Optimal. Leaf size=243 \[ \frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]
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Rubi [A]
time = 0.28, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {715, 648, 632,
212, 642} \begin {gather*} -\frac {\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+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 )}{c^4 \sqrt {b^2-4 a c}}+\frac {e (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {e^2 x \left (-c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )}{c^3}+\frac {e^3 x^2 (4 c d-b e)}{2 c^2}+\frac {e^4 x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 715
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{a+b x+c x^2} \, dx &=\int \left (\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right )}{c^3}+\frac {e^3 (4 c d-b e) x}{c^2}+\frac {e^4 x^2}{c}+\frac {c^3 d^4-6 a c^2 d^2 e^2-a b^2 e^4+a c e^3 (4 b d+a e)+e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}+\frac {\int \frac {c^3 d^4-6 a c^2 d^2 e^2-a b^2 e^4+a c e^3 (4 b d+a e)+e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}+\frac {\left (e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (-b e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+2 c \left (c^3 d^4-6 a c^2 d^2 e^2-a b^2 e^4+a c e^3 (4 b d+a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4}\\ &=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}+\frac {e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4}\\ &=\frac {e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x}{c^3}+\frac {e^3 (4 c d-b e) x^2}{2 c^2}+\frac {e^4 x^3}{3 c}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}+\frac {e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 240, normalized size = 0.99 \begin {gather*} \frac {6 c e^2 \left (6 c^2 d^2+b^2 e^2-c e (4 b d+a e)\right ) x+3 c^2 e^3 (4 c d-b e) x^2+2 c^3 e^4 x^3+\frac {6 \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+3 e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \log (a+x (b+c x))}{6 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.94, size = 297, normalized size = 1.22
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{3} c^{2} e^{2} x^{3}+\frac {1}{2} b c \,e^{2} x^{2}-2 c^{2} d e \,x^{2}+a c \,e^{2} x -b^{2} e^{2} x +4 b c d e x -6 d^{2} c^{2} x \right )}{c^{3}}+\frac {\frac {\left (2 a b c \,e^{4}-4 d \,e^{3} c^{2} a -b^{3} e^{4}+4 b^{2} d \,e^{3} c -6 d^{2} e^{2} b \,c^{2}+4 d^{3} e \,c^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (e^{4} a^{2} c -a \,b^{2} e^{4}+4 a b c d \,e^{3}-6 d^{2} e^{2} c^{2} a +d^{4} c^{3}-\frac {\left (2 a b c \,e^{4}-4 d \,e^{3} c^{2} a -b^{3} e^{4}+4 b^{2} d \,e^{3} c -6 d^{2} e^{2} b \,c^{2}+4 d^{3} e \,c^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{3}}\) | \(297\) |
risch | \(\text {Expression too large to display}\) | \(9468\) |
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 [A]
time = 2.80, size = 797, normalized size = 3.28 \begin {gather*} \left [\frac {36 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} x e^{2} + 3 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\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 ) + {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x\right )} e^{4} + 12 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{2} - 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d x\right )} e^{3} + 3 \, {\left (4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} e - 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e^{2} + 4 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e^{3} - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac {36 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{2} x e^{2} - 6 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\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 ) + {\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x\right )} e^{4} + 12 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{2} - 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d x\right )} e^{3} + 3 \, {\left (4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} e - 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e^{2} + 4 \, {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e^{3} - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1556 vs.
\(2 (238) = 476\).
time = 4.99, size = 1556, normalized size = 6.40 \begin {gather*} x^{2} \left (- \frac {b e^{4}}{2 c^{2}} + \frac {2 d e^{3}}{c}\right ) + x \left (- \frac {a e^{4}}{c^{2}} + \frac {b^{2} e^{4}}{c^{3}} - \frac {4 b d e^{3}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a^{2} b c e^{4} + 8 a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 4 a b^{2} c d e^{3} + 6 a b c^{2} d^{2} e^{2} + 4 a c^{4} \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) - 8 a c^{3} d^{3} e - b^{2} c^{3} \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) + b c^{3} d^{4}}{2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}} \right )} + \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a^{2} b c e^{4} + 8 a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 4 a b^{2} c d e^{3} + 6 a b c^{2} d^{2} e^{2} + 4 a c^{4} \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) - 8 a c^{3} d^{3} e - b^{2} c^{3} \left (\frac {e \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{2 c^{4} \cdot \left (4 a c - b^{2}\right )}\right ) + b c^{3} d^{4}}{2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}} \right )} + \frac {e^{4} x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 265, normalized size = 1.09 \begin {gather*} \frac {2 \, c^{2} x^{3} e^{4} + 12 \, c^{2} d x^{2} e^{3} + 36 \, c^{2} d^{2} x e^{2} - 3 \, b c x^{2} e^{4} - 24 \, b c d x e^{3} + 6 \, b^{2} x e^{4} - 6 \, a c x e^{4}}{6 \, c^{3}} + \frac {{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 367, normalized size = 1.51 \begin {gather*} x\,\left (\frac {b\,\left (\frac {b\,e^4}{c^2}-\frac {4\,d\,e^3}{c}\right )}{c}-\frac {a\,e^4}{c^2}+\frac {6\,d^2\,e^2}{c}\right )-x^2\,\left (\frac {b\,e^4}{2\,c^2}-\frac {2\,d\,e^3}{c}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (8\,a^2\,b\,c^2\,e^4-16\,a^2\,c^3\,d\,e^3-6\,a\,b^3\,c\,e^4+20\,a\,b^2\,c^2\,d\,e^3-24\,a\,b\,c^3\,d^2\,e^2+16\,a\,c^4\,d^3\,e+b^5\,e^4-4\,b^4\,c\,d\,e^3+6\,b^3\,c^2\,d^2\,e^2-4\,b^2\,c^3\,d^3\,e\right )}{2\,\left (4\,a\,c^5-b^2\,c^4\right )}+\frac {e^4\,x^3}{3\,c}+\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (2\,a^2\,c^2\,e^4-4\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-4\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{c^4\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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