Optimal. Leaf size=55 \[ -\frac {c d-2 b e}{c^3 (b+c x)}+\frac {b (c d-b e)}{2 c^3 (b+c x)^2}+\frac {e \log (b+c x)}{c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ -\frac {c d-2 b e}{c^3 (b+c x)}+\frac {b (c d-b e)}{2 c^3 (b+c x)^2}+\frac {e \log (b+c x)}{c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {b (-c d+b e)}{c^2 (b+c x)^3}+\frac {c d-2 b e}{c^2 (b+c x)^2}+\frac {e}{c^2 (b+c x)}\right ) \, dx\\ &=\frac {b (c d-b e)}{2 c^3 (b+c x)^2}-\frac {c d-2 b e}{c^3 (b+c x)}+\frac {e \log (b+c x)}{c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 0.98 \[ \frac {3 b^2 e-b c (d-4 e x)+2 e (b+c x)^2 \log (b+c x)-2 c^2 d x}{2 c^3 (b+c x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 79, normalized size = 1.44 \[ -\frac {b c d - 3 \, b^{2} e + 2 \, {\left (c^{2} d - 2 \, b c e\right )} x - 2 \, {\left (c^{2} e x^{2} + 2 \, b c e x + b^{2} e\right )} \log \left (c x + b\right )}{2 \, {\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 55, normalized size = 1.00 \[ \frac {e \log \left ({\left | c x + b \right |}\right )}{c^{3}} - \frac {2 \, {\left (c d - 2 \, b e\right )} x + \frac {b c d - 3 \, b^{2} e}{c}}{2 \, {\left (c x + b\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 70, normalized size = 1.27 \[ -\frac {b^{2} e}{2 \left (c x +b \right )^{2} c^{3}}+\frac {b d}{2 \left (c x +b \right )^{2} c^{2}}+\frac {2 b e}{\left (c x +b \right ) c^{3}}-\frac {d}{\left (c x +b \right ) c^{2}}+\frac {e \ln \left (c x +b \right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.91, size = 63, normalized size = 1.15 \[ -\frac {b c d - 3 \, b^{2} e + 2 \, {\left (c^{2} d - 2 \, b c e\right )} x}{2 \, {\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} + \frac {e \log \left (c x + b\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.05, size = 63, normalized size = 1.15 \[ \frac {\frac {3\,b^2\,e-b\,c\,d}{2\,c^3}+\frac {x\,\left (2\,b\,e-c\,d\right )}{c^2}}{b^2+2\,b\,c\,x+c^2\,x^2}+\frac {e\,\ln \left (b+c\,x\right )}{c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.36, size = 63, normalized size = 1.15 \[ \frac {3 b^{2} e - b c d + x \left (4 b c e - 2 c^{2} d\right )}{2 b^{2} c^{3} + 4 b c^{4} x + 2 c^{5} x^{2}} + \frac {e \log {\left (b + c x \right )}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________