3.1895 \(\int (a+b x) (d+e x)^5 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=92 \[ -\frac{3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac{3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac{b^3 (d+e x)^9}{9 e^4} \]

[Out]

-((b*d - a*e)^3*(d + e*x)^6)/(6*e^4) + (3*b*(b*d - a*e)^2*(d + e*x)^7)/(7*e^4) - (3*b^2*(b*d - a*e)*(d + e*x)^
8)/(8*e^4) + (b^3*(d + e*x)^9)/(9*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.154948, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ -\frac{3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac{3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac{b^3 (d+e x)^9}{9 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

-((b*d - a*e)^3*(d + e*x)^6)/(6*e^4) + (3*b*(b*d - a*e)^2*(d + e*x)^7)/(7*e^4) - (3*b^2*(b*d - a*e)*(d + e*x)^
8)/(8*e^4) + (b^3*(d + e*x)^9)/(9*e^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^5 \, dx\\ &=\int \left (\frac{(-b d+a e)^3 (d+e x)^5}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^6}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^7}{e^3}+\frac{b^3 (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac{(b d-a e)^3 (d+e x)^6}{6 e^4}+\frac{3 b (b d-a e)^2 (d+e x)^7}{7 e^4}-\frac{3 b^2 (b d-a e) (d+e x)^8}{8 e^4}+\frac{b^3 (d+e x)^9}{9 e^4}\\ \end{align*}

Mathematica [B]  time = 0.0408716, size = 267, normalized size = 2.9 \[ \frac{1}{7} b e^3 x^7 \left (3 a^2 e^2+15 a b d e+10 b^2 d^2\right )+\frac{1}{6} e^2 x^6 \left (15 a^2 b d e^2+a^3 e^3+30 a b^2 d^2 e+10 b^3 d^3\right )+d e x^5 \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{4} d^2 x^4 \left (30 a^2 b d e^2+10 a^3 e^3+15 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{3} a d^3 x^3 \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^2 d^4 x^2 (5 a e+3 b d)+a^3 d^5 x+\frac{1}{8} b^2 e^4 x^8 (3 a e+5 b d)+\frac{1}{9} b^3 e^5 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

a^3*d^5*x + (a^2*d^4*(3*b*d + 5*a*e)*x^2)/2 + (a*d^3*(3*b^2*d^2 + 15*a*b*d*e + 10*a^2*e^2)*x^3)/3 + (d^2*(b^3*
d^3 + 15*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 10*a^3*e^3)*x^4)/4 + d*e*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^
3*e^3)*x^5 + (e^2*(10*b^3*d^3 + 30*a*b^2*d^2*e + 15*a^2*b*d*e^2 + a^3*e^3)*x^6)/6 + (b*e^3*(10*b^2*d^2 + 15*a*
b*d*e + 3*a^2*e^2)*x^7)/7 + (b^2*e^4*(5*b*d + 3*a*e)*x^8)/8 + (b^3*e^5*x^9)/9

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 394, normalized size = 4.3 \begin{align*}{\frac{{b}^{3}{e}^{5}{x}^{9}}{9}}+{\frac{ \left ( \left ( a{e}^{5}+5\,bd{e}^{4} \right ){b}^{2}+2\,{b}^{2}{e}^{5}a \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){b}^{2}+2\, \left ( a{e}^{5}+5\,bd{e}^{4} \right ) ab+{a}^{2}b{e}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){b}^{2}+2\, \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ) ab+ \left ( a{e}^{5}+5\,bd{e}^{4} \right ){a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){b}^{2}+2\, \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ) ab+ \left ( 5\,ad{e}^{4}+10\,b{d}^{2}{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){b}^{2}+2\, \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ) ab+ \left ( 10\,a{d}^{2}{e}^{3}+10\,b{d}^{3}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{d}^{5}{b}^{2}+2\, \left ( 5\,a{d}^{4}e+b{d}^{5} \right ) ab+ \left ( 10\,a{d}^{3}{e}^{2}+5\,b{d}^{4}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{5}b+ \left ( 5\,a{d}^{4}e+b{d}^{5} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{5}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/9*b^3*e^5*x^9+1/8*((a*e^5+5*b*d*e^4)*b^2+2*b^2*e^5*a)*x^8+1/7*((5*a*d*e^4+10*b*d^2*e^3)*b^2+2*(a*e^5+5*b*d*e
^4)*a*b+a^2*b*e^5)*x^7+1/6*((10*a*d^2*e^3+10*b*d^3*e^2)*b^2+2*(5*a*d*e^4+10*b*d^2*e^3)*a*b+(a*e^5+5*b*d*e^4)*a
^2)*x^6+1/5*((10*a*d^3*e^2+5*b*d^4*e)*b^2+2*(10*a*d^2*e^3+10*b*d^3*e^2)*a*b+(5*a*d*e^4+10*b*d^2*e^3)*a^2)*x^5+
1/4*((5*a*d^4*e+b*d^5)*b^2+2*(10*a*d^3*e^2+5*b*d^4*e)*a*b+(10*a*d^2*e^3+10*b*d^3*e^2)*a^2)*x^4+1/3*(a*d^5*b^2+
2*(5*a*d^4*e+b*d^5)*a*b+(10*a*d^3*e^2+5*b*d^4*e)*a^2)*x^3+1/2*(2*a^2*d^5*b+(5*a*d^4*e+b*d^5)*a^2)*x^2+a^3*d^5*
x

