Optimal. Leaf size=372 \[ -\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2} \]
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Rubi [A]
time = 0.55, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 907, 648,
632, 212, 642} \begin {gather*} \frac {\left (a^3 c d^2-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )+a b^2 e (2 b d+3 c e)+b^4 \left (-e^2\right )\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (-a^3 c d (3 b d+4 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )-a b^3 e (2 b d+5 c e)+b^5 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\log (x) \left (-c \left (a d^2-3 c e^2\right )+b^2 d^2+2 b c d e\right )}{c^3 d^4}-\frac {e^4 \log (d+e x) \left (5 a d^2-e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {e^4}{d^3 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {b d+2 c e}{c^2 d^3 x}-\frac {1}{2 c d^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 907
Rule 1583
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^5 (d+e x)^2} \, dx &=\int \frac {1}{x^3 (d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{c d^2 x^3}+\frac {-b d-2 c e}{c^2 d^3 x^2}+\frac {b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )}{c^3 d^4 x}+\frac {e^5}{d^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac {e^5 \left (-5 a d^2+e (4 b d-3 c e)\right )}{d^4 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {-\left (a b d-b^2 e+a c e\right ) \left (a b^2 d-2 a^2 c d-b^3 e+3 a b c e\right )+a \left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) x}{c^3 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {-\left (a b d-b^2 e+a c e\right ) \left (a b^2 d-2 a^2 c d-b^3 e+3 a b c e\right )+a \left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) x}{c+b x+a x^2} \, dx}{c^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{c^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 e^2-a^3 c d (3 b d+4 c e)-a b^3 e (2 b d+5 c e)+a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^2 d^2+2 b c d e-c \left (a d^2-3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2-e (4 b d-3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (a^3 c d^2-b^4 e^2+a b^2 e (2 b d+3 c e)-a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 370, normalized size = 0.99 \begin {gather*} -\frac {1}{2 c d^2 x^2}+\frac {b d+2 c e}{c^2 d^3 x}+\frac {e^4}{d^3 \left (a d^2+e (-b d+c e)\right ) (d+e x)}+\frac {\left (-b^5 e^2+a^3 c d (3 b d+4 c e)+a b^3 e (2 b d+5 c e)-a^2 b \left (b^2 d^2+8 b c d e+5 c^2 e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{c^3 \sqrt {-b^2+4 a c} \left (a d^2+e (-b d+c e)\right )^2}+\frac {\left (b^2 d^2+2 b c d e+c \left (-a d^2+3 c e^2\right )\right ) \log (x)}{c^3 d^4}-\frac {e^4 \left (5 a d^2+e (-4 b d+3 c e)\right ) \log (d+e x)}{d^4 \left (a d^2+e (-b d+c e)\right )^2}-\frac {\left (-a^3 c d^2+b^4 e^2-a b^2 e (2 b d+3 c e)+a^2 \left (b^2 d^2+4 b c d e+c^2 e^2\right )\right ) \log (c+x (b+a x))}{2 c^3 \left (a d^2+e (-b d+c e)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 455, normalized size = 1.22
method | result | size |
default | \(\frac {e^{4}}{d^{3} \left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right )}-\frac {e^{4} \left (5 a \,d^{2}-4 d e b +3 c \,e^{2}\right ) \ln \left (e x +d \right )}{d^{4} \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}+\frac {\frac {\left (a^{4} c \,d^{2}-a^{3} b^{2} d^{2}-4 a^{3} b c d e -a^{3} c^{2} e^{2}+2 a^{2} b^{3} d e +3 a^{2} b^{2} c \,e^{2}-a \,b^{4} e^{2}\right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (2 a^{3} b c \,d^{2}+2 a^{3} c^{2} d e -a^{2} b^{3} d^{2}-6 a^{2} b^{2} c d e -3 e^{2} c^{2} a^{2} b +2 a \,b^{4} d e +4 a c \,e^{2} b^{3}-e^{2} b^{5}-\frac {\left (a^{4} c \,d^{2}-a^{3} b^{2} d^{2}-4 a^{3} b c d e -a^{3} c^{2} e^{2}+2 a^{2} b^{3} d e +3 a^{2} b^{2} c \,e^{2}-a \,b^{4} e^{2}\right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} c^{3}}-\frac {1}{2 c \,d^{2} x^{2}}-\frac {-b d -2 c e}{c^{2} d^{3} x}+\frac {\left (-a c \,d^{2}+b^{2} d^{2}+2 b c d e +3 c^{2} e^{2}\right ) \ln \left (x \right )}{c^{3} d^{4}}\) | \(455\) |
risch | \(\text {Expression too large to display}\) | \(3253\) |
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 [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.
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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.
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Giac [A]
time = 3.93, size = 587, normalized size = 1.58 \begin {gather*} \frac {{\left (a^{2} b^{3} d^{2} e^{2} - 3 \, a^{3} b c d^{2} e^{2} - 2 \, a b^{4} d e^{3} + 8 \, a^{2} b^{2} c d e^{3} - 4 \, a^{3} c^{2} d e^{3} + b^{5} e^{4} - 5 \, a b^{3} c e^{4} + 5 \, a^{2} b c^{2} e^{4}\right )} \arctan \left (-\frac {{\left (2 \, a d - \frac {2 \, a d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (a^{2} b^{2} d^{2} - a^{3} c d^{2} - 2 \, a b^{3} d e + 4 \, a^{2} b c d e + b^{4} e^{2} - 3 \, a b^{2} c e^{2} + a^{2} c^{2} e^{2}\right )} \log \left (-a + \frac {2 \, a d}{x e + d} - \frac {a d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (a^{2} c^{3} d^{4} - 2 \, a b c^{3} d^{3} e + b^{2} c^{3} d^{2} e^{2} + 2 \, a c^{4} d^{2} e^{2} - 2 \, b c^{4} d e^{3} + c^{5} e^{4}\right )}} + \frac {e^{9}}{{\left (a d^{5} e^{5} - b d^{4} e^{6} + c d^{3} e^{7}\right )} {\left (x e + d\right )}} + \frac {{\left (b^{2} d^{2} e - a c d^{2} e + 2 \, b c d e^{2} + 3 \, c^{2} e^{3}\right )} e^{\left (-1\right )} \log \left ({\left | -\frac {d}{x e + d} + 1 \right |}\right )}{c^{3} d^{4}} + \frac {2 \, b c d e + 5 \, c^{2} e^{2} - \frac {2 \, {\left (b c d^{2} e^{2} + 3 \, c^{2} d e^{3}\right )} e^{\left (-1\right )}}{x e + d}}{2 \, c^{3} d^{4} {\left (\frac {d}{x e + d} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 45.61, size = 2500, normalized size = 6.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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