Optimal. Leaf size=260 \[ \frac{3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac{(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32271, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac{(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{d+e x} \, dx &=\int \left (\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^2}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^3}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^4}{e^6}+\frac{c^3 (d+e x)^5}{e^6}\right ) \, dx\\ &=-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 x}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{2 e^7}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^3}{3 e^7}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^4}{4 e^7}-\frac{3 c^2 (2 c d-b e) (d+e x)^5}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7}+\frac{\left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.159707, size = 308, normalized size = 1.18 \[ \frac{e x \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+c^3 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{60 e^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.044, size = 546, normalized size = 2.1 \begin{align*}{\frac{3\,b{x}^{5}{c}^{2}}{5\,e}}+{\frac{{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}+6\,{\frac{abc{d}^{2}x}{{e}^{3}}}-3\,{\frac{ab{x}^{2}cd}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) abc{d}^{3}}{{e}^{4}}}+3\,{\frac{b{a}^{2}x}{e}}-{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{6}}{{e}^{7}}}+{\frac{{c}^{3}{x}^{6}}{6\,e}}+{\frac{3\,{a}^{2}{x}^{2}c}{2\,e}}+{\frac{{x}^{4}{c}^{3}{d}^{2}}{4\,{e}^{3}}}-{\frac{{x}^{2}{b}^{3}d}{2\,{e}^{2}}}+{\frac{3\,{x}^{4}{b}^{2}c}{4\,e}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2\,e}}-3\,{\frac{\ln \left ( ex+d \right ){a}^{2}bd}{{e}^{2}}}+2\,{\frac{a{x}^{3}bc}{e}}+{\frac{3\,a{x}^{4}{c}^{2}}{4\,e}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}+{\frac{{x}^{3}{b}^{3}}{3\,e}}-3\,{\frac{a{b}^{2}dx}{{e}^{2}}}-3\,{\frac{a{c}^{2}{d}^{3}x}{{e}^{4}}}-3\,{\frac{{b}^{2}c{d}^{3}x}{{e}^{4}}}-{\frac{3\,b{x}^{2}{c}^{2}{d}^{3}}{2\,{e}^{4}}}-3\,{\frac{cd{a}^{2}x}{{e}^{2}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}-{\frac{3\,b{x}^{4}{c}^{2}d}{4\,{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}c{d}^{2}}{{e}^{3}}}-{\frac{{b}^{2}c{x}^{3}d}{{e}^{2}}}+{\frac{3\,a{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{3}}}+{\frac{3\,{b}^{2}{x}^{2}c{d}^{2}}{2\,{e}^{3}}}+3\,{\frac{{d}^{4}b{c}^{2}x}{{e}^{5}}}-{\frac{a{x}^{3}{c}^{2}d}{{e}^{2}}}+{\frac{b{x}^{3}{c}^{2}{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{4}}{{e}^{5}}}+3\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{4}}{{e}^{5}}}-3\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{5}}{{e}^{6}}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.968411, size = 541, normalized size = 2.08 \begin{align*} \frac{10 \, c^{3} e^{5} x^{6} - 12 \,{\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02334, size = 829, normalized size = 3.19 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} - 12 \,{\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, a^{2} b e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.1198, size = 376, normalized size = 1.45 \begin{align*} \frac{c^{3} x^{6}}{6 e} + \frac{x^{5} \left (3 b c^{2} e - c^{3} d\right )}{5 e^{2}} + \frac{x^{4} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 3 b c^{2} d e + c^{3} d^{2}\right )}{4 e^{3}} + \frac{x^{3} \left (6 a b c e^{3} - 3 a c^{2} d e^{2} + b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - c^{3} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 6 a b c d e^{3} + 3 a c^{2} d^{2} e^{2} - b^{3} d e^{3} + 3 b^{2} c d^{2} e^{2} - 3 b c^{2} d^{3} e + c^{3} d^{4}\right )}{2 e^{5}} + \frac{x \left (3 a^{2} b e^{5} - 3 a^{2} c d e^{4} - 3 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 3 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 3 b^{2} c d^{3} e^{2} + 3 b c^{2} d^{4} e - c^{3} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12113, size = 621, normalized size = 2.39 \begin{align*}{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} + 45 \, a c^{2} x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} - 180 \, a c^{2} d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} + 120 \, a b c x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} - 180 \, a b c d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3} + 360 \, a b c d^{2} x e^{3} + 90 \, a b^{2} x^{2} e^{5} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a b^{2} d x e^{4} - 180 \, a^{2} c d x e^{4} + 180 \, a^{2} b x e^{5}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]