Optimal. Leaf size=59 \[ \frac {-c d^2-a e^2}{4 e^3 (d+e x)^4}+\frac {2 c d}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711}
\begin {gather*} -\frac {a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac {c}{2 e^3 (d+e x)^2}+\frac {2 c d}{3 e^3 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^5} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^5}-\frac {2 c d}{e^2 (d+e x)^4}+\frac {c}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {c d^2+a e^2}{4 e^3 (d+e x)^4}+\frac {2 c d}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 40, normalized size = 0.68 \begin {gather*} -\frac {3 a e^2+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 52, normalized size = 0.88
method | result | size |
gosper | \(-\frac {6 c \,x^{2} e^{2}+4 c d e x +3 e^{2} a +c \,d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(40\) |
risch | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {c d x}{3 e^{2}}-\frac {3 e^{2} a +c \,d^{2}}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(44\) |
norman | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {c d x}{3 e^{2}}-\frac {3 a \,e^{3}+d^{2} e c}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(45\) |
default | \(\frac {2 c d}{3 e^{3} \left (e x +d \right )^{3}}-\frac {e^{2} a +c \,d^{2}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {c}{2 e^{3} \left (e x +d \right )^{2}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 69, normalized size = 1.17 \begin {gather*} -\frac {6 \, c x^{2} e^{2} + 4 \, c d x e + c d^{2} + 3 \, a e^{2}}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.29, size = 68, normalized size = 1.15 \begin {gather*} -\frac {4 \, c d x e + c d^{2} + 3 \, {\left (2 \, c x^{2} + a\right )} e^{2}}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.27, size = 80, normalized size = 1.36 \begin {gather*} \frac {- 3 a e^{2} - c d^{2} - 4 c d e x - 6 c e^{2} x^{2}}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.13, size = 59, normalized size = 1.00 \begin {gather*} -\frac {1}{12} \, {\left (\frac {6 \, c e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {8 \, c d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, c d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac {3 \, a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 77, normalized size = 1.31 \begin {gather*} -\frac {\frac {c\,d^2+3\,a\,e^2}{12\,e^3}+\frac {c\,x^2}{2\,e}+\frac {c\,d\,x}{3\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________