Optimal. Leaf size=181 \[ \frac{3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac{e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0923914, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {646, 43} \[ \frac{3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac{3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac{(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac{e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^3 \, dx\\ &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac{(b d-a e)^3 \left (a b+b^2 x\right )^{2 p}}{b^3}+\frac{3 e (b d-a e)^2 \left (a b+b^2 x\right )^{1+2 p}}{b^4}+\frac{3 e^2 (b d-a e) \left (a b+b^2 x\right )^{2+2 p}}{b^5}+\frac{e^3 \left (a b+b^2 x\right )^{3+2 p}}{b^6}\right ) \, dx\\ &=\frac{(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac{3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac{3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac{e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0857858, size = 107, normalized size = 0.59 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p \left (\frac{6 e^2 (a+b x)^2 (b d-a e)}{2 p+3}+\frac{3 e (a+b x) (b d-a e)^2}{p+1}+\frac{2 (b d-a e)^3}{2 p+1}+\frac{e^3 (a+b x)^3}{p+2}\right )}{2 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 405, normalized size = 2.2 \begin{align*} -{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( -4\,{b}^{3}{e}^{3}{p}^{3}{x}^{3}-12\,{b}^{3}d{e}^{2}{p}^{3}{x}^{2}-12\,{b}^{3}{e}^{3}{p}^{2}{x}^{3}+6\,a{b}^{2}{e}^{3}{p}^{2}{x}^{2}-12\,{b}^{3}{d}^{2}e{p}^{3}x-42\,{b}^{3}d{e}^{2}{p}^{2}{x}^{2}-11\,{b}^{3}{e}^{3}p{x}^{3}+12\,a{b}^{2}d{e}^{2}{p}^{2}x+9\,a{b}^{2}{e}^{3}p{x}^{2}-4\,{b}^{3}{d}^{3}{p}^{3}-48\,{b}^{3}{d}^{2}e{p}^{2}x-42\,{b}^{3}d{e}^{2}p{x}^{2}-3\,{x}^{3}{b}^{3}{e}^{3}-6\,{a}^{2}b{e}^{3}px+6\,a{b}^{2}{d}^{2}e{p}^{2}+30\,a{b}^{2}d{e}^{2}px+3\,{x}^{2}a{b}^{2}{e}^{3}-18\,{b}^{3}{d}^{3}{p}^{2}-57\,{b}^{3}{d}^{2}epx-12\,{x}^{2}{b}^{3}d{e}^{2}-6\,{a}^{2}bd{e}^{2}p-3\,x{a}^{2}b{e}^{3}+21\,a{b}^{2}{d}^{2}ep+12\,xa{b}^{2}d{e}^{2}-26\,{b}^{3}{d}^{3}p-18\,x{b}^{3}{d}^{2}e+3\,{a}^{3}{e}^{3}-12\,d{e}^{2}{a}^{2}b+18\,a{b}^{2}{d}^{2}e-12\,{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{4} \left ( 4\,{p}^{4}+20\,{p}^{3}+35\,{p}^{2}+25\,p+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12158, size = 373, normalized size = 2.06 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} d^{3}}{b{\left (2 \, p + 1\right )}} + \frac{3 \,{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} d^{2} e}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{3 \,{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} d e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )}{\left (b x + a\right )}^{2 \, p} e^{3}}{2 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.72555, size = 1053, normalized size = 5.82 \begin{align*} \frac{{\left (4 \, a b^{3} d^{3} p^{3} + 12 \, a b^{3} d^{3} - 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} - 3 \, a^{4} e^{3} +{\left (4 \, b^{4} e^{3} p^{3} + 12 \, b^{4} e^{3} p^{2} + 11 \, b^{4} e^{3} p + 3 \, b^{4} e^{3}\right )} x^{4} + 2 \,{\left (6 \, b^{4} d e^{2} + 2 \,{\left (3 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{3} + 3 \,{\left (7 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p^{2} +{\left (21 \, b^{4} d e^{2} + a b^{3} e^{3}\right )} p\right )} x^{3} + 6 \,{\left (3 \, a b^{3} d^{3} - a^{2} b^{2} d^{2} e\right )} p^{2} + 3 \,{\left (6 \, b^{4} d^{2} e + 4 \,{\left (b^{4} d^{2} e + a b^{3} d e^{2}\right )} p^{3} + 2 \,{\left (8 \, b^{4} d^{2} e + 5 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p^{2} +{\left (19 \, b^{4} d^{2} e + 4 \, a b^{3} d e^{2} - a^{2} b^{2} e^{3}\right )} p\right )} x^{2} +{\left (26 \, a b^{3} d^{3} - 21 \, a^{2} b^{2} d^{2} e + 6 \, a^{3} b d e^{2}\right )} p + 2 \,{\left (6 \, b^{4} d^{3} + 2 \,{\left (b^{4} d^{3} + 3 \, a b^{3} d^{2} e\right )} p^{3} + 3 \,{\left (3 \, b^{4} d^{3} + 7 \, a b^{3} d^{2} e - 2 \, a^{2} b^{2} d e^{2}\right )} p^{2} +{\left (13 \, b^{4} d^{3} + 18 \, a b^{3} d^{2} e - 12 \, a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23718, size = 1710, normalized size = 9.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]