Optimal. Leaf size=74 \[ \frac {e^2 x}{c^2 d^2}-\frac {\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac {2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 74, 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}{c^3 d^3 (a e+c d x)}+\frac {2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}+\frac {e^2 x}{c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^2}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {e^2}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^2}{c^2 d^2 (a e+c d x)^2}+\frac {2 \left (c d^2 e-a e^3\right )}{c^2 d^2 (a e+c d x)}\right ) \, dx\\ &=\frac {e^2 x}{c^2 d^2}-\frac {\left (c d^2-a e^2\right )^2}{c^3 d^3 (a e+c d x)}+\frac {2 e \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^3 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 65, normalized size = 0.88 \begin {gather*} \frac {c d e^2 x-\frac {\left (c d^2-a e^2\right )^2}{a e+c d x}+2 \left (c d^2 e-a e^3\right ) \log (a e+c d x)}{c^3 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.71, size = 86, normalized size = 1.16
method | result | size |
default | \(\frac {e^{2} x}{d^{2} c^{2}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{c^{3} d^{3} \left (c d x +a e \right )}-\frac {2 e \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{3} d^{3}}\) | \(86\) |
risch | \(\frac {e^{2} x}{d^{2} c^{2}}-\frac {a^{2} e^{4}}{c^{3} d^{3} \left (c d x +a e \right )}+\frac {2 a \,e^{2}}{c^{2} d \left (c d x +a e \right )}-\frac {d}{c \left (c d x +a e \right )}-\frac {2 e^{3} \ln \left (c d x +a e \right ) a}{c^{3} d^{3}}+\frac {2 e \ln \left (c d x +a e \right )}{c^{2} d}\) | \(114\) |
norman | \(\frac {\frac {e^{3} x^{3}}{c d}-\frac {2 a^{2} e^{4}-a c \,d^{2} e^{2}+c^{2} d^{4}}{d^{2} c^{3}}-\frac {\left (2 a^{2} e^{6}-a c \,d^{2} e^{4}+2 c^{2} d^{4} e^{2}\right ) x}{c^{3} d^{3} e}}{\left (c d x +a e \right ) \left (e x +d \right )}-\frac {2 e \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{3} d^{3}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 88, normalized size = 1.19 \begin {gather*} -\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{c^{4} d^{4} x + a c^{3} d^{3} e} + \frac {x e^{2}}{c^{2} d^{2}} + \frac {2 \, {\left (c d^{2} e - a e^{3}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.95, size = 113, normalized size = 1.53 \begin {gather*} -\frac {c^{2} d^{4} - a c d x e^{3} + 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 - a c d x e^{3} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4} x + a c^{3} d^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.21, size = 85, normalized size = 1.15 \begin {gather*} \frac {- a^{2} e^{4} + 2 a c d^{2} e^{2} - c^{2} d^{4}}{a c^{3} d^{3} e + c^{4} d^{4} x} + \frac {e^{2} x}{c^{2} d^{2}} - \frac {2 e \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.82, size = 85, normalized size = 1.15 \begin {gather*} \frac {x e^{2}}{c^{2} d^{2}} + \frac {2 \, {\left (c d^{2} e - a e^{3}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{3} d^{3}} - \frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{{\left (c d x + a e\right )} c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.62, size = 96, normalized size = 1.30 \begin {gather*} \frac {e^2\,x}{c^2\,d^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (2\,a\,e^3-2\,c\,d^2\,e\right )}{c^3\,d^3}-\frac {a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}{c\,d\,\left (x\,c^3\,d^3+a\,e\,c^2\,d^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________