3.1281 \(\int (a+b x) (c+d x)^7 \, dx\)

Optimal. Leaf size=38 \[ \frac{b (c+d x)^9}{9 d^2}-\frac{(c+d x)^8 (b c-a d)}{8 d^2} \]

[Out]

-((b*c - a*d)*(c + d*x)^8)/(8*d^2) + (b*(c + d*x)^9)/(9*d^2)

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Rubi [A]  time = 0.0164614, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{b (c+d x)^9}{9 d^2}-\frac{(c+d x)^8 (b c-a d)}{8 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)^7,x]

[Out]

-((b*c - a*d)*(c + d*x)^8)/(8*d^2) + (b*(c + d*x)^9)/(9*d^2)

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) (c+d x)^7 \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^7}{d}+\frac{b (c+d x)^8}{d}\right ) \, dx\\ &=-\frac{(b c-a d) (c+d x)^8}{8 d^2}+\frac{b (c+d x)^9}{9 d^2}\\ \end{align*}

Mathematica [B]  time = 0.0170292, size = 151, normalized size = 3.97 \[ \frac{7}{6} c^2 d^4 x^6 (3 a d+5 b c)+7 c^3 d^3 x^5 (a d+b c)+\frac{7}{4} c^4 d^2 x^4 (5 a d+3 b c)+\frac{7}{3} c^5 d x^3 (3 a d+b c)+\frac{1}{2} c^6 x^2 (7 a d+b c)+\frac{1}{8} d^6 x^8 (a d+7 b c)+c d^5 x^7 (a d+3 b c)+a c^7 x+\frac{1}{9} b d^7 x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)^7,x]

[Out]

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

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Maple [B]  time = 0.001, size = 169, normalized size = 4.5 \begin{align*}{\frac{b{d}^{7}{x}^{9}}{9}}+{\frac{ \left ( a{d}^{7}+7\,bc{d}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,ac{d}^{6}+21\,b{c}^{2}{d}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,a{c}^{2}{d}^{5}+35\,b{c}^{3}{d}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,a{c}^{3}{d}^{4}+35\,b{c}^{4}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,a{c}^{4}{d}^{3}+21\,b{c}^{5}{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,a{c}^{5}{d}^{2}+7\,b{c}^{6}d \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,a{c}^{6}d+b{c}^{7} \right ){x}^{2}}{2}}+a{c}^{7}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^7,x)

[Out]

1/9*b*d^7*x^9+1/8*(a*d^7+7*b*c*d^6)*x^8+1/7*(7*a*c*d^6+21*b*c^2*d^5)*x^7+1/6*(21*a*c^2*d^5+35*b*c^3*d^4)*x^6+1
/5*(35*a*c^3*d^4+35*b*c^4*d^3)*x^5+1/4*(35*a*c^4*d^3+21*b*c^5*d^2)*x^4+1/3*(21*a*c^5*d^2+7*b*c^6*d)*x^3+1/2*(7
*a*c^6*d+b*c^7)*x^2+a*c^7*x

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Maxima [B]  time = 0.945336, size = 220, normalized size = 5.79 \begin{align*} \frac{1}{9} \, b d^{7} x^{9} + a c^{7} x + \frac{1}{8} \,{\left (7 \, b c d^{6} + a d^{7}\right )} x^{8} +{\left (3 \, b c^{2} d^{5} + a c d^{6}\right )} x^{7} + \frac{7}{6} \,{\left (5 \, b c^{3} d^{4} + 3 \, a c^{2} d^{5}\right )} x^{6} + 7 \,{\left (b c^{4} d^{3} + a c^{3} d^{4}\right )} x^{5} + \frac{7}{4} \,{\left (3 \, b c^{5} d^{2} + 5 \, a c^{4} d^{3}\right )} x^{4} + \frac{7}{3} \,{\left (b c^{6} d + 3 \, a c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{7} + 7 \, a c^{6} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^7,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.85742, size = 378, normalized size = 9.95 \begin{align*} \frac{1}{9} x^{9} d^{7} b + \frac{7}{8} x^{8} d^{6} c b + \frac{1}{8} x^{8} d^{7} a + 3 x^{7} d^{5} c^{2} b + x^{7} d^{6} c a + \frac{35}{6} x^{6} d^{4} c^{3} b + \frac{7}{2} x^{6} d^{5} c^{2} a + 7 x^{5} d^{3} c^{4} b + 7 x^{5} d^{4} c^{3} a + \frac{21}{4} x^{4} d^{2} c^{5} b + \frac{35}{4} x^{4} d^{3} c^{4} a + \frac{7}{3} x^{3} d c^{6} b + 7 x^{3} d^{2} c^{5} a + \frac{1}{2} x^{2} c^{7} b + \frac{7}{2} x^{2} d c^{6} a + x c^{7} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/9*x^9*d^7*b + 7/8*x^8*d^6*c*b + 1/8*x^8*d^7*a + 3*x^7*d^5*c^2*b + x^7*d^6*c*a + 35/6*x^6*d^4*c^3*b + 7/2*x^6
*d^5*c^2*a + 7*x^5*d^3*c^4*b + 7*x^5*d^4*c^3*a + 21/4*x^4*d^2*c^5*b + 35/4*x^4*d^3*c^4*a + 7/3*x^3*d*c^6*b + 7
*x^3*d^2*c^5*a + 1/2*x^2*c^7*b + 7/2*x^2*d*c^6*a + x*c^7*a

