Optimal. Leaf size=295 \[ \frac{c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+8 a^3 e^5}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} d \left (35 a^2 c d^2 e^4+35 a^3 e^6+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac{6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac{e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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Rubi [A] time = 0.38205, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+8 a^3 e^5}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} d \left (35 a^2 c d^2 e^4+35 a^3 e^6+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac{6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac{e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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) \left (a+c x^2\right )^4} \, dx &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}-\frac{\int \frac{-5 c d^2-6 a e^2-5 c d e x}{(d+e x) \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{\int \frac{3 c \left (5 c^2 d^4+11 a c d^2 e^2+8 a^2 e^4\right )+3 c^2 d e \left (5 c d^2+11 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac{\int \frac{-3 c^2 \left (5 c^3 d^6+16 a c^2 d^4 e^2+19 a^2 c d^2 e^4+16 a^3 e^6\right )-3 c^3 d e \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac{\int \left (-\frac{48 a^3 c^2 e^8}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{3 c^3 \left (5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac{e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac{c \int \frac{5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac{e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac{\left (c e^7\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac{\left (c d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac{6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac{8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac{e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 0.224831, size = 265, normalized size = 0.9 \[ \frac{\frac{3 \left (a e^2+c d^2\right ) \left (19 a^2 c d e^4 x+8 a^3 e^5+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right )}{a^3 \left (a+c x^2\right )}+\frac{2 \left (a e^2+c d^2\right )^2 \left (6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 \sqrt{c} d \left (35 a^2 c d^2 e^4+35 a^3 e^6+21 a c^2 d^4 e^2+5 c^3 d^6\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{8 \left (a e^2+c d^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}-24 e^7 \log \left (a+c x^2\right )+48 e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 941, normalized size = 3.2 \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 = 97.5941, size = 3580, normalized size = 12.14 \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 = 1.37122, size = 716, normalized size = 2.43 \begin{align*} -\frac{e^{7} \log \left (c x^{2} + a\right )}{2 \,{\left (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}\right )}} + \frac{e^{8} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac{{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \,{\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt{a c}} + \frac{8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \,{\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \,{\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \,{\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \,{\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \,{\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \,{\left (c d^{2} + a e^{2}\right )}^{4}{\left (c x^{2} + a\right )}^{3} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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