Optimal. Leaf size=94 \[ -\frac{\left (b^2-4 a c\right ) (b+2 c x)^3}{128 c^4 d^2}+\frac{3 x \left (b^2-4 a c\right )^2}{64 c^3 d^2}+\frac{\left (b^2-4 a c\right )^3}{128 c^4 d^2 (b+2 c x)}+\frac{(b+2 c x)^5}{640 c^4 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0945815, antiderivative size = 94, 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} \[ -\frac{\left (b^2-4 a c\right ) (b+2 c x)^3}{128 c^4 d^2}+\frac{3 x \left (b^2-4 a c\right )^2}{64 c^3 d^2}+\frac{\left (b^2-4 a c\right )^3}{128 c^4 d^2 (b+2 c x)}+\frac{(b+2 c x)^5}{640 c^4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^2} \, dx &=\int \left (\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2}+\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^2}{64 c^3 d^4}+\frac{(b d+2 c d x)^4}{64 c^3 d^6}\right ) \, dx\\ &=\frac{3 \left (b^2-4 a c\right )^2 x}{64 c^3 d^2}+\frac{\left (b^2-4 a c\right )^3}{128 c^4 d^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right ) (b+2 c x)^3}{128 c^4 d^2}+\frac{(b+2 c x)^5}{640 c^4 d^2}\\ \end{align*}
Mathematica [A] time = 0.0791392, size = 101, normalized size = 1.07 \[ \frac{\frac{10 x \left (48 a^2 c^2-12 a b^2 c+b^4\right )}{c^3}-\frac{20 b x^2 \left (b^2-12 a c\right )}{c^2}+\frac{5 \left (b^2-4 a c\right )^3}{c^4 (b+2 c x)}+\frac{40 x^3 \left (4 a c+b^2\right )}{c}+80 b x^4+32 c x^5}{640 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 135, normalized size = 1.4 \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{1}{64\,{c}^{3}} \left ({\frac{16\,{x}^{5}{c}^{4}}{5}}+8\,b{x}^{4}{c}^{3}+16\,{x}^{3}a{c}^{3}+4\,{b}^{2}{c}^{2}{x}^{3}+24\,{x}^{2}ab{c}^{2}-2\,{x}^{2}{b}^{3}c+48\,{a}^{2}{c}^{2}x-12\,ac{b}^{2}x+{b}^{4}x \right ) }-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{128\,{c}^{4} \left ( 2\,cx+b \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16678, size = 186, normalized size = 1.98 \begin{align*} \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{128 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} + \frac{16 \, c^{4} x^{5} + 40 \, b c^{3} x^{4} + 20 \,{\left (b^{2} c^{2} + 4 \, a c^{3}\right )} x^{3} - 10 \,{\left (b^{3} c - 12 \, a b c^{2}\right )} x^{2} + 5 \,{\left (b^{4} - 12 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} x}{320 \, c^{3} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.90991, size = 305, normalized size = 3.24 \begin{align*} \frac{64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 640 \, a b c^{4} x^{3} + 960 \, a^{2} c^{4} x^{2} + 5 \, b^{6} - 60 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3} + 160 \,{\left (b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 10 \,{\left (b^{5} c - 12 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{640 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.973318, size = 143, normalized size = 1.52 \begin{align*} \frac{b x^{4}}{8 d^{2}} + \frac{c x^{5}}{20 d^{2}} - \frac{64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}}{128 b c^{4} d^{2} + 256 c^{5} d^{2} x} + \frac{x^{3} \left (4 a c + b^{2}\right )}{16 c d^{2}} + \frac{x^{2} \left (12 a b c - b^{3}\right )}{32 c^{2} d^{2}} + \frac{x \left (48 a^{2} c^{2} - 12 a b^{2} c + b^{4}\right )}{64 c^{3} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.22794, size = 298, normalized size = 3.17 \begin{align*} \frac{{\left (\frac{15 \, b^{4} d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{120 \, a b^{2} c d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac{240 \, a^{2} c^{2} d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{5 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{20 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + 1\right )}{\left (2 \, c d x + b d\right )}^{5}}{640 \, c^{4} d^{7}} + \frac{\frac{b^{6} c^{5} d^{11}}{2 \, c d x + b d} - \frac{12 \, a b^{4} c^{6} d^{11}}{2 \, c d x + b d} + \frac{48 \, a^{2} b^{2} c^{7} d^{11}}{2 \, c d x + b d} - \frac{64 \, a^{3} c^{8} d^{11}}{2 \, c d x + b d}}{128 \, c^{9} d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]