Optimal. Leaf size=485 \[ -\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.71, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {754, 814, 648,
632, 212, 642} \begin {gather*} \frac {(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac {e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac {e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {2 c^2 d^2-3 b^2 e^2+c e (b d+8 a e)+3 c e (2 c d-b e) x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx}{\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}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {e^2 \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {e^2 (2 c d-b e) \left (-c^2 d^2-3 b^2 e^2+c e (b d+11 a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (b^2-4 a c\right ) e^4 \left (-10 c^2 d^2-3 b^2 e^2+2 c e (5 b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\int \frac {2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\left (e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right )\right ) \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 ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}+\frac {\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right )\right ) \text {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 ) \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \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 = 0.69, size = 489, normalized size = 1.01 \begin {gather*} -\frac {e^3}{2 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^2}+\frac {2 e^3 (-2 c d+b e)}{\left (c d^2+e (-b d+a e)\right )^3 (d+e x)}+\frac {-b^4 e^3+b^3 c e^2 (3 d-e x)+b^2 c e \left (4 a e^2-3 c d (d-e x)\right )+2 c^2 \left (-a^2 e^3+c^2 d^3 x+3 a c d e (d-e x)\right )+b c^2 \left (c d^2 (d-3 e x)+3 a e^2 (-3 d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right )^3 (a+x (b+c x))}+\frac {(-2 c d+b e) \left (-2 c^4 d^4+3 b^4 e^4+4 c^3 d^2 e (b d-5 a e)-4 b^2 c e^3 (b d+5 a e)+2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (c d^2+e (-b d+a e)\right )^4}+\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^4}-\frac {e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.04, size = 821, normalized size = 1.69
method | result | size |
default | \(\frac {\frac {\frac {c \left (3 a^{2} b \,e^{5} c -6 d \,e^{4} a^{2} c^{2}-a \,b^{3} e^{5}+6 a b \,c^{2} d^{2} e^{3}-4 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}-4 b^{3} c \,d^{2} e^{3}+6 b^{2} c^{2} d^{3} e^{2}-5 b \,c^{3} d^{4} e +2 c^{4} d^{5}\right ) x}{4 a c -b^{2}}-\frac {2 a^{3} c^{2} e^{5}-4 a^{2} b^{2} c \,e^{5}+7 a^{2} b \,c^{2} d \,e^{4}-4 a^{2} c^{3} d^{2} e^{3}+a \,b^{4} e^{5}+a \,b^{3} c d \,e^{4}-10 a \,b^{2} c^{2} d^{2} e^{3}+14 a b \,c^{3} d^{3} e^{2}-6 a \,c^{4} d^{4} e -b^{5} d \,e^{4}+4 b^{4} c \,d^{2} e^{3}-6 b^{3} c^{2} d^{3} e^{2}+4 b^{2} c^{3} d^{4} e -b \,c^{4} d^{5}}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (8 a^{2} c^{3} e^{5}-14 a \,b^{2} c^{2} e^{5}+40 a b \,c^{3} d \,e^{4}-40 a \,c^{4} d^{2} e^{3}+3 b^{4} c \,e^{5}-10 b^{3} c^{2} d \,e^{4}+10 b^{2} c^{3} d^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (19 a^{2} b \,c^{2} e^{5}-30 a^{2} c^{3} d \,e^{4}-17 a \,b^{3} c \,e^{5}+50 a \,b^{2} c^{2} d \,e^{4}-50 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}-10 b^{4} c d \,e^{4}+10 b^{3} c^{2} d^{2} e^{3}-5 b \,c^{4} d^{4} e +2 c^{5} d^{5}-\frac {\left (8 a^{2} c^{3} e^{5}-14 a \,b^{2} c^{2} e^{5}+40 a b \,c^{3} d \,e^{4}-40 a \,c^{4} d^{2} e^{3}+3 b^{4} c \,e^{5}-10 b^{3} c^{2} d \,e^{4}+10 b^{2} c^{3} d^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{4}}-\frac {e^{3}}{2 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}+\frac {2 e^{3} \left (b e -2 c d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {e^{3} \left (2 a c \,e^{2}-3 b^{2} e^{2}+10 b c d e -10 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{4}}\) | \(821\) |
risch | \(\text {Expression too large to display}\) | \(6775\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1612 vs.
\(2 (490) = 980\).
