Optimal. Leaf size=165 \[ -\frac{2 a e (a C+2 A c)-c x (a B e+a C d+5 A c d)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+5 A c d)}{16 a^{7/2} c^{3/2}}+\frac{x (a B e+a C d+5 A c d)}{16 a^3 c \left (a+c x^2\right )}-\frac{(d+e x) (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.135809, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1645, 639, 199, 205} \[ -\frac{2 a e (a C+2 A c)-c x (a B e+a C d+5 A c d)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+5 A c d)}{16 a^{7/2} c^{3/2}}+\frac{x (a B e+a C d+5 A c d)}{16 a^3 c \left (a+c x^2\right )}-\frac{(d+e x) (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 1645
Rule 639
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{\int \frac{-5 A c d-a (C d+B e)-2 (2 A c+a C) e x}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{(5 A c d+a C d+a B e) \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{(5 A c d+a C d+a B e) x}{16 a^3 c \left (a+c x^2\right )}+\frac{(5 A c d+a C d+a B e) \int \frac{1}{a+c x^2} \, dx}{16 a^3 c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac{2 a (2 A c+a C) e-c (5 A c d+a C d+a B e) x}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{(5 A c d+a C d+a B e) x}{16 a^3 c \left (a+c x^2\right )}+\frac{(5 A c d+a C d+a B e) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.161578, size = 171, normalized size = 1.04 \[ \frac{\frac{8 a^{5/2} \left (a^2 C e-a c (A e+B (d+e x)+C d x)+A c^2 d x\right )}{\left (a+c x^2\right )^3}+\frac{2 a^{3/2} \left (-6 a^2 C e+a c x (B e+C d)+5 A c^2 d x\right )}{\left (a+c x^2\right )^2}+\frac{3 \sqrt{a} c x (a B e+a C d+5 A c d)}{a+c x^2}+3 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+a C d+5 A c d)}{48 a^{7/2} c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 182, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ( 5\,Acd+aBe+Cad \right ) c{x}^{5}}{16\,{a}^{3}}}+{\frac{ \left ( 5\,Acd+aBe+Cad \right ){x}^{3}}{6\,{a}^{2}}}-{\frac{Ce{x}^{2}}{4\,c}}+{\frac{ \left ( 11\,Acd-aBe-Cad \right ) x}{16\,ac}}-{\frac{2\,Ace+2\,Bcd+aCe}{12\,{c}^{2}}} \right ) }+{\frac{5\,Ad}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Be}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Cd}{16\,{a}^{2}c}\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 = 1.4554, size = 1341, normalized size = 8.13 \begin{align*} \left [-\frac{24 \, C a^{4} c e x^{2} + 16 \, B a^{4} c d - 6 \,{\left (B a^{2} c^{3} e +{\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d\right )} x^{5} - 16 \,{\left (B a^{3} c^{2} e +{\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d\right )} x^{3} + 3 \,{\left ({\left (B a c^{3} e +{\left (C a c^{3} + 5 \, A c^{4}\right )} d\right )} x^{6} + B a^{4} e + 3 \,{\left (B a^{2} c^{2} e +{\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d\right )} x^{4} + 3 \,{\left (B a^{3} c e +{\left (C a^{3} c + 5 \, A a^{2} c^{2}\right )} d\right )} x^{2} +{\left (C a^{4} + 5 \, A a^{3} c\right )} d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 8 \,{\left (C a^{5} + 2 \, A a^{4} c\right )} e + 6 \,{\left (B a^{4} c e +{\left (C a^{4} c - 11 \, A a^{3} c^{2}\right )} d\right )} x}{96 \,{\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac{12 \, C a^{4} c e x^{2} + 8 \, B a^{4} c d - 3 \,{\left (B a^{2} c^{3} e +{\left (C a^{2} c^{3} + 5 \, A a c^{4}\right )} d\right )} x^{5} - 8 \,{\left (B a^{3} c^{2} e +{\left (C a^{3} c^{2} + 5 \, A a^{2} c^{3}\right )} d\right )} x^{3} - 3 \,{\left ({\left (B a c^{3} e +{\left (C a c^{3} + 5 \, A c^{4}\right )} d\right )} x^{6} + B a^{4} e + 3 \,{\left (B a^{2} c^{2} e +{\left (C a^{2} c^{2} + 5 \, A a c^{3}\right )} d\right )} x^{4} + 3 \,{\left (B a^{3} c e +{\left (C a^{3} c + 5 \, A a^{2} c^{2}\right )} d\right )} x^{2} +{\left (C a^{4} + 5 \, A a^{3} c\right )} d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 4 \,{\left (C a^{5} + 2 \, A a^{4} c\right )} e + 3 \,{\left (B a^{4} c e +{\left (C a^{4} c - 11 \, A a^{3} c^{2}\right )} d\right )} x}{48 \,{\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 127.102, size = 298, normalized size = 1.81 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (5 A c d + B a e + C a d\right ) \log{\left (- a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (5 A c d + B a e + C a d\right ) \log{\left (a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{- 8 A a^{3} c e - 8 B a^{3} c d - 4 C a^{4} e - 12 C a^{3} c e x^{2} + x^{5} \left (15 A c^{4} d + 3 B a c^{3} e + 3 C a c^{3} d\right ) + x^{3} \left (40 A a c^{3} d + 8 B a^{2} c^{2} e + 8 C a^{2} c^{2} d\right ) + x \left (33 A a^{2} c^{2} d - 3 B a^{3} c e - 3 C a^{3} c d\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15865, size = 262, normalized size = 1.59 \begin{align*} \frac{{\left (C a d + 5 \, A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c} + \frac{3 \, C a c^{3} d x^{5} + 15 \, A c^{4} d x^{5} + 3 \, B a c^{3} x^{5} e + 8 \, C a^{2} c^{2} d x^{3} + 40 \, A a c^{3} d x^{3} + 8 \, B a^{2} c^{2} x^{3} e - 12 \, C a^{3} c x^{2} e - 3 \, C a^{3} c d x + 33 \, A a^{2} c^{2} d x - 3 \, B a^{3} c x e - 8 \, B a^{3} c d - 4 \, C a^{4} e - 8 \, A a^{3} c e}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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