Optimal. Leaf size=306 \[ -\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {750, 834, 791,
632, 212} \begin {gather*} \frac {2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\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}}-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (-8 a c e-3 b^2 e+10 c x (2 c d-b e)+10 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 x (2 c d-b e) \left (-c e (a e+15 b d)+4 b^2 e^2+15 c^2 d^2\right )-b^2 \left (11 a e^3+25 c d^2 e\right )+6 b c d \left (13 a e^2+5 c d^2\right )-16 a c e \left (a e^2+5 c d^2\right )+3 b^3 d e^2}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 750
Rule 791
Rule 834
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {\int \frac {(d+e x)^2 (-10 c d+3 b e-4 c e x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {\int \frac {(d+e x) \left (-2 \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-10 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{6 \left (b^2-4 a c\right )^2}\\ &=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (20 a c^2 e^2 (2 c d-b e)-c \left (-4 c d \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-b \left (-10 c d e (2 c d-b e)-2 e \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{6 c \left (b^2-4 a c\right )^3}\\ &=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (20 a c^2 e^2 (2 c d-b e)-c \left (-4 c d \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )-b \left (-10 c d e (2 c d-b e)-2 e \left (30 c^2 d^2+3 b^2 e^2-c e (25 b d-8 a e)\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{3 c \left (b^2-4 a c\right )^3}\\ &=-\frac {(b+2 c x) (d+e x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (10 b c d-3 b^2 e-8 a c e+10 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {3 b^3 d e^2-16 a c e \left (5 c d^2+a e^2\right )+6 b c d \left (5 c d^2+13 a e^2\right )-b^2 \left (25 c d^2 e+11 a e^3\right )+2 (2 c d-b e) \left (15 c^2 d^2+4 b^2 e^2-c e (15 b d+a e)\right ) x}{3 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2+6 a c e^2\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}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 401, normalized size = 1.31 \begin {gather*} \frac {1}{6} \left (\frac {3 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {2 \left (-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {-2 b^4 e^3+b^3 c e^2 (6 d-e x)+4 c^2 \left (-6 a^2 e^3+5 c^2 d^3 x+3 a c d e^2 x\right )+2 b c^2 \left (5 c d^2 (d-3 e x)+3 a e^2 (d-e x)\right )+3 b^2 c e \left (3 a e^2+c d (-5 d+4 e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {12 (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(884\) vs.
\(2(295)=590\).
time = 0.91, size = 885, normalized size = 2.89
method | result | size |
default | \(\frac {-\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) c^{2} x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 \left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) b c \,x^{4}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (16 a c +11 b^{2}\right ) \left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) x^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (32 a^{3} c^{2} e^{3}+24 a^{2} b^{2} c \,e^{3}-96 a^{2} b \,c^{2} d \,e^{2}+17 a \,b^{4} e^{3}-102 a \,b^{3} c d \,e^{2}+240 a \,b^{2} c^{2} d^{2} e -160 c^{3} d^{3} a b -6 b^{5} d \,e^{2}+15 b^{4} d^{2} e c -10 b^{3} d^{3} c^{2}\right ) x^{2}}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (20 a^{3} b c \,e^{3}+24 a^{3} c^{2} d \,e^{2}+20 a^{2} b^{3} e^{3}-132 a^{2} b^{2} c d \,e^{2}+132 a^{2} b \,c^{2} d^{2} e -88 a^{2} c^{3} d^{3}-6 a \,b^{4} d \,e^{2}+54 a \,b^{3} c \,d^{2} e -36 a \,b^{2} c^{2} d^{3}-3 b^{5} d^{2} e +2 b^{4} c \,d^{3}\right ) x}{2 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {32 a^{4} c \,e^{3}+22 a^{3} b^{2} e^{3}-156 a^{3} b c d \,e^{2}+192 a^{3} c^{2} d^{2} e -6 a^{2} b^{3} d \,e^{2}+54 a^{2} b^{2} c \,d^{2} e -132 a^{2} b \,c^{2} d^{3}-3 a \,b^{4} d^{2} e +26 a \,b^{3} c \,d^{3}-2 b^{5} d^{3}}{6 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}-\frac {2 \left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\) | \(885\) |
risch | \(\text {Expression too large to display}\) | \(1974\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1774 vs.
\(2 (303) = 606\).
time = 2.56, size = 3568, normalized size = 11.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2057 vs.
