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

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {b (c+d x)^9}{9 d^2}-\frac {(c+d x)^8 (b c-a d)}{8 d^2} \end {gather*}

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*}

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Mathematica [B]  time = 0.02, size = 151, normalized size = 3.97 \begin {gather*} \frac {1}{2} c^6 x^2 (7 a d+b c)+\frac {7}{3} c^5 d x^3 (3 a d+b c)+\frac {7}{4} c^4 d^2 x^4 (5 a d+3 b c)+7 c^3 d^3 x^5 (a d+b c)+\frac {7}{6} c^2 d^4 x^6 (3 a d+5 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 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (c+d x)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(c + d*x)^7,x]

[Out]

IntegrateAlgebraic[(a + b*x)*(c + d*x)^7, x]

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fricas [B]  time = 1.57, size = 169, normalized size = 4.45 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.30, size = 169, normalized size = 4.45 \begin {gather*} \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 {gather*}

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

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maple [B]  time = 0.00, size = 169, normalized size = 4.45 \begin {gather*} \frac {b \,d^{7} x^{9}}{9}+a \,c^{7} x +\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} \end {gather*}

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 = 1.38, size = 163, normalized size = 4.29 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.08, size = 143, normalized size = 3.76 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.10, size = 178, normalized size = 4.68 \begin {gather*} 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 {gather*}

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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