3.1279 \(\int (a+b x)^3 (c+d x)^7 \, dx\)

Optimal. Leaf size=92 \[ -\frac{3 b^2 (c+d x)^{10} (b c-a d)}{10 d^4}+\frac{b (c+d x)^9 (b c-a d)^2}{3 d^4}-\frac{(c+d x)^8 (b c-a d)^3}{8 d^4}+\frac{b^3 (c+d x)^{11}}{11 d^4} \]

[Out]

-((b*c - a*d)^3*(c + d*x)^8)/(8*d^4) + (b*(b*c - a*d)^2*(c + d*x)^9)/(3*d^4) - (
3*b^2*(b*c - a*d)*(c + d*x)^10)/(10*d^4) + (b^3*(c + d*x)^11)/(11*d^4)

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Rubi [A]  time = 0.408671, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{3 b^2 (c+d x)^{10} (b c-a d)}{10 d^4}+\frac{b (c+d x)^9 (b c-a d)^2}{3 d^4}-\frac{(c+d x)^8 (b c-a d)^3}{8 d^4}+\frac{b^3 (c+d x)^{11}}{11 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(c + d*x)^7,x]

[Out]

-((b*c - a*d)^3*(c + d*x)^8)/(8*d^4) + (b*(b*c - a*d)^2*(c + d*x)^9)/(3*d^4) - (
3*b^2*(b*c - a*d)*(c + d*x)^10)/(10*d^4) + (b^3*(c + d*x)^11)/(11*d^4)

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Rubi in Sympy [A]  time = 39.0861, size = 80, normalized size = 0.87 \[ \frac{b^{3} \left (c + d x\right )^{11}}{11 d^{4}} + \frac{3 b^{2} \left (c + d x\right )^{10} \left (a d - b c\right )}{10 d^{4}} + \frac{b \left (c + d x\right )^{9} \left (a d - b c\right )^{2}}{3 d^{4}} + \frac{\left (c + d x\right )^{8} \left (a d - b c\right )^{3}}{8 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(d*x+c)**7,x)

[Out]

b**3*(c + d*x)**11/(11*d**4) + 3*b**2*(c + d*x)**10*(a*d - b*c)/(10*d**4) + b*(c
+ d*x)**9*(a*d - b*c)**2/(3*d**4) + (c + d*x)**8*(a*d - b*c)**3/(8*d**4)

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Mathematica [B]  time = 0.0762292, size = 360, normalized size = 3.91 \[ a^3 c^7 x+\frac{1}{3} b d^5 x^9 \left (a^2 d^2+7 a b c d+7 b^2 c^2\right )+a c^5 x^3 \left (7 a^2 d^2+7 a b c d+b^2 c^2\right )+\frac{1}{2} a^2 c^6 x^2 (7 a d+3 b c)+c d^3 x^7 \left (a^3 d^3+9 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right )+\frac{7}{2} c^2 d^2 x^6 \left (a^3 d^3+5 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )+\frac{7}{5} c^3 d x^5 \left (5 a^3 d^3+15 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{8} d^4 x^8 \left (a^3 d^3+21 a^2 b c d^2+63 a b^2 c^2 d+35 b^3 c^3\right )+\frac{1}{4} c^4 x^4 \left (35 a^3 d^3+63 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{10} b^2 d^6 x^{10} (3 a d+7 b c)+\frac{1}{11} b^3 d^7 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(c + d*x)^7,x]

[Out]

a^3*c^7*x + (a^2*c^6*(3*b*c + 7*a*d)*x^2)/2 + a*c^5*(b^2*c^2 + 7*a*b*c*d + 7*a^2
*d^2)*x^3 + (c^4*(b^3*c^3 + 21*a*b^2*c^2*d + 63*a^2*b*c*d^2 + 35*a^3*d^3)*x^4)/4
+ (7*c^3*d*(b^3*c^3 + 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 5*a^3*d^3)*x^5)/5 + (7*c
^2*d^2*(b^3*c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3)*x^6)/2 + c*d^3*(5*b^3
*c^3 + 15*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x^7 + (d^4*(35*b^3*c^3 + 63*a*b
^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*x^8)/8 + (b*d^5*(7*b^2*c^2 + 7*a*b*c*d + a^
2*d^2)*x^9)/3 + (b^2*d^6*(7*b*c + 3*a*d)*x^10)/10 + (b^3*d^7*x^11)/11

