Optimal. Leaf size=430 \[ -\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (47 a^2 c d^2 e^4-35 a^3 e^6+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}-\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5} \]
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Rubi [A] time = 0.533842, antiderivative size = 430, 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} \[ -\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (47 a^2 c d^2 e^4-35 a^3 e^6+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}-\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5} \]
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 )^4} \, dx &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{\int \frac{-5 c d^2-7 a e^2-6 c d e x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}+\frac{\int \frac{c \left (15 c^2 d^4+34 a c d^2 e^2+35 a^2 e^4\right )+4 c^2 d e \left (5 c d^2+13 a e^2\right ) x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac{\int \frac{-3 c^2 \left (5 c^3 d^6+13 a c^2 d^4 e^2+11 a^2 c d^2 e^4+35 a^3 e^6\right )-6 c^3 d e \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{(d+e x)^2 \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 ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac{\int \left (\frac{3 c^2 e^2 \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}-\frac{384 a^3 c^3 d e^8}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{3 c^3 \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}+\frac{c \int \frac{5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{\left (8 c^2 d e^7\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^5}+\frac{\left (c \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{\sqrt{c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^5}\\ \end{align*}
Mathematica [A] time = 0.43234, size = 336, normalized size = 0.78 \[ \frac{\frac{3 c \left (a e^2+c d^2\right ) \left (47 a^2 c d^2 e^4 x+a^3 e^5 (48 d-19 e x)+23 a c^2 d^4 e^2 x+5 c^3 d^6 x\right )}{a^3 \left (a+c x^2\right )}+\frac{2 c \left (a e^2+c d^2\right )^2 \left (a^2 e^3 (24 d-11 e x)+18 a c d^2 e^2 x+5 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 \sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{8 c \left (a e^2+c d^2\right )^3 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^3}-\frac{48 e^7 \left (a e^2+c d^2\right )}{d+e x}-192 c d e^7 \log \left (a+c x^2\right )+384 c d e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 1126, normalized size = 2.6 \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 [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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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 [B] time = 1.35452, size = 1156, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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