time = 0.77, size = 1612, normalized size = 3.32 \begin {gather*} -\frac {{\left (10 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} + 3 \, b^{2} e^{5} - 2 \, a c e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} + 4 \, a c^{3} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} - 12 \, a b c^{2} d^{5} e^{3} + b^{4} d^{4} e^{4} + 12 \, a b^{2} c d^{4} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} - 12 \, a^{2} b c d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}} + \frac {{\left (10 \, c^{2} d^{2} e^{4} - 10 \, b c d e^{5} + 3 \, b^{2} e^{6} - 2 \, a c e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} + 4 \, a c^{3} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} - 12 \, a b c^{2} d^{5} e^{4} + b^{4} d^{4} e^{5} + 12 \, a b^{2} c d^{4} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} - 12 \, a^{2} b c d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} + 4 \, a^{3} c d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}} - \frac {{\left (4 \, c^{5} d^{5} - 10 \, b c^{4} d^{4} e + 40 \, a c^{4} d^{3} e^{2} + 10 \, b^{3} c^{2} d^{2} e^{3} - 60 \, a b c^{3} d^{2} e^{3} - 10 \, b^{4} c d e^{4} + 60 \, a b^{2} c^{2} d e^{4} - 60 \, a^{2} c^{3} d e^{4} + 3 \, b^{5} e^{5} - 20 \, a b^{3} c e^{5} + 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} d^{8} - 4 \, a c^{5} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 16 \, a b c^{4} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 20 \, a b^{2} c^{3} d^{6} e^{2} - 16 \, a^{2} c^{4} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + 4 \, a b^{3} c^{2} d^{5} e^{3} + 48 \, a^{2} b c^{3} d^{5} e^{3} + b^{6} d^{4} e^{4} + 8 \, a b^{4} c d^{4} e^{4} - 42 \, a^{2} b^{2} c^{2} d^{4} e^{4} - 24 \, a^{3} c^{3} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 4 \, a^{2} b^{3} c d^{3} e^{5} + 48 \, a^{3} b c^{2} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 20 \, a^{3} b^{2} c d^{2} e^{6} - 16 \, a^{4} c^{2} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + 16 \, a^{4} b c d e^{7} + a^{4} b^{2} e^{8} - 4 \, a^{5} c e^{8}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b c^{4} d^{7} - 8 \, b^{2} c^{3} d^{6} e + 12 \, a c^{4} d^{6} e + 12 \, b^{3} c^{2} d^{5} e^{2} - 28 \, a b c^{3} d^{5} e^{2} - 8 \, b^{4} c d^{4} e^{3} + 29 \, a b^{2} c^{2} d^{4} e^{3} - 28 \, a^{2} c^{3} d^{4} e^{3} + 2 \, b^{5} d^{3} e^{4} - 16 \, a b^{3} c d^{3} e^{4} + 42 \, a^{2} b c^{2} d^{3} e^{4} + 3 \, a b^{4} d^{2} e^{5} - 2 \, a^{2} b^{2} c d^{2} e^{5} - 44 \, a^{3} c^{2} d^{2} e^{5} - 6 \, a^{2} b^{3} d e^{6} + 24 \, a^{3} b c d e^{6} + a^{3} b^{2} e^{7} - 4 \, a^{4} c e^{7} + 2 \, {\left (2 \, c^{5} d^{5} e^{2} - 5 \, b c^{4} d^{4} e^{3} + 10 \, b^{2} c^{3} d^{3} e^{4} - 20 \, a c^{4} d^{3} e^{4} - 10 \, b^{3} c^{2} d^{2} e^{5} + 30 \, a b c^{3} d^{2} e^{5} + 3 \, b^{4} c d e^{6} - 4 \, a b^{2} c^{2} d e^{6} - 22 \, a^{2} c^{3} d e^{6} - 3 \, a b^{3} c e^{7} + 11 \, a^{2} b c^{2} e^{7}\right )} x^{3} + {\left (8 \, c^{5} d^{6} e - 18 \, b c^{4} d^{5} e^{2} + 25 \, b^{2} c^{3} d^{4} e^{3} - 40 \, a c^{4} d^{4} e^{3} - 10 \, b^{3} c^{2} d^{3} e^{4} + 20 \, a b c^{3} d^{3} e^{4} - 11 \, b^{4} c d^{2} e^{5} + 58 \, a b^{2} c^{2} d^{2} e^{5} - 56 \, a^{2} c^{3} d^{2} e^{5} + 6 \, b^{5} d e^{6} - 20 \, a b^{3} c d e^{6} - 10 \, a^{2} b c^{2} d e^{6} - 6 \, a b^{4} e^{7} + 25 \, a^{2} b^{2} c e^{7} - 8 \, a^{3} c^{2} e^{7}\right )} x^{2} + {\left (4 \, c^{5} d^{7} - 6 \, b c^{4} d^{6} e - 4 \, b^{2} c^{3} d^{5} e^{2} + 16 \, a c^{4} d^{5} e^{2} + 25 \, b^{3} c^{2} d^{4} e^{3} - 80 \, a b c^{3} d^{4} e^{3} - 28 \, b^{4} c d^{3} e^{4} + 104 \, a b^{2} c^{2} d^{3} e^{4} - 28 \, a^{2} c^{3} d^{3} e^{4} + 9 \, b^{5} d^{2} e^{5} - 28 \, a b^{3} c d^{2} e^{5} - 14 \, a^{2} b c^{2} d^{2} e^{5} - 6 \, a b^{4} d e^{6} + 32 \, a^{2} b^{2} c d e^{6} - 40 \, a^{3} c^{2} d e^{6} - 3 \, a^{2} b^{3} e^{7} + 12 \, a^{3} b c e^{7}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{4} {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 7.48, size = 2500, normalized size = 5.15 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________