Optimal. Leaf size=166 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{d \log (x)}{a^2}+\frac{x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 0.393699, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {1663, 1646, 800, 634, 618, 206, 628} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (4 a^2 c e-2 a b (a f+3 c d)+b^3 d\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{d \log (x)}{a^2}+\frac{x^2 (a b f-2 a c e+b c d)-a b e-2 a (c d-a f)+b^2 d}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1646
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x^2+f x^4}{x \left (a+b x^2+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x+f x^2}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\left (\frac{b^2}{a}-4 c\right ) d-\frac{(b c d-2 a c e+a b f) x}{a}}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{\left (-b^2+4 a c\right ) d}{a^2 x}+\frac{b^3 d+2 a^2 c e-a b (5 c d+a f)+c \left (b^2-4 a c\right ) d x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{d \log (x)}{a^2}-\frac{\operatorname{Subst}\left (\int \frac{b^3 d+2 a^2 c e-a b (5 c d+a f)+c \left (b^2-4 a c\right ) d x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{d \log (x)}{a^2}-\frac{d \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}-\frac{\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )}\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{d \log (x)}{a^2}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\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 (b^2-4 a c\right )}\\ &=\frac{b^2 d-a b e-2 a (c d-a f)+(b c d-2 a c e+a b f) x^2}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b^3 d+4 a^2 c e-2 a b (3 c d+a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac{d \log (x)}{a^2}-\frac{d \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.469313, size = 268, normalized size = 1.61 \[ -\frac{-\frac{2 a \left (b \left (-a e+a f x^2+c d x^2\right )+2 a \left (a f-c \left (d+e x^2\right )\right )+b^2 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (4 a c \left (a e-d \sqrt{b^2-4 a c}\right )+b^2 d \sqrt{b^2-4 a c}-2 a b (a f+3 c d)+b^3 d\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-4 a c \left (d \sqrt{b^2-4 a c}+a e\right )+b^2 d \sqrt{b^2-4 a c}+2 a b (a f+3 c d)+b^3 (-d)\right )}{\left (b^2-4 a c\right )^{3/2}}-4 d \log (x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 462, normalized size = 2.8 \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{2}}}-{\frac{b{x}^{2}f}{ \left ( 2\,c{x}^{4}+2\,b{x}^{2}+2\,a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{c{x}^{2}e}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{b{x}^{2}cd}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{af}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{be}{ \left ( 2\,c{x}^{4}+2\,b{x}^{2}+2\,a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{cd}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{2}d}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{a \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-{bf\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{ce}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-3\,{\frac{bcd}{a \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{3}d}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \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 [B] time = 7.60329, size = 2333, normalized size = 14.05 \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(-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 = 19.5116, size = 306, normalized size = 1.84 \begin{align*} -\frac{{\left (b^{3} d - 6 \, a b c d - 2 \, a^{2} b f + 4 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{d \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} + \frac{d \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b^{2} c d x^{4} - 4 \, a c^{2} d x^{4} + b^{3} d x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} b f x^{2} - 4 \, a^{2} c x^{2} e + 3 \, a b^{2} d - 8 \, a^{2} c d + 4 \, a^{3} f - 2 \, a^{2} b e}{4 \,{\left (c x^{4} + b x^{2} + a\right )}{\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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