Optimal. Leaf size=113 \[ -\frac{6 c^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0875405, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1107, 614, 618, 206} \[ -\frac{6 c^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1107
Rule 614
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{6 c^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0996406, size = 106, normalized size = 0.94 \[ \frac{\frac{24 c^2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{\left (b+2 c x^2\right ) \left (-2 c \left (5 a+3 c x^4\right )+b^2-6 b c x^2\right )}{\left (a+b x^2+c x^4\right )^2}}{4 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.176, size = 141, normalized size = 1.3 \begin{align*}{\frac{2\,c{x}^{2}+b}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{c}^{2}{x}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+{\frac{3\,bc}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+6\,{\frac{{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.58014, size = 1709, normalized size = 15.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 9.5687, size = 481, normalized size = 4.26 \begin{align*} - 3 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{- 192 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + 3 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{192 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + \frac{10 a b c - b^{3} + 18 b c^{2} x^{4} + 12 c^{3} x^{6} + x^{2} \left (20 a c^{2} + 4 b^{2} c\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 27.711, size = 194, normalized size = 1.72 \begin{align*} \frac{6 \, c^{2} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 20 \, a c^{2} x^{2} - b^{3} + 10 \, a b c}{4 \,{\left (c x^{4} + b x^{2} + a\right )}^{2}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]