3.1058 \(\int (a+b x)^6 (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^8 (-2 a B e+A b e+b B d)}{8 b^3}+\frac{(a+b x)^7 (A b-a B) (b d-a e)}{7 b^3}+\frac{B e (a+b x)^9}{9 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.195088, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{(a+b x)^8 (-2 a B e+A b e+b B d)}{8 b^3}+\frac{(a+b x)^7 (A b-a B) (b d-a e)}{7 b^3}+\frac{B e (a+b x)^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [B]  time = 0.133329, size = 231, normalized size = 3.08 \[ \frac{1}{504} x \left (126 a^4 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+168 a^3 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+36 a^2 b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+252 a^5 b x (A (6 d+4 e x)+B x (4 d+3 e x))+84 a^6 (3 A (2 d+e x)+B x (3 d+2 e x))+18 a b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+b^6 x^6 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x),x]

[Out]

(x*(84*a^6*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)) + 126*a^4*b^2*x^2*(5*A*(4*d + 3*e*x) + 3*B*x*(5*d + 4*e*x)) +
252*a^5*b*x*(B*x*(4*d + 3*e*x) + A*(6*d + 4*e*x)) + 168*a^3*b^3*x^3*(3*A*(5*d + 4*e*x) + 2*B*x*(6*d + 5*e*x))
+ 36*a^2*b^4*x^4*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)) + 18*a*b^5*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7
*e*x)) + b^6*x^6*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x))))/504

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 293, normalized size = 3.9 \begin{align*}{\frac{{b}^{6}Be{x}^{9}}{9}}+{\frac{ \left ( \left ({b}^{6}A+6\,a{b}^{5}B \right ) e+{b}^{6}Bd \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ) e+ \left ({b}^{6}A+6\,a{b}^{5}B \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ) e+ \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ) d \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ) e+ \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ) d \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ) e+ \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ) d \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 6\,{a}^{5}bA+{a}^{6}B \right ) e+ \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{6}Ae+ \left ( 6\,{a}^{5}bA+{a}^{6}B \right ) d \right ){x}^{2}}{2}}+{a}^{6}Adx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d),x)

[Out]

1/9*b^6*B*e*x^9+1/8*((A*b^6+6*B*a*b^5)*e+b^6*B*d)*x^8+1/7*((6*A*a*b^5+15*B*a^2*b^4)*e+(A*b^6+6*B*a*b^5)*d)*x^7
+1/6*((15*A*a^2*b^4+20*B*a^3*b^3)*e+(6*A*a*b^5+15*B*a^2*b^4)*d)*x^6+1/5*((20*A*a^3*b^3+15*B*a^4*b^2)*e+(15*A*a
^2*b^4+20*B*a^3*b^3)*d)*x^5+1/4*((15*A*a^4*b^2+6*B*a^5*b)*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d)*x^4+1/3*((6*A*a^5*b
+B*a^6)*e+(15*A*a^4*b^2+6*B*a^5*b)*d)*x^3+1/2*(a^6*A*e+(6*A*a^5*b+B*a^6)*d)*x^2+a^6*A*d*x

