Optimal. Leaf size=102 \[ \frac {\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac {c x (4 c d-3 b e)}{e^3}+\frac {c^2 x^2}{e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac {c x (4 c d-3 b e)}{e^3}+\frac {c^2 x^2}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx &=\int \left (-\frac {c (4 c d-3 b e)}{e^3}+\frac {2 c^2 x}{e^2}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^2}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {c (4 c d-3 b e) x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 97, normalized size = 0.95 \begin {gather*} \frac {\log (d+e x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+\frac {(2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}-c e x (4 c d-3 b e)+c^2 e^2 x^2}{e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 172, normalized size = 1.69 \begin {gather*} \frac {c^{2} e^{3} x^{3} + 2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (c^{2} d e^{2} - b c e^{3}\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 3 \, b c d e^{2}\right )} x + {\left (6 \, c^{2} d^{3} - 6 \, b c d^{2} e + {\left (b^{2} + 2 \, a c\right )} d e^{2} + {\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{5} x + d e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 177, normalized size = 1.74 \begin {gather*} {\left (c^{2} - \frac {3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )} - {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 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 ) + {\left (\frac {2 \, c^{2} d^{3} e^{2}}{x e + d} - \frac {3 \, b c d^{2} e^{3}}{x e + d} + \frac {b^{2} d e^{4}}{x e + d} + \frac {2 \, a c d e^{4}}{x e + d} - \frac {a b e^{5}}{x e + d}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 166, normalized size = 1.63 \begin {gather*} \frac {c^{2} x^{2}}{e^{2}}-\frac {a b}{\left (e x +d \right ) e}+\frac {2 a c d}{\left (e x +d \right ) e^{2}}+\frac {2 a c \ln \left (e x +d \right )}{e^{2}}+\frac {b^{2} d}{\left (e x +d \right ) e^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{e^{2}}-\frac {3 b c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 b c d \ln \left (e x +d \right )}{e^{3}}+\frac {3 b c x}{e^{2}}+\frac {2 c^{2} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 c^{2} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 c^{2} d x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 117, normalized size = 1.15 \begin {gather*} \frac {2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}}{e^{5} x + d e^{4}} + \frac {c^{2} e x^{2} - {\left (4 \, c^{2} d - 3 \, b c e\right )} x}{e^{3}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 127, normalized size = 1.25 \begin {gather*} \frac {b^2\,d\,e^2-3\,b\,c\,d^2\,e-a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2}{e\,\left (x\,e^4+d\,e^3\right )}-x\,\left (\frac {4\,c^2\,d}{e^3}-\frac {3\,b\,c}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^4}+\frac {c^2\,x^2}{e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.67, size = 126, normalized size = 1.24 \begin {gather*} \frac {c^{2} x^{2}}{e^{2}} + x \left (\frac {3 b c}{e^{2}} - \frac {4 c^{2} d}{e^{3}}\right ) + \frac {- a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} - 3 b c d^{2} e + 2 c^{2} d^{3}}{d e^{4} + e^{5} x} + \frac {\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________