________________________________________________________________________________________

Maxima [B]  time = 0.972703, size = 374, normalized size = 4.07 \begin{align*} \frac{1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac{1}{8} \,{\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} +{\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(10*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a
^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*
d^3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3
)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4*e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.31261, size = 651, normalized size = 7.08 \begin{align*} \frac{1}{9} x^{9} e^{5} b^{3} + \frac{5}{8} x^{8} e^{4} d b^{3} + \frac{3}{8} x^{8} e^{5} b^{2} a + \frac{10}{7} x^{7} e^{3} d^{2} b^{3} + \frac{15}{7} x^{7} e^{4} d b^{2} a + \frac{3}{7} x^{7} e^{5} b a^{2} + \frac{5}{3} x^{6} e^{2} d^{3} b^{3} + 5 x^{6} e^{3} d^{2} b^{2} a + \frac{5}{2} x^{6} e^{4} d b a^{2} + \frac{1}{6} x^{6} e^{5} a^{3} + x^{5} e d^{4} b^{3} + 6 x^{5} e^{2} d^{3} b^{2} a + 6 x^{5} e^{3} d^{2} b a^{2} + x^{5} e^{4} d a^{3} + \frac{1}{4} x^{4} d^{5} b^{3} + \frac{15}{4} x^{4} e d^{4} b^{2} a + \frac{15}{2} x^{4} e^{2} d^{3} b a^{2} + \frac{5}{2} x^{4} e^{3} d^{2} a^{3} + x^{3} d^{5} b^{2} a + 5 x^{3} e d^{4} b a^{2} + \frac{10}{3} x^{3} e^{2} d^{3} a^{3} + \frac{3}{2} x^{2} d^{5} b a^{2} + \frac{5}{2} x^{2} e d^{4} a^{3} + x d^{5} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

1/9*x^9*e^5*b^3 + 5/8*x^8*e^4*d*b^3 + 3/8*x^8*e^5*b^2*a + 10/7*x^7*e^3*d^2*b^3 + 15/7*x^7*e^4*d*b^2*a + 3/7*x^
7*e^5*b*a^2 + 5/3*x^6*e^2*d^3*b^3 + 5*x^6*e^3*d^2*b^2*a + 5/2*x^6*e^4*d*b*a^2 + 1/6*x^6*e^5*a^3 + x^5*e*d^4*b^
3 + 6*x^5*e^2*d^3*b^2*a + 6*x^5*e^3*d^2*b*a^2 + x^5*e^4*d*a^3 + 1/4*x^4*d^5*b^3 + 15/4*x^4*e*d^4*b^2*a + 15/2*
x^4*e^2*d^3*b*a^2 + 5/2*x^4*e^3*d^2*a^3 + x^3*d^5*b^2*a + 5*x^3*e*d^4*b*a^2 + 10/3*x^3*e^2*d^3*a^3 + 3/2*x^2*d
^5*b*a^2 + 5/2*x^2*e*d^4*a^3 + x*d^5*a^3

