Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0566795, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^8 (b+2 c x)^8}+\frac{-b^2+4 a c}{8 c^2 d^8 (b+2 c x)^6}+\frac{1}{16 c^2 d^8 (b+2 c x)^4}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3}\\ \end{align*}
Mathematica [A] time = 0.0319239, size = 59, normalized size = 0.81 \[ \frac{42 \left (b^2-4 a c\right ) (b+2 c x)^2-15 \left (b^2-4 a c\right )^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{8}} \left ( -{\frac{1}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{224\,{c}^{3} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{4\,ac-{b}^{2}}{80\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.13351, size = 238, normalized size = 3.26 \begin{align*} -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.00608, size = 379, normalized size = 5.19 \begin{align*} -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.75623, size = 187, normalized size = 2.56 \begin{align*} - \frac{30 a^{2} c^{2} + 6 a b^{2} c + b^{4} + 140 b c^{3} x^{3} + 70 c^{4} x^{4} + x^{2} \left (84 a c^{3} + 84 b^{2} c^{2}\right ) + x \left (84 a b c^{2} + 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21734, size = 117, normalized size = 1.6 \begin{align*} -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \,{\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]