\(2 (308) = 616\).
time = 14.11, size = 2057, normalized size = 6.72 \begin {gather*} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (x + \frac {- 256 a^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + 256 a^{3} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) - 96 a^{2} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + 16 a b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + 6 a b^{2} c e^{3} - 12 a b c^{2} d e^{2} - b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + b^{4} e^{3} - 12 b^{3} c d e^{2} + 30 b^{2} c^{2} d^{2} e - 20 b c^{3} d^{3}}{12 a b c^{2} e^{3} - 24 a c^{3} d e^{2} + 2 b^{3} c e^{3} - 24 b^{2} c^{2} d e^{2} + 60 b c^{3} d^{2} e - 40 c^{4} d^{3}} \right )} - \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (x + \frac {256 a^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) - 256 a^{3} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + 96 a^{2} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) - 16 a b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + 6 a b^{2} c e^{3} - 12 a b c^{2} d e^{2} + b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) + b^{4} e^{3} - 12 b^{3} c d e^{2} + 30 b^{2} c^{2} d^{2} e - 20 b c^{3} d^{3}}{12 a b c^{2} e^{3} - 24 a c^{3} d e^{2} + 2 b^{3} c e^{3} - 24 b^{2} c^{2} d e^{2} + 60 b c^{3} d^{2} e - 40 c^{4} d^{3}} \right )} + \frac {- 32 a^{4} c e^{3} - 22 a^{3} b^{2} e^{3} + 156 a^{3} b c d e^{2} - 192 a^{3} c^{2} d^{2} e + 6 a^{2} b^{3} d e^{2} - 54 a^{2} b^{2} c d^{2} e + 132 a^{2} b c^{2} d^{3} + 3 a b^{4} d^{2} e - 26 a b^{3} c d^{3} + 2 b^{5} d^{3} + x^{5} \left (- 36 a b c^{3} e^{3} + 72 a c^{4} d e^{2} - 6 b^{3} c^{2} e^{3} + 72 b^{2} c^{3} d e^{2} - 180 b c^{4} d^{2} e + 120 c^{5} d^{3}\right ) + x^{4} \left (- 90 a b^{2} c^{2} e^{3} + 180 a b c^{3} d e^{2} - 15 b^{4} c e^{3} + 180 b^{3} c^{2} d e^{2} - 450 b^{2} c^{3} d^{2} e + 300 b c^{4} d^{3}\right ) + x^{3} \left (- 96 a^{2} b c^{2} e^{3} + 192 a^{2} c^{3} d e^{2} - 82 a b^{3} c e^{3} + 324 a b^{2} c^{2} d e^{2} - 480 a b c^{3} d^{2} e + 320 a c^{4} d^{3} - 11 b^{5} e^{3} + 132 b^{4} c d e^{2} - 330 b^{3} c^{2} d^{2} e + 220 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 96 a^{3} c^{2} e^{3} - 72 a^{2} b^{2} c e^{3} + 288 a^{2} b c^{2} d e^{2} - 51 a b^{4} e^{3} + 306 a b^{3} c d e^{2} - 720 a b^{2} c^{2} d^{2} e + 480 a b c^{3} d^{3} + 18 b^{5} d e^{2} - 45 b^{4} c d^{2} e + 30 b^{3} c^{2} d^{3}\right ) + x \left (- 60 a^{3} b c e^{3} - 72 a^{3} c^{2} d e^{2} - 60 a^{2} b^{3} e^{3} + 396 a^{2} b^{2} c d e^{2} - 396 a^{2} b c^{2} d^{2} e + 264 a^{2} c^{3} d^{3} + 18 a b^{4} d e^{2} - 162 a b^{3} c d^{2} e + 108 a b^{2} c^{2} d^{3} + 9 b^{5} d^{2} e - 6 b^{4} c d^{3}\right )}{384 a^{6} c^{3} - 288 a^{5} b^{2} c^{2} + 72 a^{4} b^{4} c - 6 a^{3} b^{6} + x^{6} \cdot \left (384 a^{3} c^{6} - 288 a^{2} b^{2} c^{5} + 72 a b^{4} c^{4} - 6 b^{6} c^{3}\right ) + x^{5} \cdot \left (1152 a^{3} b c^{5} - 864 a^{2} b^{3} c^{4} + 216 a b^{5} c^{3} - 18 b^{7} c^{2}\right ) + x^{4} \cdot \left (1152 a^{4} c^{5} + 288 a^{3} b^{2} c^{4} - 648 a^{2} b^{4} c^{3} + 198 a b^{6} c^{2} - 18 b^{8} c\right ) + x^{3} \cdot \left (2304 a^{4} b c^{4} - 1344 a^{3} b^{3} c^{3} + 144 a^{2} b^{5} c^{2} + 36 a b^{7} c - 6 b^{9}\right ) + x^{2} \cdot \left (1152 a^{5} c^{4} + 288 a^{4} b^{2} c^{3} - 648 a^{3} b^{4} c^{2} + 198 a^{2} b^{6} c - 18 a b^{8}\right ) + x \left (1152 a^{5} b c^{3} - 864 a^{4} b^{3} c^{2} + 216 a^{3} b^{5} c - 18 a^{2} b^{7}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 828 vs.
