Optimal. Leaf size=65 \[ \frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0877685, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^4 (c+d x)^2 \, dx &=\int \left (\frac{(b c-a d)^2 (a+b x)^4}{b^2}+\frac{2 d (b c-a d) (a+b x)^5}{b^2}+\frac{d^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac{d (b c-a d) (a+b x)^6}{3 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}\\ \end{align*}
Mathematica [B] time = 0.0257322, size = 148, normalized size = 2.28 \[ \frac{1}{5} b^2 x^5 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a b x^4 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+a^3 c x^2 (a d+2 b c)+a^4 c^2 x+\frac{1}{3} b^3 d x^6 (2 a d+b c)+\frac{1}{7} b^4 d^2 x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.002, size = 163, normalized size = 2.5 \begin{align*}{\frac{{b}^{4}{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( 4\,a{b}^{3}{d}^{2}+2\,{b}^{4}cd \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{b}^{2}{a}^{2}{d}^{2}+8\,a{b}^{3}cd+{b}^{4}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{3}b{d}^{2}+12\,{b}^{2}{a}^{2}cd+4\,a{b}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{4}{d}^{2}+8\,{a}^{3}bcd+6\,{b}^{2}{a}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{4}cd+4\,{a}^{3}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{4}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.96925, size = 211, normalized size = 3.25 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac{1}{3} \,{\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} +{\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} +{\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.72042, size = 363, normalized size = 5.58 \begin{align*} \frac{1}{7} x^{7} d^{2} b^{4} + \frac{1}{3} x^{6} d c b^{4} + \frac{2}{3} x^{6} d^{2} b^{3} a + \frac{1}{5} x^{5} c^{2} b^{4} + \frac{8}{5} x^{5} d c b^{3} a + \frac{6}{5} x^{5} d^{2} b^{2} a^{2} + x^{4} c^{2} b^{3} a + 3 x^{4} d c b^{2} a^{2} + x^{4} d^{2} b a^{3} + 2 x^{3} c^{2} b^{2} a^{2} + \frac{8}{3} x^{3} d c b a^{3} + \frac{1}{3} x^{3} d^{2} a^{4} + 2 x^{2} c^{2} b a^{3} + x^{2} d c a^{4} + x c^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.089372, size = 168, normalized size = 2.58 \begin{align*} a^{4} c^{2} x + \frac{b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac{2 a b^{3} d^{2}}{3} + \frac{b^{4} c d}{3}\right ) + x^{5} \left (\frac{6 a^{2} b^{2} d^{2}}{5} + \frac{8 a b^{3} c d}{5} + \frac{b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac{a^{4} d^{2}}{3} + \frac{8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.05064, size = 230, normalized size = 3.54 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + \frac{1}{3} \, b^{4} c d x^{6} + \frac{2}{3} \, a b^{3} d^{2} x^{6} + \frac{1}{5} \, b^{4} c^{2} x^{5} + \frac{8}{5} \, a b^{3} c d x^{5} + \frac{6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac{8}{3} \, a^{3} b c d x^{3} + \frac{1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]