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Maple [B]  time = 0.002, size = 385, normalized size = 4.2 \[{\frac{{b}^{3}{d}^{7}{x}^{11}}{11}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{7}+7\,{b}^{3}c{d}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{7}+21\,a{b}^{2}c{d}^{6}+21\,{b}^{3}{c}^{2}{d}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{3}{d}^{7}+21\,{a}^{2}bc{d}^{6}+63\,a{b}^{2}{c}^{2}{d}^{5}+35\,{b}^{3}{c}^{3}{d}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{3}c{d}^{6}+63\,{a}^{2}b{c}^{2}{d}^{5}+105\,a{b}^{2}{c}^{3}{d}^{4}+35\,{b}^{3}{c}^{4}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{3}{c}^{2}{d}^{5}+105\,{a}^{2}b{c}^{3}{d}^{4}+105\,a{b}^{2}{c}^{4}{d}^{3}+21\,{b}^{3}{c}^{5}{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{3}{c}^{3}{d}^{4}+105\,{a}^{2}b{c}^{4}{d}^{3}+63\,a{b}^{2}{c}^{5}{d}^{2}+7\,{b}^{3}{c}^{6}d \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{3}{c}^{4}{d}^{3}+63\,{a}^{2}b{c}^{5}{d}^{2}+21\,a{b}^{2}{c}^{6}d+{b}^{3}{c}^{7} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{3}{c}^{5}{d}^{2}+21\,{a}^{2}b{c}^{6}d+3\,a{b}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{3}{c}^{6}d+3\,{a}^{2}b{c}^{7} \right ){x}^{2}}{2}}+{a}^{3}{c}^{7}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(d*x+c)^7,x)

[Out]

1/11*b^3*d^7*x^11+1/10*(3*a*b^2*d^7+7*b^3*c*d^6)*x^10+1/9*(3*a^2*b*d^7+21*a*b^2*
c*d^6+21*b^3*c^2*d^5)*x^9+1/8*(a^3*d^7+21*a^2*b*c*d^6+63*a*b^2*c^2*d^5+35*b^3*c^
3*d^4)*x^8+1/7*(7*a^3*c*d^6+63*a^2*b*c^2*d^5+105*a*b^2*c^3*d^4+35*b^3*c^4*d^3)*x
^7+1/6*(21*a^3*c^2*d^5+105*a^2*b*c^3*d^4+105*a*b^2*c^4*d^3+21*b^3*c^5*d^2)*x^6+1
/5*(35*a^3*c^3*d^4+105*a^2*b*c^4*d^3+63*a*b^2*c^5*d^2+7*b^3*c^6*d)*x^5+1/4*(35*a
^3*c^4*d^3+63*a^2*b*c^5*d^2+21*a*b^2*c^6*d+b^3*c^7)*x^4+1/3*(21*a^3*c^5*d^2+21*a
^2*b*c^6*d+3*a*b^2*c^7)*x^3+1/2*(7*a^3*c^6*d+3*a^2*b*c^7)*x^2+a^3*c^7*x

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Maxima [A]  time = 1.35881, size = 508, normalized size = 5.52 \[ \frac{1}{11} \, b^{3} d^{7} x^{11} + a^{3} c^{7} x + \frac{1}{10} \,{\left (7 \, b^{3} c d^{6} + 3 \, a b^{2} d^{7}\right )} x^{10} + \frac{1}{3} \,{\left (7 \, b^{3} c^{2} d^{5} + 7 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (35 \, b^{3} c^{3} d^{4} + 63 \, a b^{2} c^{2} d^{5} + 21 \, a^{2} b c d^{6} + a^{3} d^{7}\right )} x^{8} +{\left (5 \, b^{3} c^{4} d^{3} + 15 \, a b^{2} c^{3} d^{4} + 9 \, a^{2} b c^{2} d^{5} + a^{3} c d^{6}\right )} x^{7} + \frac{7}{2} \,{\left (b^{3} c^{5} d^{2} + 5 \, a b^{2} c^{4} d^{3} + 5 \, a^{2} b c^{3} d^{4} + a^{3} c^{2} d^{5}\right )} x^{6} + \frac{7}{5} \,{\left (b^{3} c^{6} d + 9 \, a b^{2} c^{5} d^{2} + 15 \, a^{2} b c^{4} d^{3} + 5 \, a^{3} c^{3} d^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{7} + 21 \, a b^{2} c^{6} d + 63 \, a^{2} b c^{5} d^{2} + 35 \, a^{3} c^{4} d^{3}\right )} x^{4} +{\left (a b^{2} c^{7} + 7 \, a^{2} b c^{6} d + 7 \, a^{3} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c^{7} + 7 \, a^{3} c^{6} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*(d*x + c)^7,x, algorithm="maxima")

