Optimal. Leaf size=125 \[ \frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}}-\frac{3 (d+e x) (a e-c d x) (a B e+A c d)}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.0582483, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {805, 723, 205} \[ \frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}}-\frac{3 (d+e x) (a e-c d x) (a B e+A c d)}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 805
Rule 723
Rule 205
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a B-A c x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}+\frac{(3 (A c d+a B e)) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a B-A c x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{3 (A c d+a B e) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\left (3 (A c d+a B e) \left (c d^2+a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-A c x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac{3 (A c d+a B e) (a e-c d x) (d+e x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 (A c d+a B e) \left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.107883, size = 186, normalized size = 1.49 \[ \frac{-a^2 e^2 (4 A e+12 B d+5 B e x)+3 a c d e x (A e+B d)+3 A c^2 d^3 x}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{a^2 e^2 (A e+3 B d+B e x)-a c d (3 A e (d+e x)+B d (d+3 e x))+A c^2 d^3 x}{4 a c^2 \left (a+c x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{8 a^{5/2} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 260, normalized size = 2.1 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Aacd{e}^{2}+3\,A{d}^{3}{c}^{2}-5\,{a}^{2}B{e}^{3}+3\,Bac{d}^{2}e \right ){x}^{3}}{8\,{a}^{2}c}}-{\frac{{e}^{2} \left ( Ae+3\,Bd \right ){x}^{2}}{2\,c}}-{\frac{ \left ( 3\,Aacd{e}^{2}-5\,A{d}^{3}{c}^{2}+3\,{a}^{2}B{e}^{3}+3\,Bac{d}^{2}e \right ) x}{8\,a{c}^{2}}}-{\frac{aA{e}^{3}+3\,Ac{d}^{2}e+3\,aBd{e}^{2}+Bc{d}^{3}}{4\,{c}^{2}}} \right ) }+{\frac{3\,Ad{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{3}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{e}^{3}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,B{d}^{2}e}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.9381, size = 1544, normalized size = 12.35 \begin{align*} \left [-\frac{4 \, B a^{3} c^{2} d^{3} + 12 \, A a^{3} c^{2} d^{2} e + 12 \, B a^{4} c d e^{2} + 4 \, A a^{4} c e^{3} - 2 \,{\left (3 \, A a c^{4} d^{3} + 3 \, B a^{2} c^{3} d^{2} e + 3 \, A a^{2} c^{3} d e^{2} - 5 \, B a^{3} c^{2} e^{3}\right )} x^{3} + 8 \,{\left (3 \, B a^{3} c^{2} d e^{2} + A a^{3} c^{2} e^{3}\right )} x^{2} + 3 \,{\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} +{\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (5 \, A a^{2} c^{3} d^{3} - 3 \, B a^{3} c^{2} d^{2} e - 3 \, A a^{3} c^{2} d e^{2} - 3 \, B a^{4} c e^{3}\right )} x}{16 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac{2 \, B a^{3} c^{2} d^{3} + 6 \, A a^{3} c^{2} d^{2} e + 6 \, B a^{4} c d e^{2} + 2 \, A a^{4} c e^{3} -{\left (3 \, A a c^{4} d^{3} + 3 \, B a^{2} c^{3} d^{2} e + 3 \, A a^{2} c^{3} d e^{2} - 5 \, B a^{3} c^{2} e^{3}\right )} x^{3} + 4 \,{\left (3 \, B a^{3} c^{2} d e^{2} + A a^{3} c^{2} e^{3}\right )} x^{2} - 3 \,{\left (A a^{2} c^{2} d^{3} + B a^{3} c d^{2} e + A a^{3} c d e^{2} + B a^{4} e^{3} +{\left (A c^{4} d^{3} + B a c^{3} d^{2} e + A a c^{3} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{4} + 2 \,{\left (A a c^{3} d^{3} + B a^{2} c^{2} d^{2} e + A a^{2} c^{2} d e^{2} + B a^{3} c e^{3}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, A a^{2} c^{3} d^{3} - 3 \, B a^{3} c^{2} d^{2} e - 3 \, A a^{3} c^{2} d e^{2} - 3 \, B a^{4} c e^{3}\right )} x}{8 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.6199, size = 466, normalized size = 3.73 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log{\left (- \frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right ) \log{\left (\frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (A c d + B a e\right )}{3 A a c d e^{2} + 3 A c^{2} d^{3} + 3 B a^{2} e^{3} + 3 B a c d^{2} e} + x \right )}}{16} - \frac{2 A a^{3} e^{3} + 6 A a^{2} c d^{2} e + 6 B a^{3} d e^{2} + 2 B a^{2} c d^{3} + x^{3} \left (- 3 A a c^{2} d e^{2} - 3 A c^{3} d^{3} + 5 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e\right ) + x^{2} \left (4 A a^{2} c e^{3} + 12 B a^{2} c d e^{2}\right ) + x \left (3 A a^{2} c d e^{2} - 5 A a c^{2} d^{3} + 3 B a^{3} e^{3} + 3 B a^{2} c d^{2} e\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17474, size = 315, normalized size = 2.52 \begin{align*} \frac{3 \,{\left (A c^{2} d^{3} + B a c d^{2} e + A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{3 \, A c^{3} d^{3} x^{3} + 3 \, B a c^{2} d^{2} x^{3} e + 3 \, A a c^{2} d x^{3} e^{2} + 5 \, A a c^{2} d^{3} x - 5 \, B a^{2} c x^{3} e^{3} - 12 \, B a^{2} c d x^{2} e^{2} - 3 \, B a^{2} c d^{2} x e - 2 \, B a^{2} c d^{3} - 4 \, A a^{2} c x^{2} e^{3} - 3 \, A a^{2} c d x e^{2} - 6 \, A a^{2} c d^{2} e - 3 \, B a^{3} x e^{3} - 6 \, B a^{3} d e^{2} - 2 \, A a^{3} e^{3}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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