Optimal. Leaf size=17 \[ -\frac {c^2}{2 e (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ -\frac {c^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac {c^2}{(d+e x)^3} \, dx\\ &=c^2 \int \frac {1}{(d+e x)^3} \, dx\\ &=-\frac {c^2}{2 e (d+e x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {c^2}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 27, normalized size = 1.59 \[ -\frac {c^{2}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 66, normalized size = 3.88 \[ -\frac {{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{2 \, {\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 16, normalized size = 0.94 \[ -\frac {c^{2}}{2 \left (e x +d \right )^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 27, normalized size = 1.59 \[ -\frac {c^{2}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 26, normalized size = 1.53 \[ -\frac {c^2}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 27, normalized size = 1.59 \[ - \frac {c^{2}}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________