[Out]

1/11*b^3*d^7*x^11 + a^3*c^7*x + 1/10*(7*b^3*c*d^6 + 3*a*b^2*d^7)*x^10 + 1/3*(7*b
^3*c^2*d^5 + 7*a*b^2*c*d^6 + a^2*b*d^7)*x^9 + 1/8*(35*b^3*c^3*d^4 + 63*a*b^2*c^2
*d^5 + 21*a^2*b*c*d^6 + a^3*d^7)*x^8 + (5*b^3*c^4*d^3 + 15*a*b^2*c^3*d^4 + 9*a^2
*b*c^2*d^5 + a^3*c*d^6)*x^7 + 7/2*(b^3*c^5*d^2 + 5*a*b^2*c^4*d^3 + 5*a^2*b*c^3*d
^4 + a^3*c^2*d^5)*x^6 + 7/5*(b^3*c^6*d + 9*a*b^2*c^5*d^2 + 15*a^2*b*c^4*d^3 + 5*
a^3*c^3*d^4)*x^5 + 1/4*(b^3*c^7 + 21*a*b^2*c^6*d + 63*a^2*b*c^5*d^2 + 35*a^3*c^4
*d^3)*x^4 + (a*b^2*c^7 + 7*a^2*b*c^6*d + 7*a^3*c^5*d^2)*x^3 + 1/2*(3*a^2*b*c^7 +
7*a^3*c^6*d)*x^2

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Fricas [A]  time = 0.189904, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} d^{7} b^{3} + \frac{7}{10} x^{10} d^{6} c b^{3} + \frac{3}{10} x^{10} d^{7} b^{2} a + \frac{7}{3} x^{9} d^{5} c^{2} b^{3} + \frac{7}{3} x^{9} d^{6} c b^{2} a + \frac{1}{3} x^{9} d^{7} b a^{2} + \frac{35}{8} x^{8} d^{4} c^{3} b^{3} + \frac{63}{8} x^{8} d^{5} c^{2} b^{2} a + \frac{21}{8} x^{8} d^{6} c b a^{2} + \frac{1}{8} x^{8} d^{7} a^{3} + 5 x^{7} d^{3} c^{4} b^{3} + 15 x^{7} d^{4} c^{3} b^{2} a + 9 x^{7} d^{5} c^{2} b a^{2} + x^{7} d^{6} c a^{3} + \frac{7}{2} x^{6} d^{2} c^{5} b^{3} + \frac{35}{2} x^{6} d^{3} c^{4} b^{2} a + \frac{35}{2} x^{6} d^{4} c^{3} b a^{2} + \frac{7}{2} x^{6} d^{5} c^{2} a^{3} + \frac{7}{5} x^{5} d c^{6} b^{3} + \frac{63}{5} x^{5} d^{2} c^{5} b^{2} a + 21 x^{5} d^{3} c^{4} b a^{2} + 7 x^{5} d^{4} c^{3} a^{3} + \frac{1}{4} x^{4} c^{7} b^{3} + \frac{21}{4} x^{4} d c^{6} b^{2} a + \frac{63}{4} x^{4} d^{2} c^{5} b a^{2} + \frac{35}{4} x^{4} d^{3} c^{4} a^{3} + x^{3} c^{7} b^{2} a + 7 x^{3} d c^{6} b a^{2} + 7 x^{3} d^{2} c^{5} a^{3} + \frac{3}{2} x^{2} c^{7} b a^{2} + \frac{7}{2} x^{2} d c^{6} a^{3} + x c^{7} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*(d*x + c)^7,x, algorithm="fricas")

[Out]

