Optimal. Leaf size=94 \[ -\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^2}{2 e^3}-\frac {c^2 d x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {\left (c d^2+a e^2\right )^2 \log (d+e x)}{e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711}
\begin {gather*} \frac {\left (a e^2+c d^2\right )^2 \log (d+e x)}{e^5}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {c x^2 \left (2 a e^2+c d^2\right )}{2 e^3}-\frac {c^2 d x^3}{3 e^2}+\frac {c^2 x^4}{4 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^2}{d+e x} \, dx &=\int \left (-\frac {c d \left (c d^2+2 a e^2\right )}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x}{e^3}-\frac {c^2 d x^2}{e^2}+\frac {c^2 x^3}{e}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^2}{2 e^3}-\frac {c^2 d x^3}{3 e^2}+\frac {c^2 x^4}{4 e}+\frac {\left (c d^2+a e^2\right )^2 \log (d+e x)}{e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 79, normalized size = 0.84 \begin {gather*} \frac {c e x \left (12 a e^2 (-2 d+e x)+c \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (c d^2+a e^2\right )^2 \log (d+e x)}{12 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.48, size = 96, normalized size = 1.02
method | result | size |
default | \(-\frac {c \left (-\frac {c \,x^{4} e^{3}}{4}+\frac {c d \,e^{2} x^{3}}{3}-\frac {\left (2 e^{2} a +c \,d^{2}\right ) x^{2} e}{2}+d \left (2 e^{2} a +c \,d^{2}\right ) x \right )}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(96\) |
norman | \(\frac {c^{2} x^{4}}{4 e}+\frac {c \left (2 e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {c^{2} d \,x^{3}}{3 e^{2}}-\frac {c d \left (2 e^{2} a +c \,d^{2}\right ) x}{e^{4}}+\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(101\) |
risch | \(\frac {c^{2} x^{4}}{4 e}-\frac {c^{2} d \,x^{3}}{3 e^{2}}+\frac {c a \,x^{2}}{e}+\frac {c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {2 c a d x}{e^{2}}-\frac {c^{2} d^{3} x}{e^{4}}+\frac {\ln \left (e x +d \right ) a^{2}}{e}+\frac {2 \ln \left (e x +d \right ) a c \,d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) c^{2} d^{4}}{e^{5}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 99, normalized size = 1.05 \begin {gather*} {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, {\left (c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{2} - 12 \, {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.64, size = 99, normalized size = 1.05 \begin {gather*} \frac {1}{12} \, {\left (6 \, c^{2} d^{2} x^{2} e^{2} - 12 \, c^{2} d^{3} x e + 3 \, {\left (c^{2} x^{4} + 4 \, a c x^{2}\right )} e^{4} - 4 \, {\left (c^{2} d x^{3} + 6 \, a c d x\right )} e^{3} + 12 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.12, size = 88, normalized size = 0.94 \begin {gather*} - \frac {c^{2} d x^{3}}{3 e^{2}} + \frac {c^{2} x^{4}}{4 e} + x^{2} \left (\frac {a c}{e} + \frac {c^{2} d^{2}}{2 e^{3}}\right ) + x \left (- \frac {2 a c d}{e^{2}} - \frac {c^{2} d^{3}}{e^{4}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.02, size = 100, normalized size = 1.06 \begin {gather*} {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 12 \, a c x^{2} e^{3} - 24 \, a c d x e^{2}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 106, normalized size = 1.13 \begin {gather*} x^2\,\left (\frac {c^2\,d^2}{2\,e^3}+\frac {a\,c}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{e^5}+\frac {c^2\,x^4}{4\,e}-\frac {c^2\,d\,x^3}{3\,e^2}-\frac {d\,x\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,a\,c}{e}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________