Optimal. Leaf size=110 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0685907, antiderivative size = 110, normalized size of antiderivative = 1., 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 = {697} \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^6}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^5}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^4}-\frac{4 c^2 d}{e^4 (d+e x)^3}+\frac{c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c d \left (c d^2+a e^2\right )}{e^5 (d+e x)^4}-\frac{2 c \left (3 c d^2+a e^2\right )}{3 e^5 (d+e x)^3}+\frac{2 c^2 d}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0374355, size = 90, normalized size = 0.82 \[ -\frac{3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{15 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 119, normalized size = 1.1 \begin{align*}{\frac{cd \left ( a{e}^{2}+c{d}^{2} \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{2\,c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.33863, size = 204, normalized size = 1.85 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80105, size = 312, normalized size = 2.84 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.3188, size = 160, normalized size = 1.45 \begin{align*} - \frac{3 a^{2} e^{4} + a c d^{2} e^{2} + 3 c^{2} d^{4} + 30 c^{2} d e^{3} x^{3} + 15 c^{2} e^{4} x^{4} + x^{2} \left (10 a c e^{4} + 30 c^{2} d^{2} e^{2}\right ) + x \left (5 a c d e^{3} + 15 c^{2} d^{3} e\right )}{15 d^{5} e^{5} + 75 d^{4} e^{6} x + 150 d^{3} e^{7} x^{2} + 150 d^{2} e^{8} x^{3} + 75 d e^{9} x^{4} + 15 e^{10} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29556, size = 132, normalized size = 1.2 \begin{align*} -\frac{{\left (15 \, c^{2} x^{4} e^{4} + 30 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 15 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c x^{2} e^{4} + 5 \, a c d x e^{3} + a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]