1/11*x^11*d^7*b^3 + 7/10*x^10*d^6*c*b^3 + 3/10*x^10*d^7*b^2*a + 7/3*x^9*d^5*c^2*
b^3 + 7/3*x^9*d^6*c*b^2*a + 1/3*x^9*d^7*b*a^2 + 35/8*x^8*d^4*c^3*b^3 + 63/8*x^8*
d^5*c^2*b^2*a + 21/8*x^8*d^6*c*b*a^2 + 1/8*x^8*d^7*a^3 + 5*x^7*d^3*c^4*b^3 + 15*
x^7*d^4*c^3*b^2*a + 9*x^7*d^5*c^2*b*a^2 + x^7*d^6*c*a^3 + 7/2*x^6*d^2*c^5*b^3 +
35/2*x^6*d^3*c^4*b^2*a + 35/2*x^6*d^4*c^3*b*a^2 + 7/2*x^6*d^5*c^2*a^3 + 7/5*x^5*
d*c^6*b^3 + 63/5*x^5*d^2*c^5*b^2*a + 21*x^5*d^3*c^4*b*a^2 + 7*x^5*d^4*c^3*a^3 +
1/4*x^4*c^7*b^3 + 21/4*x^4*d*c^6*b^2*a + 63/4*x^4*d^2*c^5*b*a^2 + 35/4*x^4*d^3*c
^4*a^3 + x^3*c^7*b^2*a + 7*x^3*d*c^6*b*a^2 + 7*x^3*d^2*c^5*a^3 + 3/2*x^2*c^7*b*a
^2 + 7/2*x^2*d*c^6*a^3 + x*c^7*a^3

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Sympy [A]  time = 0.271954, size = 427, normalized size = 4.64 \[ a^{3} c^{7} x + \frac{b^{3} d^{7} x^{11}}{11} + x^{10} \left (\frac{3 a b^{2} d^{7}}{10} + \frac{7 b^{3} c d^{6}}{10}\right ) + x^{9} \left (\frac{a^{2} b d^{7}}{3} + \frac{7 a b^{2} c d^{6}}{3} + \frac{7 b^{3} c^{2} d^{5}}{3}\right ) + x^{8} \left (\frac{a^{3} d^{7}}{8} + \frac{21 a^{2} b c d^{6}}{8} + \frac{63 a b^{2} c^{2} d^{5}}{8} + \frac{35 b^{3} c^{3} d^{4}}{8}\right ) + x^{7} \left (a^{3} c d^{6} + 9 a^{2} b c^{2} d^{5} + 15 a b^{2} c^{3} d^{4} + 5 b^{3} c^{4} d^{3}\right ) + x^{6} \left (\frac{7 a^{3} c^{2} d^{5}}{2} + \frac{35 a^{2} b c^{3} d^{4}}{2} + \frac{35 a b^{2} c^{4} d^{3}}{2} + \frac{7 b^{3} c^{5} d^{2}}{2}\right ) + x^{5} \left (7 a^{3} c^{3} d^{4} + 21 a^{2} b c^{4} d^{3} + \frac{63 a b^{2} c^{5} d^{2}}{5} + \frac{7 b^{3} c^{6} d}{5}\right ) + x^{4} \left (\frac{35 a^{3} c^{4} d^{3}}{4} + \frac{63 a^{2} b c^{5} d^{2}}{4} + \frac{21 a b^{2} c^{6} d}{4} + \frac{b^{3} c^{7}}{4}\right ) + x^{3} \left (7 a^{3} c^{5} d^{2} + 7 a^{2} b c^{6} d + a b^{2} c^{7}\right ) + x^{2} \left (\frac{7 a^{3} c^{6} d}{2} + \frac{3 a^{2} b c^{7}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(d*x+c)**7,x)

[Out]

a**3*c**7*x + b**3*d**7*x**11/11 + x**10*(3*a*b**2*d**7/10 + 7*b**3*c*d**6/10) +
x**9*(a**2*b*d**7/3 + 7*a*b**2*c*d**6/3 + 7*b**3*c**2*d**5/3) + x**8*(a**3*d**7
/8 + 21*a**2*b*c*d**6/8 + 63*a*b**2*c**2*d**5/8 + 35*b**3*c**3*d**4/8) + x**7*(a
**3*c*d**6 + 9*a**2*b*c**2*d**5 + 15*a*b**2*c**3*d**4 + 5*b**3*c**4*d**3) + x**6
*(7*a**3*c**2*d**5/2 + 35*a**2*b*c**3*d**4/2 + 35*a*b**2*c**4*d**3/2 + 7*b**3*c*
*5*d**2/2) + x**5*(7*a**3*c**3*d**4 + 21*a**2*b*c**4*d**3 + 63*a*b**2*c**5*d**2/
5 + 7*b**3*c**6*d/5) + x**4*(35*a**3*c**4*d**3/4 + 63*a**2*b*c**5*d**2/4 + 21*a*
b**2*c**6*d/4 + b**3*c**7/4) + x**3*(7*a**3*c**5*d**2 + 7*a**2*b*c**6*d + a*b**2
*c**7) + x**2*(7*a**3*c**6*d/2 + 3*a**2*b*c**7/2)

