Optimal. Leaf size=343 \[ \frac {\left (-b^2 c \left (3 a d^2-c e^2\right )+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {x (2 a d+b e)}{a^2 e^3}+\frac {\left (-4 a^2 c^3 d e-b^3 c \left (5 a d^2-c e^2\right )+8 a b^2 c^2 d e+a b c^2 \left (5 a d^2-3 c e^2\right )+b^5 d^2-2 b^4 c d e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^5}{e^4 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {d^4 \log (d+e x) \left (3 a d^2-e (4 b d-5 c e)\right )}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {x^2}{2 a e^2} \]
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Rubi [A] time = 0.91, antiderivative size = 343, 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} \begin {gather*} \frac {\left (-b^2 c \left (3 a d^2-c e^2\right )+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-2 b^3 c d e+b^4 d^2\right ) \log \left (a x^2+b x+c\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (-4 a^2 c^3 d e+8 a b^2 c^2 d e-b^3 c \left (5 a d^2-c e^2\right )+a b c^2 \left (5 a d^2-3 c e^2\right )-2 b^4 c d e+b^5 d^2\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 )^2}-\frac {x (2 a d+b e)}{a^2 e^3}+\frac {d^5}{e^4 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {d^4 \log (d+e x) \left (3 a d^2-e (4 b d-5 c e)\right )}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {x^2}{2 a e^2} \end {gather*}
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^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx &=\int \frac {x^5}{(d+e x)^2 \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {-2 a d-b e}{a^2 e^3}+\frac {x}{a e^2}+\frac {d^5}{e^3 \left (-a d^2+e (b d-c e)\right ) (d+e x)^2}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2 (d+e x)}+\frac {c (b d-c e) \left (b^2 d-2 a c d-b c e\right )+\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) x}{a^2 \left (a d^2-e (b d-c e)\right )^2 \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\int \frac {c (b d-c e) \left (b^2 d-2 a c d-b c e\right )+\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) x}{c+b x+a x^2} \, dx}{a^2 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\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 )^2}-\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\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 )^2}\\ &=-\frac {(2 a d+b e) x}{a^2 e^3}+\frac {x^2}{2 a e^2}+\frac {d^5}{e^4 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^5 d^2-2 b^4 c d e+8 a b^2 c^2 d e-4 a^2 c^3 d e+a b c^2 \left (5 a d^2-3 c e^2\right )-b^3 c \left (5 a d^2-c e^2\right )\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 )^2}+\frac {d^4 \left (3 a d^2-e (4 b d-5 c e)\right ) \log (d+e x)}{e^4 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^4 d^2-2 b^3 c d e+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )-b^2 c \left (3 a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a^3 \left (a d^2-e (b d-c e)\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 338, normalized size = 0.99 \begin {gather*} \frac {\left (b^2 c \left (c e^2-3 a d^2\right )+4 a b c^2 d e+a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right ) \log (x (a x+b)+c)}{2 a^3 \left (a d^2+e (c e-b d)\right )^2}-\frac {x (2 a d+b e)}{a^2 e^3}-\frac {\left (-4 a^2 c^3 d e+b^3 c \left (c e^2-5 a d^2\right )+8 a b^2 c^2 d e+a b c^2 \left (5 a d^2-3 c e^2\right )+b^5 d^2-2 b^4 c d 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 )^2}+\frac {d^5}{e^4 (d+e x) \left (a d^2+e (c e-b d)\right )}+\frac {\log (d+e x) \left (3 a d^6+d^4 e (5 c e-4 b d)\right )}{e^4 \left (a d^2+e (c e-b d)\right )^2}+\frac {x^2}{2 a e^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] 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 = 0.42, size = 565, normalized size = 1.65 \begin {gather*} \frac {d^{5} e^{4}}{{\left (a d^{2} e^{8} - b d e^{9} + c e^{10}\right )} {\left (x e + d\right )}} + \frac {{\left (b^{5} d^{2} e^{2} - 5 \, a b^{3} c d^{2} e^{2} + 5 \, a^{2} b c^{2} d^{2} e^{2} - 2 \, b^{4} c d e^{3} + 8 \, a b^{2} c^{2} d e^{3} - 4 \, a^{2} c^{3} d e^{3} + b^{3} c^{2} e^{4} - 3 \, a b c^{3} 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^{5} d^{4} - 2 \, a^{4} b d^{3} e + a^{3} b^{2} d^{2} e^{2} + 2 \, a^{4} c d^{2} e^{2} - 2 \, a^{3} b c d e^{3} + a^{3} c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (a^{2} - \frac {2 \, {\left (3 \, a^{2} d e + a b e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )}}{2 \, a^{3}} + \frac {{\left (b^{4} d^{2} - 3 \, a b^{2} c d^{2} + a^{2} c^{2} d^{2} - 2 \, b^{3} c d e + 4 \, a b c^{2} d e + b^{2} c^{2} e^{2} - a c^{3} 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^{5} d^{4} - 2 \, a^{4} b d^{3} e + a^{3} b^{2} d^{2} e^{2} + 2 \, a^{4} c d^{2} e^{2} - 2 \, a^{3} b c d e^{3} + a^{3} c^{2} e^{4}\right )}} - \frac {{\left (3 \, a^{2} d^{2} + 2 \, a b d e + b^{2} e^{2} - a c e^{2}\right )} e^{\left (-4\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 943, normalized size = 2.75 \begin {gather*} -\frac {5 b \,c^{2} d^{2} \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 )^{2} \sqrt {4 a c -b^{2}}\, a}+\frac {4 c^{3} d 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 )^{2} \sqrt {4 a c -b^{2}}\, a}+\frac {5 b^{3} c \,d^{2} \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 )^{2} \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {8 b^{2} c^{2} d 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 )^{2} \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {3 b \,c^{3} e^{2} \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 )^{2} \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{5} d^{2} \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 )^{2} \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {2 b^{4} c d 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 )^{2} \sqrt {4 a c -b^{2}}\, a^{3}}-\frac {b^{3} c^{2} e^{2} \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 )^{2} \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {3 a \,d^{6} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} e^{4}}+\frac {c^{2} d^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a}-\frac {3 b^{2} c \,d^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{2}}+\frac {2 b \,c^{2} d e \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{2}}-\frac {c^{3} e^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{2}}+\frac {b^{4} d^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{3}}-\frac {b^{3} c d e \ln \left (a \,x^{2}+b x +c \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{3}}+\frac {b^{2} c^{2} e^{2} \ln \left (a \,x^{2}+b x +c \right )}{2 \left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} a^{3}}-\frac {4 b \,d^{5} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} e^{3}}+\frac {5 c \,d^{4} \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} e^{2}}+\frac {d^{5}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right ) e^{4}}+\frac {x^{2}}{2 a \,e^{2}}-\frac {2 d x}{a \,e^{3}}-\frac {b x}{a^{2} e^{2}} \end {gather*}
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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mupad [B] time = 8.04, size = 3503, normalized size = 10.21
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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