Optimal. Leaf size=212 \[ \frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac{4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]
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Rubi [A] time = 0.220795, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {741, 823, 801, 635, 205, 260} \[ \frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac{4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 741
Rule 823
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+c x^2\right )^3} \, dx &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac{\int \frac{-3 c d^2-4 a e^2-3 c d e x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\int \frac{c \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )+c^2 d e \left (3 c d^2+7 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\int \left (\frac{8 a^2 c e^6}{\left (c d^2+a e^2\right ) (d+e x)}+\frac{c^2 \left (3 c^2 d^5+10 a c d^3 e^2+15 a^2 d e^4-8 a^2 e^5 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{c \int \frac{3 c^2 d^5+10 a c d^3 e^2+15 a^2 d e^4-8 a^2 e^5 x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac{\left (c e^5\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{\left (c 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 \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\sqrt{c} 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} \left (c d^2+a e^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac{e^5 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.146377, size = 180, normalized size = 0.85 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (a e+c d x)}{a \left (a+c x^2\right )^2}-4 e^5 \log \left (a+c x^2\right )+8 e^5 \log (d+e x)}{8 \left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 532, normalized size = 2.5 \begin{align*}{\frac{7\,{c}^{2}d{x}^{3}{e}^{4}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{c}^{3}{d}^{3}{x}^{3}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,{c}^{4}{d}^{5}{x}^{3}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{c{x}^{2}a{e}^{5}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{{c}^{2}{x}^{2}{d}^{2}{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{9\,acdx{e}^{4}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{c}^{2}{d}^{3}x{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{c}^{3}{d}^{5}x}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,{a}^{2}{e}^{5}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{ac{d}^{2}{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{{c}^{2}{d}^{4}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}-{\frac{{e}^{5}\ln \left ( c{x}^{2}+a \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{15\,d{e}^{4}c}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{3}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{c}^{3}{d}^{5}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{5}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}} \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 = 24.1499, size = 2045, normalized size = 9.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.37224, size = 462, normalized size = 2.18 \begin{align*} -\frac{e^{5} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{e^{6} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \,{\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt{a c}} + \frac{2 \, a^{2} c^{2} d^{4} e + 8 \, a^{3} c d^{2} e^{3} + 6 \, a^{4} e^{5} +{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 4 \,{\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} +{\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x}{8 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (c x^{2} + a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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