Optimal. Leaf size=198 \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{d e^2 x \left (7 a e^2+3 c d^2\right )}{8 a^2 c^2}+\frac{d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left (a+c x^2\right )}{2 c^3}-\frac{(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.185442, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {739, 819, 774, 635, 205, 260} \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{d e^2 x \left (7 a e^2+3 c d^2\right )}{8 a^2 c^2}+\frac{d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left (a+c x^2\right )}{2 c^3}-\frac{(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 739
Rule 819
Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}+\frac{\int \frac{(d+e x)^3 \left (3 c d^2+4 a e^2-c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\int \frac{(d+e x) \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4-c d e \left (3 c d^2+7 a e^2\right ) x\right )}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac{(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{\int \frac{a c d e^2 \left (3 c d^2+7 a e^2\right )+c d \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )+c \left (-c d^2 e \left (3 c d^2+7 a e^2\right )+e \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right ) x}{a+c x^2} \, dx}{8 a^2 c^3}\\ &=-\frac{d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac{(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{e^5 \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac{d e^2 \left (3 c d^2+7 a e^2\right ) x}{8 a^2 c^2}-\frac{(a e-c d x) (d+e x)^4}{4 a c \left (a+c x^2\right )^2}-\frac{(d+e x)^2 \left (2 a e \left (c d^2+2 a e^2\right )-c d \left (3 c d^2+5 a e^2\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.152277, size = 199, normalized size = 1.01 \[ \frac{-\frac{2 \left (-5 a^2 c d e^3 (2 d+e x)+a^3 e^5+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )}{a \left (a+c x^2\right )^2}+\frac{-5 a^2 c d e^3 (8 d+5 e x)+8 a^3 e^5+10 a c^2 d^3 e^2 x+3 c^3 d^5 x}{a^2 \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+4 e^5 \log \left (a+c x^2\right )}{8 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 233, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{d \left ( 25\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}-3\,{c}^{2}{d}^{4} \right ){x}^{3}}{8\,{a}^{2}c}}+{\frac{{e}^{3} \left ( a{e}^{2}-5\,c{d}^{2} \right ){x}^{2}}{{c}^{2}}}-{\frac{5\,d \left ( 3\,{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) x}{8\,a{c}^{2}}}+{\frac{e \left ( 3\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ) }{4\,{c}^{3}}} \right ) }+{\frac{{e}^{5}\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{3}}}+{\frac{15\,d{e}^{4}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{3}{e}^{2}}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{5}}{8\,{a}^{2}}\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 [A] time = 2.09844, size = 1443, normalized size = 7.29 \begin{align*} \left [-\frac{20 \, a^{3} c^{2} d^{4} e + 40 \, a^{4} c d^{2} e^{3} - 12 \, a^{5} e^{5} - 2 \,{\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 16 \,{\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} +{\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} +{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \,{\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 10 \,{\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 8 \,{\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{16 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac{10 \, a^{3} c^{2} d^{4} e + 20 \, a^{4} c d^{2} e^{3} - 6 \, a^{5} e^{5} -{\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} - 25 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 8 \,{\left (5 \, a^{3} c^{2} d^{2} e^{3} - a^{4} c e^{5}\right )} x^{2} -{\left (3 \, a^{2} c^{2} d^{5} + 10 \, a^{3} c d^{3} e^{2} + 15 \, a^{4} d e^{4} +{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 15 \, a^{2} c^{2} d e^{4}\right )} x^{4} + 2 \,{\left (3 \, a c^{3} d^{5} + 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) - 5 \,{\left (a^{2} c^{3} d^{5} - 2 \, a^{3} c^{2} d^{3} e^{2} - 3 \, a^{4} c d e^{4}\right )} x - 4 \,{\left (a^{3} c^{2} e^{5} x^{4} + 2 \, a^{4} c e^{5} x^{2} + a^{5} e^{5}\right )} \log \left (c x^{2} + a\right )}{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 = 4.43216, size = 520, normalized size = 2.63 \begin{align*} \left (\frac{e^{5}}{2 c^{3}} - \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log{\left (x + \frac{16 a^{3} c^{3} \left (\frac{e^{5}}{2 c^{3}} - \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \left (\frac{e^{5}}{2 c^{3}} + \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log{\left (x + \frac{16 a^{3} c^{3} \left (\frac{e^{5}}{2 c^{3}} + \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} - \frac{- 6 a^{4} e^{5} + 20 a^{3} c d^{2} e^{3} + 10 a^{2} c^{2} d^{4} e + x^{3} \left (25 a^{2} c^{2} d e^{4} - 10 a c^{3} d^{3} e^{2} - 3 c^{4} d^{5}\right ) + x^{2} \left (- 8 a^{3} c e^{5} + 40 a^{2} c^{2} d^{2} e^{3}\right ) + x \left (15 a^{3} c d e^{4} + 10 a^{2} c^{2} d^{3} e^{2} - 5 a c^{3} d^{5}\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26101, size = 279, normalized size = 1.41 \begin{align*} \frac{e^{5} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 25 \, a^{2} c d e^{4}\right )} x^{3} - 8 \,{\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2} + 5 \,{\left (a c^{2} d^{5} - 2 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4}\right )} x - \frac{2 \,{\left (5 \, a^{2} c^{2} d^{4} e + 10 \, a^{3} c d^{2} e^{3} - 3 \, a^{4} e^{5}\right )}}{c}}{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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