Optimal. Leaf size=680 \[ -\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3} \]
[Out]
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Rubi [A]
time = 0.67, antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6857, 1890,
1182, 1176, 631, 210, 1179, 642, 1262, 649, 211, 266} \begin {gather*} -\frac {\sqrt {c} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}-\frac {4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 211
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1890
Rule 6857
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx &=\int \left (\frac {e^4}{\left (c d^4+a e^4\right ) (d+e x)^3}+\frac {4 c d^3 e^4}{\left (c d^4+a e^4\right )^2 (d+e x)^2}+\frac {2 c d^2 e^4 \left (5 c d^4-3 a e^4\right )}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \left (\frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \text {Subst}\left (\int \frac {-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c^2 d^2 e^3 \left (5 c d^4-3 a e^4\right )\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 738, normalized size = 1.09 \begin {gather*} \frac {-4 a^{3/4} e^3 \left (c d^4+a e^4\right )^2-32 a^{3/4} c d^3 e^3 \left (c d^4+a e^4\right ) (d+e x)-2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9-6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4+24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8-2 a^{9/4} e^9\right ) (d+e x)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9+6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4-24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8+2 a^{9/4} e^9\right ) (d+e x)^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+16 a^{3/4} c d^2 e^3 \left (5 c d^4-3 a e^4\right ) (d+e x)^2 \log (d+e x)-\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+4 a^{3/4} c d^2 e^3 \left (-5 c d^4+3 a e^4\right ) (d+e x)^2 \log \left (a+c x^4\right )}{8 a^{3/4} \left (c d^4+a e^4\right )^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.27, size = 446, normalized size = 0.66
method | result | size |
default | \(-\frac {e^{3}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {4 c \,d^{3} e^{3}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}-\frac {2 d^{2} e^{3} c \left (3 e^{4} a -5 d^{4} c \right ) \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{3}}+\frac {c \left (\frac {\left (3 a^{2} d \,e^{8}-12 a c \,d^{5} e^{4}+c^{2} d^{9}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a^{2} e^{9}+12 a c \,d^{4} e^{5}-3 c^{2} e \,d^{8}\right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (-10 a c \,d^{3} e^{6}+6 c^{2} d^{7} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+\frac {\left (6 a c \,d^{2} e^{7}-10 e^{3} c^{2} d^{6}\right ) \ln \left (c \,x^{4}+a \right )}{4 c}\right )}{\left (e^{4} a +d^{4} c \right )^{3}}\) | \(446\) |
risch | \(\frac {-\frac {4 d^{3} e^{4} c x}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}-\frac {\left (e^{4} a +9 d^{4} c \right ) e^{3}}{2 \left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}\right )}}{\left (e x +d \right )^{2}}-\frac {6 d^{2} e^{7} c \ln \left (e x +d \right ) a}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {10 d^{6} e^{3} c^{2} \ln \left (e x +d \right )}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{6} e^{12}+3 a^{5} c \,d^{4} e^{8}+3 a^{4} c^{2} d^{8} e^{4}+a^{3} c^{3} d^{12}\right ) \textit {\_Z}^{4}+\left (-24 e^{7} d^{2} c \,a^{4}+40 a^{3} c^{2} d^{6} e^{3}\right ) \textit {\_Z}^{3}+\left (2 a^{3} c \,e^{6}+42 a^{2} c^{2} d^{4} e^{2}\right ) \textit {\_Z}^{2}+12 a \,c^{2} d^{2} e \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{7} e^{22}+17 a^{6} c \,d^{4} e^{18}+18 a^{5} c^{2} d^{8} e^{14}+2 