________________________________________________________________________________________

Sympy [B]  time = 0.105917, size = 308, normalized size = 3.35 \begin{align*} a^{3} d^{5} x + \frac{b^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac{3 a b^{2} e^{5}}{8} + \frac{5 b^{3} d e^{4}}{8}\right ) + x^{7} \left (\frac{3 a^{2} b e^{5}}{7} + \frac{15 a b^{2} d e^{4}}{7} + \frac{10 b^{3} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{5}}{6} + \frac{5 a^{2} b d e^{4}}{2} + 5 a b^{2} d^{2} e^{3} + \frac{5 b^{3} d^{3} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} b d^{2} e^{3} + 6 a b^{2} d^{3} e^{2} + b^{3} d^{4} e\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{3}}{2} + \frac{15 a^{2} b d^{3} e^{2}}{2} + \frac{15 a b^{2} d^{4} e}{4} + \frac{b^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{2}}{3} + 5 a^{2} b d^{4} e + a b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{3} d^{4} e}{2} + \frac{3 a^{2} b d^{5}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

a**3*d**5*x + b**3*e**5*x**9/9 + x**8*(3*a*b**2*e**5/8 + 5*b**3*d*e**4/8) + x**7*(3*a**2*b*e**5/7 + 15*a*b**2*
d*e**4/7 + 10*b**3*d**2*e**3/7) + x**6*(a**3*e**5/6 + 5*a**2*b*d*e**4/2 + 5*a*b**2*d**2*e**3 + 5*b**3*d**3*e**
2/3) + x**5*(a**3*d*e**4 + 6*a**2*b*d**2*e**3 + 6*a*b**2*d**3*e**2 + b**3*d**4*e) + x**4*(5*a**3*d**2*e**3/2 +
15*a**2*b*d**3*e**2/2 + 15*a*b**2*d**4*e/4 + b**3*d**5/4) + x**3*(10*a**3*d**3*e**2/3 + 5*a**2*b*d**4*e + a*b
**2*d**5) + x**2*(5*a**3*d**4*e/2 + 3*a**2*b*d**5/2)

________________________________________________________________________________________

Giac [B]  time = 1.14466, size = 393, normalized size = 4.27 \begin{align*} \frac{1}{9} \, b^{3} x^{9} e^{5} + \frac{5}{8} \, b^{3} d x^{8} e^{4} + \frac{10}{7} \, b^{3} d^{2} x^{7} e^{3} + \frac{5}{3} \, b^{3} d^{3} x^{6} e^{2} + b^{3} d^{4} x^{5} e + \frac{1}{4} \, b^{3} d^{5} x^{4} + \frac{3}{8} \, a b^{2} x^{8} e^{5} + \frac{15}{7} \, a b^{2} d x^{7} e^{4} + 5 \, a b^{2} d^{2} x^{6} e^{3} + 6 \, a b^{2} d^{3} x^{5} e^{2} + \frac{15}{4} \, a b^{2} d^{4} x^{4} e + a b^{2} d^{5} x^{3} + \frac{3}{7} \, a^{2} b x^{7} e^{5} + \frac{5}{2} \, a^{2} b d x^{6} e^{4} + 6 \, a^{2} b d^{2} x^{5} e^{3} + \frac{15}{2} \, a^{2} b d^{3} x^{4} e^{2} + 5 \, a^{2} b d^{4} x^{3} e + \frac{3}{2} \, a^{2} b d^{5} x^{2} + \frac{1}{6} \, a^{3} x^{6} e^{5} + a^{3} d x^{5} e^{4} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{3} d^{4} x^{2} e + a^{3} d^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

1/9*b^3*x^9*e^5 + 5/8*b^3*d*x^8*e^4 + 10/7*b^3*d^2*x^7*e^3 + 5/3*b^3*d^3*x^6*e^2 + b^3*d^4*x^5*e + 1/4*b^3*d^5
*x^4 + 3/8*a*b^2*x^8*e^5 + 15/7*a*b^2*d*x^7*e^4 + 5*a*b^2*d^2*x^6*e^3 + 6*a*b^2*d^3*x^5*e^2 + 15/4*a*b^2*d^4*x
^4*e + a*b^2*d^5*x^3 + 3/7*a^2*b*x^7*e^5 + 5/2*a^2*b*d*x^6*e^4 + 6*a^2*b*d^2*x^5*e^3 + 15/2*a^2*b*d^3*x^4*e^2
+ 5*a^2*b*d^4*x^3*e + 3/2*a^2*b*d^5*x^2 + 1/6*a^3*x^6*e^5 + a^3*d*x^5*e^4 + 5/2*a^3*d^2*x^4*e^3 + 10/3*a^3*d^3
*x^3*e^2 + 5/2*a^3*d^4*x^2*e + a^3*d^5*x