\(\int (a+b x) (c+d x)^7 \, dx\) [1281]

Optimal result

Integrand size = 13, antiderivative size = 38 \[ \int (a+b x) (c+d x)^7 \, dx=-\frac {(b c-a d) (c+d x)^8}{8 d^2}+\frac {b (c+d x)^9}{9 d^2} \]

[Out]

-1/8*(-a*d+b*c)*(d*x+c)^8/d^2+1/9*b*(d*x+c)^9/d^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x) (c+d x)^7 \, dx=\frac {b (c+d x)^9}{9 d^2}-\frac {(c+d x)^8 (b c-a d)}{8 d^2} \]

[In]

Int[(a + b*x)*(c + d*x)^7,x]

[Out]

-1/8*((b*c - a*d)*(c + d*x)^8)/d^2 + (b*(c + d*x)^9)/(9*d^2)

Rule 45

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(38)=76\).

Time = 0.01 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.97 \[ \int (a+b x) (c+d x)^7 \, dx=a c^7 x+\frac {1}{2} c^6 (b c+7 a d) x^2+\frac {7}{3} c^5 d (b c+3 a d) x^3+\frac {7}{4} c^4 d^2 (3 b c+5 a d) x^4+7 c^3 d^3 (b c+a d) x^5+\frac {7}{6} c^2 d^4 (5 b c+3 a d) x^6+c d^5 (3 b c+a d) x^7+\frac {1}{8} d^6 (7 b c+a d) x^8+\frac {1}{9} b d^7 x^9 \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(162\) vs. \(2(34)=68\).

Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29

method	result	size
norman	\(\frac {b \,d^{7} x^{9}}{9}+\left (\frac {1}{8} a \,d^{7}+\frac {7}{8} b c \,d^{6}\right ) x^{8}+\left (a c \,d^{6}+3 b \,c^{2} d^{5}\right ) x^{7}+\left (\frac {7}{2} a \,c^{2} d^{5}+\frac {35}{6} b \,c^{3} d^{4}\right ) x^{6}+\left (7 a \,c^{3} d^{4}+7 b \,c^{4} d^{3}\right ) x^{5}+\left (\frac {35}{4} a \,c^{4} d^{3}+\frac {21}{4} b \,c^{5} d^{2}\right ) x^{4}+\left (7 a \,c^{5} d^{2}+\frac {7}{3} b \,c^{6} d \right ) x^{3}+\left (\frac {7}{2} a \,c^{6} d +\frac {1}{2} b \,c^{7}\right ) x^{2}+a \,c^{7} x\)	\(163\)
default	\(\frac {b \,d^{7} x^{9}}{9}+\frac {\left (a \,d^{7}+7 b c \,d^{6}\right ) x^{8}}{8}+\frac {\left (7 a c \,d^{6}+21 b \,c^{2} d^{5}\right ) x^{7}}{7}+\frac {\left (21 a \,c^{2} d^{5}+35 b \,c^{3} d^{4}\right ) x^{6}}{6}+\frac {\left (35 a \,c^{3} d^{4}+35 b \,c^{4} d^{3}\right ) x^{5}}{5}+\frac {\left (35 a \,c^{4} d^{3}+21 b \,c^{5} d^{2}\right ) x^{4}}{4}+\frac {\left (21 a \,c^{5} d^{2}+7 b \,c^{6} d \right ) x^{3}}{3}+\frac {\left (7 a \,c^{6} d +b \,c^{7}\right ) x^{2}}{2}+a \,c^{7} x\)	\(169\)
gosper	\(\frac {1}{9} b \,d^{7} x^{9}+\frac {1}{8} x^{8} a \,d^{7}+\frac {7}{8} x^{8} b c \,d^{6}+a c \,d^{6} x^{7}+3 b \,c^{2} d^{5} x^{7}+\frac {7}{2} x^{6} a \,c^{2} d^{5}+\frac {35}{6} x^{6} b \,c^{3} d^{4}+7 a \,c^{3} d^{4} x^{5}+7 b \,c^{4} d^{3} x^{5}+\frac {35}{4} x^{4} a \,c^{4} d^{3}+\frac {21}{4} x^{4} b \,c^{5} d^{2}+7 x^{3} a \,c^{5} d^{2}+\frac {7}{3} x^{3} b \,c^{6} d +\frac {7}{2} x^{2} a \,c^{6} d +\frac {1}{2} x^{2} b \,c^{7}+a \,c^{7} x\)	\(170\)
risch	\(\frac {1}{9} b \,d^{7} x^{9}+\frac {1}{8} x^{8} a \,d^{7}+\frac {7}{8} x^{8} b c \,d^{6}+a c \,d^{6} x^{7}+3 b \,c^{2} d^{5} x^{7}+\frac {7}{2} x^{6} a \,c^{2} d^{5}+\frac {35}{6} x^{6} b \,c^{3} d^{4}+7 a \,c^{3} d^{4} x^{5}+7 b \,c^{4} d^{3} x^{5}+\frac {35}{4} x^{4} a \,c^{4} d^{3}+\frac {21}{4} x^{4} b \,c^{5} d^{2}+7 x^{3} a \,c^{5} d^{2}+\frac {7}{3} x^{3} b \,c^{6} d +\frac {7}{2} x^{2} a \,c^{6} d +\frac {1}{2} x^{2} b \,c^{7}+a \,c^{7} x\)	\(170\)
parallelrisch	\(\frac {1}{9} b \,d^{7} x^{9}+\frac {1}{8} x^{8} a \,d^{7}+\frac {7}{8} x^{8} b c \,d^{6}+a c \,d^{6} x^{7}+3 b \,c^{2} d^{5} x^{7}+\frac {7}{2} x^{6} a \,c^{2} d^{5}+\frac {35}{6} x^{6} b \,c^{3} d^{4}+7 a \,c^{3} d^{4} x^{5}+7 b \,c^{4} d^{3} x^{5}+\frac {35}{4} x^{4} a \,c^{4} d^{3}+\frac {21}{4} x^{4} b \,c^{5} d^{2}+7 x^{3} a \,c^{5} d^{2}+\frac {7}{3} x^{3} b \,c^{6} d +\frac {7}{2} x^{2} a \,c^{6} d +\frac {1}{2} x^{2} b \,c^{7}+a \,c^{7} x\)	\(170\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29 \[ \int (a+b x) (c+d x)^7 \, dx=\frac {1}{9} \, b d^{7} x^{9} + a c^{7} x + \frac {1}{8} \, {\left (7 \, b c d^{6} + a d^{7}\right )} x^{8} + {\left (3 \, b c^{2} d^{5} + a c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b c^{3} d^{4} + 3 \, a c^{2} d^{5}\right )} x^{6} + 7 \, {\left (b c^{4} d^{3} + a c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (3 \, b c^{5} d^{2} + 5 \, a c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (b c^{6} d + 3 \, a c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{7} + 7 \, a c^{6} d\right )} x^{2} \]

