Optimal. Leaf size=1384 \[ \frac {12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac {3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {\sqrt {c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.96, antiderivative size = 1384, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac {12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac {3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {\sqrt {c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 205
Rule 260
Rule 275
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1854
Rule 1876
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac {e^8}{\left (c d^4+a e^4\right )^2 (d+e x)^3}+\frac {8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)^2}+\frac {12 c d^2 e^8 \left (3 c d^4-a e^4\right )}{\left (c d^4+a e^4\right )^4 (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 )^2}+\frac {c e^4 \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\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}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \left (\frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 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}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \left (\frac {2 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x}{a+c x^4}+\frac {-3 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}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \operatorname {Subst}\left (\int \frac {-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 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}{4 a \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 ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )^3}-\frac {\left (\sqrt {c} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac {\left (6 c^2 d^2 e^7 \left (3 c d^4-a e^4\right )\right ) \operatorname {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^4}-\frac {\left (c e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )^3}+\frac {\left (c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\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} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\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} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 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 {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 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 )^4}-\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 )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac {\left (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {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 )^4}-\frac {\left (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {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 )^4}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\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}{16 \sqrt {2} a^{5/4} \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 {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\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}{16 \sqrt {2} a^{5/4} \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 {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\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}{16 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 {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\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}{16 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 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 )^4}-\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 )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}+\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \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 {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \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\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 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 )^4}-\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 )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\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 )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\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 )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}\\ \end {align*}
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Mathematica [A] time = 1.36, size = 996, normalized size = 0.72 \[ \frac {384 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7-96 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7-\frac {256 c d^3 \left (c d^4+a e^4\right ) e^7}{d+e x}-\frac {16 \left (c d^4+a e^4\right )^2 e^7}{(d+e x)^2}+\frac {8 c \left (c d^4+a e^4\right ) \left (c^2 x \left (d^2-3 e x d+6 e^2 x^2\right ) d^7+2 a c e^3 \left (5 d^3-6 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right ) d^3-a^2 e^7 \left (6 d^2-3 e x d+e^2 x^2\right )\right )}{a \left (c x^4+a\right )}-\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}-4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt {2} \sqrt {a} c^{11/4} e^2 d^{11}+9 \sqrt {2} a c^{9/4} e^4 d^9-44 a^{5/4} c^2 e^5 d^8+36 \sqrt {2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt {2} a^2 c^{5/4} e^8 d^5+84 a^{9/4} c e^9 d^4-30 \sqrt {2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt {2} a^3 \sqrt [4]{c} e^{12} d-4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}+4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt {2} \sqrt {a} c^{11/4} e^2 d^{11}+9 \sqrt {2} a c^{9/4} e^4 d^9+44 a^{5/4} c^2 e^5 d^8+36 \sqrt {2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt {2} a^2 c^{5/4} e^8 d^5-84 a^{9/4} c e^9 d^4-30 \sqrt {2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt {2} a^3 \sqrt [4]{c} e^{12} d+4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt {c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{a^{7/4}}+\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt {c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^4} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 1488, normalized size = 1.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 2121, normalized size = 1.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.60, size = 1394, normalized size = 1.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.04, size = 3256, normalized size = 2.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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