Optimal. Leaf size=771 \[ \frac{-2 c x (2 c d-b e) \left (c^2 e^2 \left (38 a^2 e^2-32 a b d e+7 b^2 d^2\right )+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right )+2 b^2 c^2 e \left (43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right )-4 b c^3 d \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )-64 a^3 c^3 e^5-2 b^3 c^2 d e^2 \left (5 a e^2+17 c d^2\right )+b^4 c e^3 \left (c d^2-23 a e^2\right )+b^5 c d e^4+2 b^6 e^5}{2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (6 a^2 b d e^2-4 a^3 e^3-4 a b^2 d^2 e+b^3 d^3\right )+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 7.29987, antiderivative size = 771, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {740, 822, 800, 634, 618, 206, 628} \[ \frac{-2 c x (2 c d-b e) \left (c^2 e^2 \left (38 a^2 e^2-32 a b d e+7 b^2 d^2\right )+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right )+2 b^2 c^2 e \left (43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right )-4 b c^3 d \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )-64 a^3 c^3 e^5-2 b^3 c^2 d e^2 \left (5 a e^2+17 c d^2\right )+b^4 c e^3 \left (c d^2-23 a e^2\right )+b^5 c d e^4+2 b^6 e^5}{2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (6 a^2 b d e^2-4 a^3 e^3-4 a b^2 d^2 e+b^3 d^3\right )+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 740
Rule 822
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{\int \frac{10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)+5 c e (2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\int \frac{3 \left (20 c^4 d^4+2 b^4 e^4-2 c^3 d^2 e (15 b d-22 a e)+b^2 c e^3 (3 b d-16 a e)+2 c^2 e^2 \left (2 b^2 d^2-11 a b d e+16 a^2 e^2\right )\right )+3 c e (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}-\frac{\int \frac{6 \left (20 c^6 d^6-b^6 e^6-2 c^5 d^4 e (25 b d-32 a e)-b^4 c e^5 (b d-12 a e)+2 c^4 d^2 e^2 \left (17 b^2 d^2-48 a b d e+38 a^2 e^2\right )-b^2 c^2 e^4 \left (b^2 d^2-11 a b d e+48 a^2 e^2\right )-c^3 e^3 \left (b^3 d^3-10 a b^2 d^2 e+38 a^2 b d e^2-64 a^3 e^3\right )\right )+6 c e (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{6 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}-\frac{\int \left (-\frac{6 \left (b^2-4 a c\right )^3 e^8}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{6 \left (20 c^7 d^7+b^7 e^7-13 a b^5 c e^7+59 a^2 b^3 c^2 e^7-102 a^3 b c^3 e^7-14 c^6 d^5 e (5 b d-6 a e)+14 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-35 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )+c \left (b^2-4 a c\right )^3 e^7 x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{6 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac{e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{\int \frac{20 c^7 d^7+b^7 e^7-13 a b^5 c e^7+59 a^2 b^3 c^2 e^7-102 a^3 b c^3 e^7-14 c^6 d^5 e (5 b d-6 a e)+14 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-35 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )+c \left (b^2-4 a c\right )^3 e^7 x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac{e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^7 \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^4}-\frac{\left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac{e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (10 c^2 d^2-3 b^2 e^2-c e (5 b d-12 a e)\right )-c (2 c d-b e) \left (10 c^2 d^2-3 b^2 e^2-2 c e (5 b d-11 a e)\right ) x}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )^2}+\frac{b^5 c d e^4+2 b^6 e^5-64 a^3 c^3 e^5+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (17 c d^2+5 a e^2\right )-4 b c^3 d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right )+2 b^2 c^2 e \left (25 c^2 d^4+48 a c d^2 e^2+43 a^2 e^4\right )-2 c (2 c d-b e) \left (10 c^4 d^4+b^4 e^4+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+c^2 e^2 \left (7 b^2 d^2-32 a b d e+38 a^2 e^2\right )\right ) x}{2 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}+\frac{\left (40 c^7 d^7+b^7 e^7-14 a b^5 c e^7+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-28 c^6 d^5 e (5 b d-6 a e)+28 c^5 d^3 e^2 \left (6 b^2 d^2-15 a b d e+10 a^2 e^2\right )-70 c^4 d e^3 \left (b^3 d^3-4 a b^2 d^2 e+6 a^2 b d e^2-4 a^3 e^3\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (c d^2-b d e+a e^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}\\ \end{align*}
Mathematica [A] time = 3.6818, size = 769, normalized size = 1. \[ \frac{1}{6} \left (\frac{4 c^2 \left (6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right )+b^2 c e \left (c d (4 e x-15 d)-23 a e^2\right )+2 b c^2 \left (11 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )+b^3 c e^2 (2 d+3 e x)+3 b^4 e^3}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )^2}+\frac{3 \left (2 b^2 c^2 e \left (-43 a^2 e^4+2 a c d e^2 (5 e x-24 d)+c^2 d^3 (34 e x-25 d)\right )+4 b c^3 \left (19 a^2 e^4 (d-e x)+16 a c d^2 e^2 (d-3 e x)+5 c^2 d^4 (d-5 e x)\right )+8 c^3 \left (19 a^2 c d e^4 x+8 a^3 e^5+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right )+2 b^3 c^2 e^2 \left (a e^2 (5 d+11 e x)+c d^2 (17 d-e x)\right )-b^4 c e^3 \left (c d (d+2 e x)-23 a e^2\right )-b^5 c e^4 (d+2 e x)-2 b^6 e^5\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )^3}+\frac{6 \left (28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (6 a^2 b d e^2-4 a^3 e^3-4 a b^2 d^2 e+b^3 d^3\right )+70 a^2 b^3 c^2 e^7-140 a^3 b c^3 e^7-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2} \left (e (a e-b d)+c d^2\right )^4}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^3 \left (e (b d-a e)-c d^2\right )}+\frac{6 e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4}-\frac{3 e^7 \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^4}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.222, size = 14396, normalized size = 18.7 \begin{align*} \text{output 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 = 1.27588, size = 4361, normalized size = 5.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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