Optimal. Leaf size=262 \[ \frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.42, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {832, 814, 648,
632, 212, 642} \begin {gather*} -\frac {\left (-30 a^2 b c^2 e+12 a^2 c^3 d+20 a b^3 c e-12 a b^2 c^2 d-3 b^5 e+2 b^4 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 a c e-3 b^2 e+2 b c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {x^2 \left (8 a c e-3 b^2 e+2 b c d\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {x \left (11 a b c e-6 a c^2 d-3 b^3 e+2 b^2 c d\right )}{c^3 \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 832
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^2 \left (-3 a (2 c d-b e)-\left (2 b c d-3 b^2 e+8 a c e\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e}{c^2}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x}{c}-\frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {a \left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right )+\left (b^2-4 a c\right ) \left (2 b c d-3 b^2 e+2 a c e\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )}\\ &=\frac {\left (2 b^2 c d-6 a c^2 d-3 b^3 e+11 a b c e\right ) x}{c^3 \left (b^2-4 a c\right )}-\frac {\left (2 b c d-3 b^2 e+8 a c e\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^3 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 c d-12 a b^2 c^2 d+12 a^2 c^3 d-3 b^5 e+20 a b^3 c e-30 a^2 b c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (2 b c d-3 b^2 e+2 a c e\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 249, normalized size = 0.95 \begin {gather*} \frac {2 c (c d-2 b e) x+c^2 e x^2+\frac {2 \left (2 a^3 c^2 e+b^4 (-c d+b e) x+a b^2 \left (b^2 e+4 c^2 d x-b c (d+5 e x)\right )+a^2 c \left (-4 b^2 e-2 c^2 d x+b c (3 d+5 e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 \left (-2 b^4 c d+12 a b^2 c^2 d-12 a^2 c^3 d+3 b^5 e-20 a b^3 c e+30 a^2 b c^2 e\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 (-2 b c d+3 b^2 e-2 a c e\right ) \log (a+x (b+c x))}{2 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.98, size = 349, normalized size = 1.33
method | result | size |
default | \(-\frac {-\frac {1}{2} c e \,x^{2}+2 b e x -c d x}{c^{3}}+\frac {\frac {-\frac {\left (5 a^{2} b \,c^{2} e -2 a^{2} c^{3} d -5 a \,b^{3} c e +4 a \,b^{2} c^{2} d +b^{5} e -b^{4} c d \right ) x}{c \left (4 a c -b^{2}\right )}-\frac {a \left (2 a^{2} c^{2} e -4 a \,b^{2} c e +3 a b \,c^{2} d +b^{4} e -b^{3} d c \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} c^{2} e +14 a \,b^{2} c e -8 a b \,c^{2} d -3 b^{4} e +2 b^{3} d c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (11 a^{2} b c e -6 a^{2} c^{2} d -3 a \,b^{3} e +2 a \,b^{2} c d -\frac {\left (-8 a^{2} c^{2} e +14 a \,b^{2} c e -8 a b \,c^{2} d -3 b^{4} e +2 b^{3} d c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{3}}\) | \(349\) |
risch | \(\text {Expression too large to display}\) | \(4104\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 830 vs.
\(2 (266) = 532\).
time = 1.35, size = 1680, normalized size = 6.41 \begin {gather*} \left [\frac {2 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d x^{3} + 2 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d x^{2} - 2 \, {\left (b^{6} c - 9 \, a b^{4} c^{2} + 26 \, a^{2} b^{2} c^{3} - 24 \, a^{3} c^{4}\right )} d x + {\left (2 \, {\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 6 \, a^{2} c^{4}\right )} d x^{2} + 2 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} d x + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 6 \, a^{3} c^{3}\right )} d - {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\right )} x\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (a b^{5} c - 7 \, a^{2} b^{3} c^{2} + 12 \, a^{3} b c^{3}\right )} d + {\left (2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} + 2 \, {\left (b^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x\right )} e - {\left (2 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d x + 2 \, {\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} d - {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}, \frac {2 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d x^{3} + 2 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d x^{2} - 2 \, {\left (b^{6} c - 9 \, a b^{4} c^{2} + 26 \, a^{2} b^{2} c^{3} - 24 \, a^{3} c^{4}\right )} d x - 2 \, {\left (2 \, {\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 6 \, a^{2} c^{4}\right )} d x^{2} + 2 \, {\left (b^{5} c - 6 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} d x + 2 \, {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 6 \, a^{3} c^{3}\right )} d - {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\right )} x\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (a b^{5} c - 7 \, a^{2} b^{3} c^{2} + 12 \, a^{3} b c^{3}\right )} d + {\left (2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} + 2 \, {\left (b^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x\right )} e - {\left (2 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d x^{2} + 2 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d x + 2 \, {\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} d - {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1572 vs.
