Optimal. Leaf size=240 \[ \frac{\log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )}{2 c^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (4 a B d e \left (c d^2-a e^2\right )-A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^2 \left (-a B e^2+4 A c d e+6 B c d^2\right )}{2 c^2}+\frac{e x \left (-a A e^3-4 a B d e^2+6 A c d^2 e+4 B c d^3\right )}{c^2}+\frac{e^3 x^3 (A e+4 B d)}{3 c}+\frac{B e^4 x^4}{4 c} \]
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Rubi [A] time = 0.261257, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {801, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )}{2 c^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (4 a B d e \left (c d^2-a e^2\right )-A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^2 \left (-a B e^2+4 A c d e+6 B c d^2\right )}{2 c^2}+\frac{e x \left (-a A e^3-4 a B d e^2+6 A c d^2 e+4 B c d^3\right )}{c^2}+\frac{e^3 x^3 (A e+4 B d)}{3 c}+\frac{B e^4 x^4}{4 c} \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^4}{a+c x^2} \, dx &=\int \left (\frac{e \left (4 B c d^3+6 A c d^2 e-4 a B d e^2-a A e^3\right )}{c^2}+\frac{e^2 \left (6 B c d^2+4 A c d e-a B e^2\right ) x}{c^2}+\frac{e^3 (4 B d+A e) x^2}{c}+\frac{B e^4 x^3}{c}+\frac{-4 a B d e \left (c d^2-a e^2\right )+A \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )+\left (4 A c d e \left (c d^2-a e^2\right )+B \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{e \left (4 B c d^3+6 A c d^2 e-4 a B d e^2-a A e^3\right ) x}{c^2}+\frac{e^2 \left (6 B c d^2+4 A c d e-a B e^2\right ) x^2}{2 c^2}+\frac{e^3 (4 B d+A e) x^3}{3 c}+\frac{B e^4 x^4}{4 c}+\frac{\int \frac{-4 a B d e \left (c d^2-a e^2\right )+A \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )+\left (4 A c d e \left (c d^2-a e^2\right )+B \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) x}{a+c x^2} \, dx}{c^2}\\ &=\frac{e \left (4 B c d^3+6 A c d^2 e-4 a B d e^2-a A e^3\right ) x}{c^2}+\frac{e^2 \left (6 B c d^2+4 A c d e-a B e^2\right ) x^2}{2 c^2}+\frac{e^3 (4 B d+A e) x^3}{3 c}+\frac{B e^4 x^4}{4 c}-\frac{\left (4 a B d e \left (c d^2-a e^2\right )-A \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}+\frac{\left (4 A c d e \left (c d^2-a e^2\right )+B \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}\\ &=\frac{e \left (4 B c d^3+6 A c d^2 e-4 a B d e^2-a A e^3\right ) x}{c^2}+\frac{e^2 \left (6 B c d^2+4 A c d e-a B e^2\right ) x^2}{2 c^2}+\frac{e^3 (4 B d+A e) x^3}{3 c}+\frac{B e^4 x^4}{4 c}-\frac{\left (4 a B d e \left (c d^2-a e^2\right )-A \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{\left (4 A c d e \left (c d^2-a e^2\right )+B \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )\right ) \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.193508, size = 217, normalized size = 0.9 \[ \frac{6 \log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )+c e x \left (-12 a A e^3-6 a B e^2 (8 d+e x)+4 A c e \left (18 d^2+6 d e x+e^2 x^2\right )+B c \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )}{12 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (a e^2-c d^2\right )\right )}{\sqrt{a} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 345, normalized size = 1.4 \begin{align*}{\frac{B{e}^{4}{x}^{4}}{4\,c}}+{\frac{A{e}^{4}{x}^{3}}{3\,c}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,c}}+2\,{\frac{A{e}^{3}{x}^{2}d}{c}}-{\frac{B{e}^{4}{x}^{2}a}{2\,{c}^{2}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{c}}-{\frac{Aa{e}^{4}x}{{c}^{2}}}+6\,{\frac{A{e}^{2}{d}^{2}x}{c}}-4\,{\frac{Bad{e}^{3}x}{{c}^{2}}}+4\,{\frac{Be{d}^{3}x}{c}}-2\,{\frac{\ln \left ( c{x}^{2}+a \right ) Ada{e}^{3}}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+a \right ) A{d}^{3}e}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{e}^{4}{a}^{2}}{2\,{c}^{3}}}-3\,{\frac{\ln \left ( c{x}^{2}+a \right ) Ba{d}^{2}{e}^{2}}{{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{4}}{2\,c}}+{\frac{A{a}^{2}{e}^{4}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{aA{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{A{d}^{4}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+4\,{\frac{B{a}^{2}d{e}^{3}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }-4\,{\frac{aB{d}^{3}e}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \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 [A] time = 1.95071, size = 1180, normalized size = 4.92 \begin{align*} \left [\frac{3 \, B a c^{2} e^{4} x^{4} + 4 \,{\left (4 \, B a c^{2} d e^{3} + A a c^{2} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B a c^{2} d^{2} e^{2} + 4 \, A a c^{2} d e^{3} - B a^{2} c e^{4}\right )} x^{2} - 6 \,{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 12 \,{\left (4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 4 \, B a^{2} c d e^{3} - A a^{2} c e^{4}\right )} x + 6 \,{\left (B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}, \frac{3 \, B a c^{2} e^{4} x^{4} + 4 \,{\left (4 \, B a c^{2} d e^{3} + A a c^{2} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B a c^{2} d^{2} e^{2} + 4 \, A a c^{2} d e^{3} - B a^{2} c e^{4}\right )} x^{2} + 12 \,{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 12 \,{\left (4 \, B a c^{2} d^{3} e + 6 \, A a c^{2} d^{2} e^{2} - 4 \, B a^{2} c d e^{3} - A a^{2} c e^{4}\right )} x + 6 \,{\left (B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.19954, size = 901, normalized size = 3.75 \begin{align*} \frac{B e^{4} x^{4}}{4 c} + \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{4 A a^{2} c d e^{3} - 4 A a c^{2} d^{3} e - B a^{3} e^{4} + 6 B a^{2} c d^{2} e^{2} - B a c^{2} d^{4} + 2 a c^{3} \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right )}{A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} + A c^{3} d^{4} + 4 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{4 A a^{2} c d e^{3} - 4 A a c^{2} d^{3} e - B a^{3} e^{4} + 6 B a^{2} c d^{2} e^{2} - B a c^{2} d^{4} + 2 a c^{3} \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right )}{A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} + A c^{3} d^{4} + 4 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \frac{x^{3} \left (A e^{4} + 4 B d e^{3}\right )}{3 c} - \frac{x^{2} \left (- 4 A c d e^{3} + B a e^{4} - 6 B c d^{2} e^{2}\right )}{2 c^{2}} - \frac{x \left (A a e^{4} - 6 A c d^{2} e^{2} + 4 B a d e^{3} - 4 B c d^{3} e\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18022, size = 331, normalized size = 1.38 \begin{align*} \frac{{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{{\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, B c^{3} x^{4} e^{4} + 16 \, B c^{3} d x^{3} e^{3} + 36 \, B c^{3} d^{2} x^{2} e^{2} + 48 \, B c^{3} d^{3} x e + 4 \, A c^{3} x^{3} e^{4} + 24 \, A c^{3} d x^{2} e^{3} + 72 \, A c^{3} d^{2} x e^{2} - 6 \, B a c^{2} x^{2} e^{4} - 48 \, B a c^{2} d x e^{3} - 12 \, A a c^{2} x e^{4}}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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