Optimal. Leaf size=63 \[ \frac {c^2 d^2 x}{e^2}-\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} -\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac {c^2 d^2 x}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^2} \, dx\\ &=\int \left (\frac {c^2 d^2}{e^2}+\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^2}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {c^2 d^2 x}{e^2}-\frac {\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac {2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 59, normalized size = 0.94 \begin {gather*} \frac {c^2 d^2 e x-\frac {\left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (-c d^2+a e^2\right ) \log (d+e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.76, size = 75, normalized size = 1.19
method | result | size |
default | \(\frac {c^{2} d^{2} x}{e^{2}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{e^{3} \left (e x +d \right )}+\frac {2 c d \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(75\) |
risch | \(\frac {c^{2} d^{2} x}{e^{2}}-\frac {e \,a^{2}}{e x +d}+\frac {2 a c \,d^{2}}{e \left (e x +d \right )}-\frac {c^{2} d^{4}}{e^{3} \left (e x +d \right )}+\frac {2 c d \ln \left (e x +d \right ) a}{e}-\frac {2 c^{2} d^{3} \ln \left (e x +d \right )}{e^{3}}\) | \(92\) |
norman | \(\frac {e \,c^{2} d^{2} x^{4}-\frac {d^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+4 c^{2} d^{4}\right )}{e^{3}}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) x^{2}}{e}-\frac {d \left (2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+10 c^{2} d^{4}\right ) x}{e^{2}}}{\left (e x +d \right )^{3}}+\frac {2 c d \left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 73, normalized size = 1.16 \begin {gather*} c^{2} d^{2} x e^{\left (-2\right )} - 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\right ) - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{x e^{4} + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.88, size = 103, normalized size = 1.63 \begin {gather*} \frac {c^{2} d^{3} x e - c^{2} d^{4} - a^{2} e^{4} + {\left (c^{2} d^{2} x^{2} + 2 \, a c d^{2}\right )} e^{2} - 2 \, {\left (c^{2} d^{3} x e + c^{2} d^{4} - a c d x e^{3} - a c d^{2} e^{2}\right )} \log \left (x e + d\right )}{x e^{4} + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.20, size = 71, normalized size = 1.13 \begin {gather*} \frac {c^{2} d^{2} x}{e^{2}} + \frac {2 c d \left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{4} + 2 a c d^{2} e^{2} - c^{2} d^{4}}{d e^{3} + e^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.87, size = 73, normalized size = 1.16 \begin {gather*} c^{2} d^{2} x e^{\left (-2\right )} - 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.61, size = 83, normalized size = 1.32 \begin {gather*} \frac {c^2\,d^2\,x}{e^2}-\frac {a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}{e\,\left (x\,e^3+d\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________