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Sympy [B]  time = 0.089849, size = 178, normalized size = 4.68 \begin{align*} a c^{7} x + \frac{b d^{7} x^{9}}{9} + x^{8} \left (\frac{a d^{7}}{8} + \frac{7 b c d^{6}}{8}\right ) + x^{7} \left (a c d^{6} + 3 b c^{2} d^{5}\right ) + x^{6} \left (\frac{7 a c^{2} d^{5}}{2} + \frac{35 b c^{3} d^{4}}{6}\right ) + x^{5} \left (7 a c^{3} d^{4} + 7 b c^{4} d^{3}\right ) + x^{4} \left (\frac{35 a c^{4} d^{3}}{4} + \frac{21 b c^{5} d^{2}}{4}\right ) + x^{3} \left (7 a c^{5} d^{2} + \frac{7 b c^{6} d}{3}\right ) + x^{2} \left (\frac{7 a c^{6} d}{2} + \frac{b c^{7}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**7,x)

[Out]

a*c**7*x + b*d**7*x**9/9 + x**8*(a*d**7/8 + 7*b*c*d**6/8) + x**7*(a*c*d**6 + 3*b*c**2*d**5) + x**6*(7*a*c**2*d
**5/2 + 35*b*c**3*d**4/6) + x**5*(7*a*c**3*d**4 + 7*b*c**4*d**3) + x**4*(35*a*c**4*d**3/4 + 21*b*c**5*d**2/4)
+ x**3*(7*a*c**5*d**2 + 7*b*c**6*d/3) + x**2*(7*a*c**6*d/2 + b*c**7/2)

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Giac [B]  time = 1.07522, size = 228, normalized size = 6. \begin{align*} \frac{1}{9} \, b d^{7} x^{9} + \frac{7}{8} \, b c d^{6} x^{8} + \frac{1}{8} \, a d^{7} x^{8} + 3 \, b c^{2} d^{5} x^{7} + a c d^{6} x^{7} + \frac{35}{6} \, b c^{3} d^{4} x^{6} + \frac{7}{2} \, a c^{2} d^{5} x^{6} + 7 \, b c^{4} d^{3} x^{5} + 7 \, a c^{3} d^{4} x^{5} + \frac{21}{4} \, b c^{5} d^{2} x^{4} + \frac{35}{4} \, a c^{4} d^{3} x^{4} + \frac{7}{3} \, b c^{6} d x^{3} + 7 \, a c^{5} d^{2} x^{3} + \frac{1}{2} \, b c^{7} x^{2} + \frac{7}{2} \, a c^{6} d x^{2} + a c^{7} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^7,x, algorithm="giac")

[Out]

1/9*b*d^7*x^9 + 7/8*b*c*d^6*x^8 + 1/8*a*d^7*x^8 + 3*b*c^2*d^5*x^7 + a*c*d^6*x^7 + 35/6*b*c^3*d^4*x^6 + 7/2*a*c
^2*d^5*x^6 + 7*b*c^4*d^3*x^5 + 7*a*c^3*d^4*x^5 + 21/4*b*c^5*d^2*x^4 + 35/4*a*c^4*d^3*x^4 + 7/3*b*c^6*d*x^3 + 7
*a*c^5*d^2*x^3 + 1/2*b*c^7*x^2 + 7/2*a*c^6*d*x^2 + a*c^7*x