Optimal. Leaf size=300 \[ \frac{3 \sqrt{c} \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac{3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac{6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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Rubi [A] time = 0.356834, antiderivative size = 300, 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} \[ \frac{3 \sqrt{c} \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d x \left (3 a e^2+c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}+\frac{3 e \left (c d^2-a e^2\right ) \left (5 a e^2+c d^2\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^5 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^4}+\frac{6 c d e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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)^2 \left (a+c x^2\right )^3} \, dx &=\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{\int \frac{-3 c d^2-5 a e^2-4 c d e x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{\int \frac{3 c \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )+6 c^2 d e \left (c d^2+3 a e^2\right ) x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{\int \left (\frac{3 c e^2 \left (-c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac{48 a^2 c^2 d e^6}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac{3 c^2 \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6-16 a^2 c d e^5 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac{(3 c) \int \frac{c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6-16 a^2 c d e^5 x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=\frac{3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac{\left (6 c^2 d e^5\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac{\left (3 c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=\frac{3 e \left (c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac{a e+c d x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (c d^2-5 a e^2\right )-3 c d \left (c d^2+3 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac{3 \sqrt{c} \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^4}+\frac{6 c d e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac{3 c d e^5 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 0.388818, size = 241, normalized size = 0.8 \[ \frac{\frac{c \left (a e^2+c d^2\right ) \left (a^2 e^3 (16 d-7 e x)+12 a c d^2 e^2 x+3 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{3 \sqrt{c} \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 c \left (a e^2+c d^2\right )^2 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^2}-\frac{8 e^5 \left (a e^2+c d^2\right )}{d+e x}-24 c d e^5 \log \left (a+c x^2\right )+48 c d e^5 \log (d+e x)}{8 \left (a e^2+c d^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 680, normalized size = 2.3 \begin{align*} \text{result too large to display} \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 = 67.5698, size = 4594, normalized size = 15.31 \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 = 1.36292, size = 768, normalized size = 2.56 \begin{align*} -\frac{3 \, c d e^{5} \log \left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{3 \,{\left (c^{4} d^{6} e^{2} + 5 \, a c^{3} d^{4} e^{4} + 15 \, a^{2} c^{2} d^{2} e^{6} - 5 \, a^{3} c e^{8}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{8 \,{\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt{a c}} - \frac{e^{11}}{{\left (c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}\right )}{\left (x e + d\right )}} + \frac{3 \, c^{5} d^{5} e + 14 \, a c^{4} d^{3} e^{3} - 29 \, a^{2} c^{3} d e^{5} - \frac{{\left (9 \, c^{5} d^{6} e^{2} + 41 \, a c^{4} d^{4} e^{4} - 121 \, a^{2} c^{3} d^{2} e^{6} + 7 \, a^{3} c^{2} e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac{{\left (9 \, c^{5} d^{7} e^{3} + 45 \, a c^{4} d^{5} e^{5} - 145 \, a^{2} c^{3} d^{3} e^{7} - 21 \, a^{3} c^{2} d e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{3 \,{\left (c^{5} d^{8} e^{4} + 6 \, a c^{4} d^{6} e^{6} - 20 \, a^{2} c^{3} d^{4} e^{8} - 22 \, a^{3} c^{2} d^{2} e^{10} + 3 \, a^{4} c e^{12}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \,{\left (c d^{2} + a e^{2}\right )}^{4} a^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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