Optimal. Leaf size=205 \[ -\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac{\log (x) (a e+b d)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac{1}{2 a d x^2} \]
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Rubi [A] time = 0.470497, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1251, 893, 634, 618, 206, 628} \[ -\frac{\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac{\log (x) (a e+b d)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac{1}{2 a d x^2} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 893
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a d x^2}+\frac{-b d-a e}{a^2 d^2 x}+\frac{e^4}{d^2 \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x}{a^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a d x^2}-\frac{(b d+a e) \log (x)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{2 a d x^2}-\frac{(b d+a e) \log (x)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b c d-b^2 e+a c e\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{2 a d x^2}-\frac{(b d+a e) \log (x)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{1}{2 a d x^2}-\frac{\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac{(b d+a e) \log (x)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.338795, size = 331, normalized size = 1.61 \[ \frac{1}{4} \left (\frac{\left (b^2 \left (e \sqrt{b^2-4 a c}-c d\right )-b c \left (d \sqrt{b^2-4 a c}+3 a e\right )+a c \left (2 c d-e \sqrt{b^2-4 a c}\right )+b^3 e\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}+\frac{\left (b^2 \left (e \sqrt{b^2-4 a c}+c d\right )+b c \left (3 a e-d \sqrt{b^2-4 a c}\right )-a c \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 (-e)\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}-\frac{4 \log (x) (a e+b d)}{a^2 d^2}+\frac{2 e^3 \log \left (d+e x^2\right )}{d^2 e (a e-b d)+c d^4}-\frac{2}{a d x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 430, normalized size = 2.1 \begin{align*}{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) a}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{3\,ecb}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{d{c}^{2}}{ \left ( a{e}^{2}-deb+c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}cd}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{2\,ad{x}^{2}}}-{\frac{\ln \left ( x \right ) e}{{d}^{2}a}}-{\frac{\ln \left ( x \right ) b}{d{a}^{2}}}+{\frac{{e}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{2} \left ( a{e}^{2}-deb+c{d}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15765, size = 320, normalized size = 1.56 \begin{align*} \frac{{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )}} + \frac{e^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )}} + \frac{{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (b d + a e\right )} \log \left (x^{2}\right )}{2 \, a^{2} d^{2}} + \frac{b d x^{2} + a x^{2} e - a d}{2 \, a^{2} d^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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