\(2 (262) = 524\).
time = 4.56, size = 1572, normalized size = 6.00 \begin {gather*} x \left (- \frac {2 b e}{c^{3}} + \frac {d}{c^{2}}\right ) + \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} e - 17 a^{2} b^{2} c e + 10 a^{2} b c^{2} d + 16 a^{2} c^{5} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) + 3 a b^{4} e - 2 a b^{3} c d - 8 a b^{2} c^{4} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) + b^{4} c^{3} \left (- \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right )}{30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d} \right )} + \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} e - 17 a^{2} b^{2} c e + 10 a^{2} b c^{2} d + 16 a^{2} c^{5} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) + 3 a b^{4} e - 2 a b^{3} c d - 8 a b^{2} c^{4} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right ) + b^{4} c^{3} \left (\frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c e - 3 b^{2} e + 2 b c d}{2 c^{4}}\right )}{30 a^{2} b c^{2} e - 12 a^{2} c^{3} d - 20 a b^{3} c e + 12 a b^{2} c^{2} d + 3 b^{5} e - 2 b^{4} c d} \right )} + \frac {- 2 a^{3} c^{2} e + 4 a^{2} b^{2} c e - 3 a^{2} b c^{2} d - a b^{4} e + a b^{3} c d + x \left (- 5 a^{2} b c^{2} e + 2 a^{2} c^{3} d + 5 a b^{3} c e - 4 a b^{2} c^{2} d - b^{5} e + b^{4} c d\right )}{4 a^{2} c^{5} - a b^{2} c^{4} + x^{2} \cdot \left (4 a c^{6} - b^{2} c^{5}\right ) + x \left (4 a b c^{5} - b^{3} c^{4}\right )} + \frac {e x^{2}}{2 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.32, size = 297, normalized size = 1.13 \begin {gather*} \frac {{\left (2 \, b^{4} c d - 12 \, a b^{2} c^{2} d + 12 \, a^{2} c^{3} d - 3 \, b^{5} e + 20 \, a b^{3} c e - 30 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, b c d - 3 \, b^{2} e + 2 \, a c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac {a b^{3} c d - 3 \, a^{2} b c^{2} d - a b^{4} e + 4 \, a^{2} b^{2} c e - 2 \, a^{3} c^{2} e + {\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.98, size = 427, normalized size = 1.63 \begin {gather*} x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )-\frac {\frac {a\,\left (2\,e\,a^2\,c^2-4\,e\,a\,b^2\,c+3\,d\,a\,b\,c^2+e\,b^4-d\,b^3\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}+\frac {e\,x^2}{2\,c^2}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,e\,a^4\,c^4-288\,e\,a^3\,b^2\,c^3+128\,d\,a^3\,b\,c^4+168\,e\,a^2\,b^4\,c^2-96\,d\,a^2\,b^3\,c^3-38\,e\,a\,b^6\,c+24\,d\,a\,b^5\,c^2+3\,e\,b^8-2\,d\,b^7\,c\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}+\frac {\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^3-4\,a\,b\,c^4}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (30\,e\,a^2\,b\,c^2-12\,d\,a^2\,c^3-20\,e\,a\,b^3\,c+12\,d\,a\,b^2\,c^2+3\,e\,b^5-2\,d\,b^4\,c\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________