Optimal. Leaf size=218 \[ -\frac {\left (-2 a b c d+a c^2 e+b^3 d-b^2 c e\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac {x (a d+b e)}{a^2 e^2}-\frac {\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac {x^2}{2 a e} \]
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Rubi [A] time = 0.40, antiderivative size = 218, 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 = {1569, 1628, 634, 618, 206, 628} \[ -\frac {\left (-2 a b c d+a c^2 e-b^2 c e+b^3 d\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac {\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e-b^3 c e+b^4 d\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {x (a d+b e)}{a^2 e^2}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac {x^2}{2 a e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1569
Rule 1628
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx &=\int \frac {x^4}{(d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {-a d-b e}{a^2 e^2}+\frac {x}{a e}+\frac {d^4}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {-c \left (b^2 d-a c d-b c e\right )-\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) x}{a^2 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {(a d+b e) x}{a^2 e^2}+\frac {x^2}{2 a e}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}+\frac {\int \frac {-c \left (b^2 d-a c d-b c e\right )-\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) x}{c+b x+a x^2} \, dx}{a^2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {(a d+b e) x}{a^2 e^2}+\frac {x^2}{2 a e}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )}+\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {(a d+b e) x}{a^2 e^2}+\frac {x^2}{2 a e}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{a^3 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {(a d+b e) x}{a^2 e^2}+\frac {x^2}{2 a e}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-b^3 c e+3 a b c^2 e\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )}-\frac {\left (b^3 d-2 a b c d-b^2 c e+a c^2 e\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 218, normalized size = 1.00 \[ \frac {\left (2 a b c d-a c^2 e+b^3 (-d)+b^2 c e\right ) \log (x (a x+b)+c)}{2 a^3 \left (a d^2+e (c e-b d)\right )}-\frac {x (a d+b e)}{a^2 e^2}+\frac {\left (2 a^2 c^2 d-4 a b^2 c d+3 a b c^2 e+b^4 d-b^3 c e\right ) \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a^3 \sqrt {4 a c-b^2} \left (a d^2+e (c e-b d)\right )}+\frac {d^4 \log (d+e x)}{e^3 \left (a d^2+e (c e-b d)\right )}+\frac {x^2}{2 a e} \]
Antiderivative was successfully verified.
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fricas [A] time = 35.52, size = 798, normalized size = 3.66 \[ \left [\frac {2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{4} \log \left (e x + d\right ) + {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )} x^{2} + {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d e^{3} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right ) - 2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{3} e - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d e^{3} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} e^{4}\right )} x - {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d e^{3} - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e^{4}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )}}, \frac {2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{4} \log \left (e x + d\right ) + {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{2} e^{2} - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d e^{3} + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e^{4}\right )} x^{2} - 2 \, {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d e^{3} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{3} e - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d e^{3} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} e^{4}\right )} x - {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d e^{3} - {\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e^{4}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{4} b^{2} - 4 \, a^{5} c\right )} d^{2} e^{3} - {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} d e^{4} + {\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} e^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 224, normalized size = 1.03 \[ \frac {d^{4} \log \left ({\left | x e + d \right |}\right )}{a d^{2} e^{3} - b d e^{4} + c e^{5}} - \frac {{\left (b^{3} d - 2 \, a b c d - b^{2} c e + a c^{2} e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} d^{2} - a^{3} b d e + a^{3} c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (a x^{2} e - 2 \, a d x - 2 \, b x e\right )} e^{\left (-2\right )}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 512, normalized size = 2.35 \[ \frac {2 c^{2} d \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {4 b^{2} c d \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {3 b \,c^{2} e \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {b^{4} d \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}-\frac {b^{3} c e \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {b c d \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) a^{2}}-\frac {c^{2} e \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right ) a^{2}}-\frac {b^{3} d \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right ) a^{3}}+\frac {b^{2} c e \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right ) a^{3}}+\frac {d^{4} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) e^{3}}+\frac {x^{2}}{2 a e}-\frac {d x}{a \,e^{2}}-\frac {b x}{a^{2} e} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 2051, normalized size = 9.41 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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