[In]

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

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (32) = 64\).

Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 4.68 \[ \int (a+b x) (c+d x)^7 \, dx=a c^{7} x + \frac {b d^{7} x^{9}}{9} + x^{8} \left (\frac {a d^{7}}{8} + \frac {7 b c d^{6}}{8}\right ) + x^{7} \left (a c d^{6} + 3 b c^{2} d^{5}\right ) + x^{6} \cdot \left (\frac {7 a c^{2} d^{5}}{2} + \frac {35 b c^{3} d^{4}}{6}\right ) + x^{5} \cdot \left (7 a c^{3} d^{4} + 7 b c^{4} d^{3}\right ) + x^{4} \cdot \left (\frac {35 a c^{4} d^{3}}{4} + \frac {21 b c^{5} d^{2}}{4}\right ) + x^{3} \cdot \left (7 a c^{5} d^{2} + \frac {7 b c^{6} d}{3}\right ) + x^{2} \cdot \left (\frac {7 a c^{6} d}{2} + \frac {b c^{7}}{2}\right ) \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.29 \[ \int (a+b x) (c+d x)^7 \, dx=\frac {1}{9} \, b d^{7} x^{9} + a c^{7} x + \frac {1}{8} \, {\left (7 \, b c d^{6} + a d^{7}\right )} x^{8} + {\left (3 \, b c^{2} d^{5} + a c d^{6}\right )} x^{7} + \frac {7}{6} \, {\left (5 \, b c^{3} d^{4} + 3 \, a c^{2} d^{5}\right )} x^{6} + 7 \, {\left (b c^{4} d^{3} + a c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (3 \, b c^{5} d^{2} + 5 \, a c^{4} d^{3}\right )} x^{4} + \frac {7}{3} \, {\left (b c^{6} d + 3 \, a c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{7} + 7 \, a c^{6} d\right )} x^{2} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (34) = 68\).

Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.45 \[ \int (a+b x) (c+d x)^7 \, dx=\frac {1}{9} \, b d^{7} x^{9} + \frac {7}{8} \, b c d^{6} x^{8} + \frac {1}{8} \, a d^{7} x^{8} + 3 \, b c^{2} d^{5} x^{7} + a c d^{6} x^{7} + \frac {35}{6} \, b c^{3} d^{4} x^{6} + \frac {7}{2} \, a c^{2} d^{5} x^{6} + 7 \, b c^{4} d^{3} x^{5} + 7 \, a c^{3} d^{4} x^{5} + \frac {21}{4} \, b c^{5} d^{2} x^{4} + \frac {35}{4} \, a c^{4} d^{3} x^{4} + \frac {7}{3} \, b c^{6} d x^{3} + 7 \, a c^{5} d^{2} x^{3} + \frac {1}{2} \, b c^{7} x^{2} + \frac {7}{2} \, a c^{6} d x^{2} + a c^{7} x \]

[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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.76 \[ \int (a+b x) (c+d x)^7 \, dx=x^2\,\left (\frac {b\,c^7}{2}+\frac {7\,a\,d\,c^6}{2}\right )+x^8\,\left (\frac {a\,d^7}{8}+\frac {7\,b\,c\,d^6}{8}\right )+\frac {b\,d^7\,x^9}{9}+a\,c^7\,x+\frac {7\,c^5\,d\,x^3\,\left (3\,a\,d+b\,c\right )}{3}+c\,d^5\,x^7\,\left (a\,d+3\,b\,c\right )+7\,c^3\,d^3\,x^5\,\left (a\,d+b\,c\right )+\frac {7\,c^4\,d^2\,x^4\,\left (5\,a\,d+3\,b\,c\right )}{4}+\frac {7\,c^2\,d^4\,x^6\,\left (3\,a\,d+5\,b\,c\right )}{6} \]

[In]

int((a + b*x)*(c + d*x)^7,x)

[Out]

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