\(2 (303) = 606\).
time = 0.79, size = 828, normalized size = 2.71 \begin {gather*} -\frac {2 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {120 \, c^{5} d^{3} x^{5} - 180 \, b c^{4} d^{2} x^{5} e + 300 \, b c^{4} d^{3} x^{4} + 72 \, b^{2} c^{3} d x^{5} e^{2} + 72 \, a c^{4} d x^{5} e^{2} - 450 \, b^{2} c^{3} d^{2} x^{4} e + 220 \, b^{2} c^{3} d^{3} x^{3} + 320 \, a c^{4} d^{3} x^{3} - 6 \, b^{3} c^{2} x^{5} e^{3} - 36 \, a b c^{3} x^{5} e^{3} + 180 \, b^{3} c^{2} d x^{4} e^{2} + 180 \, a b c^{3} d x^{4} e^{2} - 330 \, b^{3} c^{2} d^{2} x^{3} e - 480 \, a b c^{3} d^{2} x^{3} e + 30 \, b^{3} c^{2} d^{3} x^{2} + 480 \, a b c^{3} d^{3} x^{2} - 15 \, b^{4} c x^{4} e^{3} - 90 \, a b^{2} c^{2} x^{4} e^{3} + 132 \, b^{4} c d x^{3} e^{2} + 324 \, a b^{2} c^{2} d x^{3} e^{2} + 192 \, a^{2} c^{3} d x^{3} e^{2} - 45 \, b^{4} c d^{2} x^{2} e - 720 \, a b^{2} c^{2} d^{2} x^{2} e - 6 \, b^{4} c d^{3} x + 108 \, a b^{2} c^{2} d^{3} x + 264 \, a^{2} c^{3} d^{3} x - 11 \, b^{5} x^{3} e^{3} - 82 \, a b^{3} c x^{3} e^{3} - 96 \, a^{2} b c^{2} x^{3} e^{3} + 18 \, b^{5} d x^{2} e^{2} + 306 \, a b^{3} c d x^{2} e^{2} + 288 \, a^{2} b c^{2} d x^{2} e^{2} + 9 \, b^{5} d^{2} x e - 162 \, a b^{3} c d^{2} x e - 396 \, a^{2} b c^{2} d^{2} x e + 2 \, b^{5} d^{3} - 26 \, a b^{3} c d^{3} + 132 \, a^{2} b c^{2} d^{3} - 51 \, a b^{4} x^{2} e^{3} - 72 \, a^{2} b^{2} c x^{2} e^{3} - 96 \, a^{3} c^{2} x^{2} e^{3} + 18 \, a b^{4} d x e^{2} + 396 \, a^{2} b^{2} c d x e^{2} - 72 \, a^{3} c^{2} d x e^{2} + 3 \, a b^{4} d^{2} e - 54 \, a^{2} b^{2} c d^{2} e - 192 \, a^{3} c^{2} d^{2} e - 60 \, a^{2} b^{3} x e^{3} - 60 \, a^{3} b c x e^{3} + 6 \, a^{2} b^{3} d e^{2} + 156 \, a^{3} b c d e^{2} - 22 \, a^{3} b^{2} e^{3} - 32 \, a^{4} c e^{3}}{6 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 1126, normalized size = 3.68 \begin {gather*} \frac {\frac {x\,\left (20\,a^3\,b\,c\,e^3+24\,a^3\,c^2\,d\,e^2+20\,a^2\,b^3\,e^3-132\,a^2\,b^2\,c\,d\,e^2+132\,a^2\,b\,c^2\,d^2\,e-88\,a^2\,c^3\,d^3-6\,a\,b^4\,d\,e^2+54\,a\,b^3\,c\,d^2\,e-36\,a\,b^2\,c^2\,d^3-3\,b^5\,d^2\,e+2\,b^4\,c\,d^3\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {-32\,a^4\,c\,e^3-22\,a^3\,b^2\,e^3+156\,a^3\,b\,c\,d\,e^2-192\,a^3\,c^2\,d^2\,e+6\,a^2\,b^3\,d\,e^2-54\,a^2\,b^2\,c\,d^2\,e+132\,a^2\,b\,c^2\,d^3+3\,a\,b^4\,d^2\,e-26\,a\,b^3\,c\,d^3+2\,b^5\,d^3}{6\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^2\,\left (32\,a^3\,c^2\,e^3+24\,a^2\,b^2\,c\,e^3-96\,a^2\,b\,c^2\,d\,e^2+17\,a\,b^4\,e^3-102\,a\,b^3\,c\,d\,e^2+240\,a\,b^2\,c^2\,d^2\,e-160\,a\,b\,c^3\,d^3-6\,b^5\,d\,e^2+15\,b^4\,c\,d^2\,e-10\,b^3\,c^2\,d^3\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^3\,\left (11\,b^2+16\,a\,c\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{6\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {c^2\,x^5\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {5\,b\,c\,x^4\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}+\frac {2\,\mathrm {atan}\left (\frac {\left (\frac {\left (b\,e-2\,c\,d\right )\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {2\,c\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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