________________________________________________________________________________________

Maxima [B]  time = 1.11793, size = 401, normalized size = 5.35 \begin{align*} \frac{1}{9} \, B b^{6} e x^{9} + A a^{6} d x + \frac{1}{8} \,{\left (B b^{6} d +{\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e\right )} x^{6} +{\left ({\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d +{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e\right )} x^{5} + \frac{1}{4} \,{\left (5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d +{\left (B a^{6} + 6 \, A a^{5} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{6} e +{\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.56782, size = 733, normalized size = 9.77 \begin{align*} \frac{1}{9} x^{9} e b^{6} B + \frac{1}{8} x^{8} d b^{6} B + \frac{3}{4} x^{8} e b^{5} a B + \frac{1}{8} x^{8} e b^{6} A + \frac{6}{7} x^{7} d b^{5} a B + \frac{15}{7} x^{7} e b^{4} a^{2} B + \frac{1}{7} x^{7} d b^{6} A + \frac{6}{7} x^{7} e b^{5} a A + \frac{5}{2} x^{6} d b^{4} a^{2} B + \frac{10}{3} x^{6} e b^{3} a^{3} B + x^{6} d b^{5} a A + \frac{5}{2} x^{6} e b^{4} a^{2} A + 4 x^{5} d b^{3} a^{3} B + 3 x^{5} e b^{2} a^{4} B + 3 x^{5} d b^{4} a^{2} A + 4 x^{5} e b^{3} a^{3} A + \frac{15}{4} x^{4} d b^{2} a^{4} B + \frac{3}{2} x^{4} e b a^{5} B + 5 x^{4} d b^{3} a^{3} A + \frac{15}{4} x^{4} e b^{2} a^{4} A + 2 x^{3} d b a^{5} B + \frac{1}{3} x^{3} e a^{6} B + 5 x^{3} d b^{2} a^{4} A + 2 x^{3} e b a^{5} A + \frac{1}{2} x^{2} d a^{6} B + 3 x^{2} d b a^{5} A + \frac{1}{2} x^{2} e a^{6} A + x d a^{6} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.100287, size = 333, normalized size = 4.44 \begin{align*} A a^{6} d x + \frac{B b^{6} e x^{9}}{9} + x^{8} \left (\frac{A b^{6} e}{8} + \frac{3 B a b^{5} e}{4} + \frac{B b^{6} d}{8}\right ) + x^{7} \left (\frac{6 A a b^{5} e}{7} + \frac{A b^{6} d}{7} + \frac{15 B a^{2} b^{4} e}{7} + \frac{6 B a b^{5} d}{7}\right ) + x^{6} \left (\frac{5 A a^{2} b^{4} e}{2} + A a b^{5} d + \frac{10 B a^{3} b^{3} e}{3} + \frac{5 B a^{2} b^{4} d}{2}\right ) + x^{5} \left (4 A a^{3} b^{3} e + 3 A a^{2} b^{4} d + 3 B a^{4} b^{2} e + 4 B a^{3} b^{3} d\right ) + x^{4} \left (\frac{15 A a^{4} b^{2} e}{4} + 5 A a^{3} b^{3} d + \frac{3 B a^{5} b e}{2} + \frac{15 B a^{4} b^{2} d}{4}\right ) + x^{3} \left (2 A a^{5} b e + 5 A a^{4} b^{2} d + \frac{B a^{6} e}{3} + 2 B a^{5} b d\right ) + x^{2} \left (\frac{A a^{6} e}{2} + 3 A a^{5} b d + \frac{B a^{6} d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)*(e*x+d),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 2.38428, size = 452, normalized size = 6.03 \begin{align*} \frac{1}{9} \, B b^{6} x^{9} e + \frac{1}{8} \, B b^{6} d x^{8} + \frac{3}{4} \, B a b^{5} x^{8} e + \frac{1}{8} \, A b^{6} x^{8} e + \frac{6}{7} \, B a b^{5} d x^{7} + \frac{1}{7} \, A b^{6} d x^{7} + \frac{15}{7} \, B a^{2} b^{4} x^{7} e + \frac{6}{7} \, A a b^{5} x^{7} e + \frac{5}{2} \, B a^{2} b^{4} d x^{6} + A a b^{5} d x^{6} + \frac{10}{3} \, B a^{3} b^{3} x^{6} e + \frac{5}{2} \, A a^{2} b^{4} x^{6} e + 4 \, B a^{3} b^{3} d x^{5} + 3 \, A a^{2} b^{4} d x^{5} + 3 \, B a^{4} b^{2} x^{5} e + 4 \, A a^{3} b^{3} x^{5} e + \frac{15}{4} \, B a^{4} b^{2} d x^{4} + 5 \, A a^{3} b^{3} d x^{4} + \frac{3}{2} \, B a^{5} b x^{4} e + \frac{15}{4} \, A a^{4} b^{2} x^{4} e + 2 \, B a^{5} b d x^{3} + 5 \, A a^{4} b^{2} d x^{3} + \frac{1}{3} \, B a^{6} x^{3} e + 2 \, A a^{5} b x^{3} e + \frac{1}{2} \, B a^{6} d x^{2} + 3 \, A a^{5} b d x^{2} + \frac{1}{2} \, A a^{6} x^{2} e + A a^{6} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d),x, algorithm="giac")

[Out]

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