 1.22 \[ \frac{1}{2} \left (\frac{2 \left (-2 a^2 e+a b \left (d-e x^2\right )-2 a c d x^2+b^2 d x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (e (b d-a e)-c d^2\right )}-\frac{d^2 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{d^2 \log \left (d+e x^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 613, normalized size = 3.7 \begin{align*} -{\frac{b{x}^{2}}{e \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-2\,{\frac{a}{e\sqrt{c{x}^{4}+b{x}^{2}+a} \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{x}^{2}cd}{{e}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}}-{\frac{bd}{{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-2\,{\frac{c{d}^{2}}{{e}^{2} \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{c \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}+\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{b}{c}}-1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{-1}}+2\,{\frac{c{d}^{2}}{e \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }\ln \left ({ \left ( 2\,{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}}}}}+2\,{\frac{c{d}^{2}}{{e}^{2} \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{c \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}-\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}}+1/2\,{\frac{b}{c}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 5.92126, size = 2815, normalized size = 16.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30515, size = 536, normalized size = 3.21 \begin{align*} \frac{d^{2} \arctan \left (-\frac{{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} - \frac{\frac{{\left (b^{4} c d^{3} - 6 \, a b^{2} c^{2} d^{3} + 8 \, a^{2} c^{3} d^{3} - b^{5} d^{2} e + 5 \, a b^{3} c d^{2} e - 4 \, a^{2} b c^{2} d^{2} e + 2 \, a b^{4} d e^{2} - 10 \, a^{2} b^{2} c d e^{2} + 8 \, a^{3} c^{2} d e^{2} - a^{2} b^{3} e^{3} + 4 \, a^{3} b c e^{3}\right )} x^{2}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}} + \frac{a b^{3} c d^{3} - 4 \, a^{2} b c^{2} d^{3} - a b^{4} d^{2} e + 2 \, a^{2} b^{2} c d^{2} e + 8 \, a^{3} c^{2} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - 12 \, a^{3} b c d e^{2} - 2 \, a^{3} b^{2} e^{3} + 8 \, a^{4} c e^{3}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}}{32 \, \sqrt{c x^{4} + b x^{2} + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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