Optimal. Leaf size=265 \[ -\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}-\frac{\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}+\frac{e}{2 a^2 d^2 x^2}-\frac{1}{4 a^2 d x^4}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.326811, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1252, 894, 639, 205, 635, 260} \[ -\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}-\frac{\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}+\frac{e}{2 a^2 d^2 x^2}-\frac{1}{4 a^2 d x^4}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1252
Rule 894
Rule 639
Rule 205
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 d x^3}-\frac{e}{a^2 d^2 x^2}+\frac{-2 c d^2+a e^2}{a^3 d^3 x}-\frac{e^7}{d^3 \left (c d^2+a e^2\right )^2 (d+e x)}+\frac{c^2 (a e+c d x)}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{c^2 \left (a e \left (c d^2+2 a e^2\right )+c d \left (2 c d^2+3 a e^2\right ) x\right )}{a^3 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a^2 d x^4}+\frac{e}{2 a^2 d^2 x^2}-\frac{\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac{c^2 \operatorname{Subst}\left (\int \frac{a e \left (c d^2+2 a e^2\right )+c d \left (2 c d^2+3 a e^2\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (c d^2+a e^2\right )^2}+\frac{c^2 \operatorname{Subst}\left (\int \frac{a e+c d x}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )}\\ &=-\frac{1}{4 a^2 d x^4}+\frac{e}{2 a^2 d^2 x^2}-\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac{\left (c^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2+a e^2\right )}+\frac{\left (c^2 e \left (c d^2+2 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )^2}+\frac{\left (c^3 d \left (2 c d^2+3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (c d^2+a e^2\right )^2}\\ &=-\frac{1}{4 a^2 d x^4}+\frac{e}{2 a^2 d^2 x^2}-\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (c d^2+a e^2\right )}+\frac{c^{3/2} e \left (c d^2+2 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (c d^2+a e^2\right )^2}-\frac{\left (2 c d^2-a e^2\right ) \log (x)}{a^3 d^3}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )^2}+\frac{c^2 d \left (2 c d^2+3 a e^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.426983, size = 278, normalized size = 1.05 \[ \frac{1}{4} \left (\frac{c^2 \left (e x^2-d\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{c^2 \left (3 a d e^2+2 c d^3\right ) \log \left (a+c x^4\right )}{a^3 \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{4 \log (x) \left (a e^2-2 c d^2\right )}{a^3 d^3}+\frac{2 e}{a^2 d^2 x^2}-\frac{1}{a^2 d x^4}-\frac{2 e^6 \log \left (d+e x^2\right )}{d^3 \left (a e^2+c d^2\right )^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.025, size = 363, normalized size = 1.4 \begin{align*}{\frac{{e}^{3}{c}^{2}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{3}{x}^{2}e{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{e}^{2}d{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{3}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{c}^{2}\ln \left ( c{x}^{4}+a \right ){e}^{2}d}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}+{\frac{{c}^{3}\ln \left ( c{x}^{4}+a \right ){d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{3}}}+{\frac{5\,{e}^{3}{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,e{d}^{2}{c}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{1}{4\,d{a}^{2}{x}^{4}}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}{a}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{a}^{3}d}}+{\frac{e}{2\,{d}^{2}{a}^{2}{x}^{2}}}-{\frac{{e}^{6}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \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 [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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09672, size = 473, normalized size = 1.78 \begin{align*} \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (c x^{4} + a\right )}{4 \,{\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}} - \frac{e^{7} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{7} e + 2 \, a c d^{5} e^{3} + a^{2} d^{3} e^{5}\right )}} + \frac{{\left (3 \, c^{3} d^{2} e + 5 \, a c^{2} e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt{a c}} - \frac{2 \, c^{4} d^{3} x^{4} + 3 \, a c^{3} d x^{4} e^{2} - a c^{3} d^{2} x^{2} e + 3 \, a c^{3} d^{3} - a^{2} c^{2} x^{2} e^{3} + 4 \, a^{2} c^{2} d e^{2}}{4 \,{\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}{\left (c x^{4} + a\right )}} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3} d^{3}} + \frac{6 \, c d^{2} x^{4} - 3 \, a x^{4} e^{2} + 2 \, a d x^{2} e - a d^{2}}{4 \, a^{3} d^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]