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GIAC/XCAS [A]  time = 0.217043, size = 567, normalized size = 6.16 \[ \frac{1}{11} \, b^{3} d^{7} x^{11} + \frac{7}{10} \, b^{3} c d^{6} x^{10} + \frac{3}{10} \, a b^{2} d^{7} x^{10} + \frac{7}{3} \, b^{3} c^{2} d^{5} x^{9} + \frac{7}{3} \, a b^{2} c d^{6} x^{9} + \frac{1}{3} \, a^{2} b d^{7} x^{9} + \frac{35}{8} \, b^{3} c^{3} d^{4} x^{8} + \frac{63}{8} \, a b^{2} c^{2} d^{5} x^{8} + \frac{21}{8} \, a^{2} b c d^{6} x^{8} + \frac{1}{8} \, a^{3} d^{7} x^{8} + 5 \, b^{3} c^{4} d^{3} x^{7} + 15 \, a b^{2} c^{3} d^{4} x^{7} + 9 \, a^{2} b c^{2} d^{5} x^{7} + a^{3} c d^{6} x^{7} + \frac{7}{2} \, b^{3} c^{5} d^{2} x^{6} + \frac{35}{2} \, a b^{2} c^{4} d^{3} x^{6} + \frac{35}{2} \, a^{2} b c^{3} d^{4} x^{6} + \frac{7}{2} \, a^{3} c^{2} d^{5} x^{6} + \frac{7}{5} \, b^{3} c^{6} d x^{5} + \frac{63}{5} \, a b^{2} c^{5} d^{2} x^{5} + 21 \, a^{2} b c^{4} d^{3} x^{5} + 7 \, a^{3} c^{3} d^{4} x^{5} + \frac{1}{4} \, b^{3} c^{7} x^{4} + \frac{21}{4} \, a b^{2} c^{6} d x^{4} + \frac{63}{4} \, a^{2} b c^{5} d^{2} x^{4} + \frac{35}{4} \, a^{3} c^{4} d^{3} x^{4} + a b^{2} c^{7} x^{3} + 7 \, a^{2} b c^{6} d x^{3} + 7 \, a^{3} c^{5} d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{7} x^{2} + \frac{7}{2} \, a^{3} c^{6} d x^{2} + a^{3} c^{7} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*(d*x + c)^7,x, algorithm="giac")

[Out]

1/11*b^3*d^7*x^11 + 7/10*b^3*c*d^6*x^10 + 3/10*a*b^2*d^7*x^10 + 7/3*b^3*c^2*d^5*
x^9 + 7/3*a*b^2*c*d^6*x^9 + 1/3*a^2*b*d^7*x^9 + 35/8*b^3*c^3*d^4*x^8 + 63/8*a*b^
2*c^2*d^5*x^8 + 21/8*a^2*b*c*d^6*x^8 + 1/8*a^3*d^7*x^8 + 5*b^3*c^4*d^3*x^7 + 15*
a*b^2*c^3*d^4*x^7 + 9*a^2*b*c^2*d^5*x^7 + a^3*c*d^6*x^7 + 7/2*b^3*c^5*d^2*x^6 +
35/2*a*b^2*c^4*d^3*x^6 + 35/2*a^2*b*c^3*d^4*x^6 + 7/2*a^3*c^2*d^5*x^6 + 7/5*b^3*
c^6*d*x^5 + 63/5*a*b^2*c^5*d^2*x^5 + 21*a^2*b*c^4*d^3*x^5 + 7*a^3*c^3*d^4*x^5 +
1/4*b^3*c^7*x^4 + 21/4*a*b^2*c^6*d*x^4 + 63/4*a^2*b*c^5*d^2*x^4 + 35/4*a^3*c^4*d
^3*x^4 + a*b^2*c^7*x^3 + 7*a^2*b*c^6*d*x^3 + 7*a^3*c^5*d^2*x^3 + 3/2*a^2*b*c^7*x
^2 + 7/2*a^3*c^6*d*x^2 + a^3*c^7*x