a^{4} c^{3} d^{12} e^{10}-7 a^{3} c^{4} d^{16} e^{6}-3 a^{2} c^{5} d^{20} e^{2}\right ) \textit {\_R}^{4}+\left (-81 a^{5} c \,d^{2} e^{17}-56 a^{4} c^{2} d^{6} e^{13}+126 a^{3} c^{3} d^{10} e^{9}+96 a^{2} c^{4} d^{14} e^{5}-5 a \,c^{5} d^{18} e \right ) \textit {\_R}^{3}+\left (9 a^{4} c \,e^{16}-38 a^{3} c^{2} d^{4} e^{12}+408 a^{2} c^{3} d^{8} e^{8}+198 a \,c^{4} d^{12} e^{4}-c^{5} d^{16}\right ) \textit {\_R}^{2}+\left (72 a^{2} c^{2} d^{2} e^{11}-16 a \,c^{3} d^{6} e^{7}+40 c^{4} d^{10} e^{3}\right ) \textit {\_R} +4 a \,c^{2} e^{10}+4 c^{3} d^{4} e^{6}\right ) x +\left (6 a^{7} d \,e^{21}+22 a^{6} c \,d^{5} e^{17}+28 a^{5} c^{2} d^{9} e^{13}+12 a^{4} c^{3} d^{13} e^{9}-2 a^{3} c^{4} d^{17} e^{5}-2 a^{2} c^{5} d^{21} e \right ) \textit {\_R}^{4}+\left (-61 a^{5} c \,d^{3} e^{16}-56 a^{4} c^{2} d^{7} e^{12}+70 a^{3} c^{3} d^{11} e^{8}+64 a^{2} c^{4} d^{15} e^{4}-a \,c^{5} d^{19}\right ) \textit {\_R}^{3}+\left (4 a^{4} c d \,e^{15}-148 a^{3} c^{2} d^{5} e^{11}+652 a^{2} c^{3} d^{9} e^{7}+36 a \,c^{4} d^{13} e^{3}\right ) \textit {\_R}^{2}+\left (-48 a^{2} c^{2} d^{3} e^{10}+208 a \,c^{3} d^{7} e^{6}\right ) \textit {\_R} +4 a \,c^{2} d \,e^{9}+4 c^{3} d^{5} e^{5}\right )\right )}{4}\) | \(867\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 779, normalized size = 1.15 \begin {gather*} \frac {c {\left (\frac {\sqrt {2} {\left (c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} - 10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} + 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {\sqrt {2} {\left (c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} + 10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {a} c^{3} d^{8} e + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} + 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} - 6 \, \sqrt {a} c^{3} d^{8} e + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} + 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} - 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left (x e + d\right )}{c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}} - \frac {8 \, c d^{3} x e^{4} + 9 \, c d^{4} e^{3} + a e^{7}}{2 \, {\left (c^{2} d^{10} + 2 \, a c d^{6} e^{4} + a^{2} d^{2} e^{8} + {\left (c^{2} d^{8} e^{2} + 2 \, a c d^{4} e^{6} + a^{2} e^{10}\right )} x^{2} + 2 \, {\left (c^{2} d^{9} e + 2 \, a c d^{5} e^{5} + a^{2} d e^{9}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.03, size = 901, normalized size = 1.32 \begin {gather*} \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 2 \, a c^{2} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (a c^{3} d^{6} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e + 9 \, \sqrt {a c} a c^{2} d^{4} e^{2} + 9 \, a^{2} c^{2} d^{2} e^{4} - 8 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{5} + \sqrt {a c} a^{2} c e^{6}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - 2 \, a c^{2} e^{3} - 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, {\left (a c^{3} d^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e - 9 \, \sqrt {a c} a c^{2} d^{4} e^{2} + 9 \, a^{2} c^{2} d^{2} e^{4} + 8 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{3} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{5} + \sqrt {a c} a^{2} c e^{6}\right )}} + \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{9} - 6 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{7} e^{2} - 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e^{4} + 10 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{8}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{12} + 3 \, a^{2} c^{3} d^{8} e^{4} + 3 \, a^{3} c^{2} d^{4} e^{8} + a^{4} c e^{12}\right )}} - \frac {{\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{9} - 6 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{7} e^{2} - 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{5} e^{4} + 10 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d^{3} e^{6} + 3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c d e^{8}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, {\left (a c^{4} d^{12} + 3 \, a^{2} c^{3} d^{8} e^{4} + 3 \, a^{3} c^{2} d^{4} e^{8} + a^{4} c e^{12}\right )}} - \frac {{\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left ({\left | c x^{4} + a \right |}\right )}{2 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{4} - 3 \, a c d^{2} e^{8}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{12} e + 3 \, a c^{2} d^{8} e^{5} + 3 \, a^{2} c d^{4} e^{9} + a^{3} e^{13}} - \frac {9 \, c^{2} d^{8} e^{3} + 10 \, a c d^{4} e^{7} + a^{2} e^{11} + 8 \, {\left (c^{2} d^{7} e^{4} + a c d^{3} e^{8}\right )} x}{2 \, {\left (c d^{4} + a e^{4}\right )}^{3} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.67, size = 1955, normalized size = 2.88 \begin {gather*} \left (\sum _{k=1}^4\ln \left (\frac {c^7\,d^5\,e^6+a\,c^6\,d\,e^{10}}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {208\,a\,c^7\,d^7\,e^7-48\,a^2\,c^6\,d^3\,e^{11}}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (-\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {976\,a^5\,c^5\,d^3\,e^{17}+896\,a^4\,c^6\,d^7\,e^{13}-1120\,a^3\,c^7\,d^{11}\,e^9-1024\,a^2\,c^8\,d^{15}\,e^5+16\,a\,c^9\,d^{19}\,e}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}-\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\,\left (\frac {384\,a^7\,c^4\,d\,e^{22}+1408\,a^6\,c^5\,d^5\,e^{18}+1792\,a^5\,c^6\,d^9\,e^{14}+768\,a^4\,c^7\,d^{13}\,e^{10}-128\,a^3\,c^8\,d^{17}\,e^6-128\,a^2\,c^9\,d^{21}\,e^2}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\frac {x\,\left (320\,a^7\,c^4\,e^{23}+1088\,a^6\,c^5\,d^4\,e^{19}+1152\,a^5\,c^6\,d^8\,e^{15}+128\,a^4\,c^7\,d^{12}\,e^{11}-448\,a^3\,c^8\,d^{16}\,e^7-192\,a^2\,c^9\,d^{20}\,e^3\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (1296\,a^5\,c^5\,d^2\,e^{18}+896\,a^4\,c^6\,d^6\,e^{14}-2016\,a^3\,c^7\,d^{10}\,e^{10}-1536\,a^2\,c^8\,d^{14}\,e^6+80\,a\,c^9\,d^{18}\,e^2\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {16\,a^4\,c^5\,d\,e^{16}-592\,a^3\,c^6\,d^5\,e^{12}+2608\,a^2\,c^7\,d^9\,e^8+144\,a\,c^8\,d^{13}\,e^4}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}+\frac {x\,\left (36\,a^4\,c^5\,e^{17}-152\,a^3\,c^6\,d^4\,e^{13}+1632\,a^2\,c^7\,d^8\,e^9+792\,a\,c^8\,d^{12}\,e^5-4\,c^9\,d^{16}\,e\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (72\,a^2\,c^6\,d^2\,e^{12}-16\,a\,c^7\,d^6\,e^8+40\,c^8\,d^{10}\,e^4\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )+\frac {x\,\left (c^7\,d^4\,e^7+a\,c^6\,e^{11}\right )}{a^4\,e^{16}+4\,a^3\,c\,d^4\,e^{12}+6\,a^2\,c^2\,d^8\,e^8+4\,a\,c^3\,d^{12}\,e^4+c^4\,d^{16}}\right )\,\mathrm {root}\left (768\,a^5\,c\,d^4\,e^8\,z^4+768\,a^4\,c^2\,d^8\,e^4\,z^4+256\,a^3\,c^3\,d^{12}\,z^4+256\,a^6\,e^{12}\,z^4-1536\,a^4\,c\,d^2\,e^7\,z^3+2560\,a^3\,c^2\,d^6\,e^3\,z^3+672\,a^2\,c^2\,d^4\,e^2\,z^2+32\,a^3\,c\,e^6\,z^2+48\,a\,c^2\,d^2\,e\,z+c^2,z,k\right )\right )-\frac {\frac {9\,c\,d^4\,e^3+a\,e^7}{2\,\left (a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8\right )}+\frac {4\,c\,d^3\,e^4\,x}{a^2\,e^8+2\,a\,c\,d^4\,e^4+c^2\,d^8}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (10\,c^2\,d^6\,e^3-6\,a\,c\,d^2\,e^7\right )}{a^3\,e^{12}+3\,a^2\,c\,d^4\,e^8+3\,a\,c^2\,d^8\,e^4+c^3\,d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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