3.1277 \(\int (a+b x)^5 (c+d x)^7 \, dx\)

Optimal. Leaf size=144 \[ -\frac{5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac{10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac{b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac{5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac{(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac{b^5 (c+d x)^{13}}{13 d^6} \]

[Out]

-((b*c - a*d)^5*(c + d*x)^8)/(8*d^6) + (5*b*(b*c - a*d)^4*(c + d*x)^9)/(9*d^6) -
(b^2*(b*c - a*d)^3*(c + d*x)^10)/d^6 + (10*b^3*(b*c - a*d)^2*(c + d*x)^11)/(11*
d^6) - (5*b^4*(b*c - a*d)*(c + d*x)^12)/(12*d^6) + (b^5*(c + d*x)^13)/(13*d^6)

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Rubi [A]  time = 0.701997, antiderivative size = 144, 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 \[ -\frac{5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac{10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac{b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac{5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac{(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac{b^5 (c+d x)^{13}}{13 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5*(c + d*x)^7,x]

[Out]

-((b*c - a*d)^5*(c + d*x)^8)/(8*d^6) + (5*b*(b*c - a*d)^4*(c + d*x)^9)/(9*d^6) -
(b^2*(b*c - a*d)^3*(c + d*x)^10)/d^6 + (10*b^3*(b*c - a*d)^2*(c + d*x)^11)/(11*
d^6) - (5*b^4*(b*c - a*d)*(c + d*x)^12)/(12*d^6) + (b^5*(c + d*x)^13)/(13*d^6)

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Rubi in Sympy [A]  time = 71.1042, size = 129, normalized size = 0.9 \[ \frac{b^{5} \left (c + d x\right )^{13}}{13 d^{6}} + \frac{5 b^{4} \left (c + d x\right )^{12} \left (a d - b c\right )}{12 d^{6}} + \frac{10 b^{3} \left (c + d x\right )^{11} \left (a d - b c\right )^{2}}{11 d^{6}} + \frac{b^{2} \left (c + d x\right )^{10} \left (a d - b c\right )^{3}}{d^{6}} + \frac{5 b \left (c + d x\right )^{9} \left (a d - b c\right )^{4}}{9 d^{6}} + \frac{\left (c + d x\right )^{8} \left (a d - b c\right )^{5}}{8 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(d*x+c)**7,x)

[Out]

b**5*(c + d*x)**13/(13*d**6) + 5*b**4*(c + d*x)**12*(a*d - b*c)/(12*d**6) + 10*b
**3*(c + d*x)**11*(a*d - b*c)**2/(11*d**6) + b**2*(c + d*x)**10*(a*d - b*c)**3/d
**6 + 5*b*(c + d*x)**9*(a*d - b*c)**4/(9*d**6) + (c + d*x)**8*(a*d - b*c)**5/(8*
d**6)

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Mathematica [B]  time = 0.122534, size = 574, normalized size = 3.99 \[ a^5 c^7 x+\frac{1}{2} a^4 c^6 x^2 (7 a d+5 b c)+\frac{1}{11} b^3 d^5 x^{11} \left (10 a^2 d^2+35 a b c d+21 b^2 c^2\right )+\frac{1}{3} a^3 c^5 x^3 \left (21 a^2 d^2+35 a b c d+10 b^2 c^2\right )+\frac{1}{2} b^2 d^4 x^{10} \left (2 a^3 d^3+14 a^2 b c d^2+21 a b^2 c^2 d+7 b^3 c^3\right )+\frac{5}{4} a^2 c^4 x^4 \left (7 a^3 d^3+21 a^2 b c d^2+14 a b^2 c^2 d+2 b^3 c^3\right )+\frac{5}{9} b d^3 x^9 \left (a^4 d^4+14 a^3 b c d^3+42 a^2 b^2 c^2 d^2+35 a b^3 c^3 d+7 b^4 c^4\right )+a c^3 x^5 \left (7 a^4 d^4+35 a^3 b c d^3+42 a^2 b^2 c^2 d^2+14 a b^3 c^3 d+b^4 c^4\right )+\frac{1}{8} d^2 x^8 \left (a^5 d^5+35 a^4 b c d^4+210 a^3 b^2 c^2 d^3+350 a^2 b^3 c^3 d^2+175 a b^4 c^4 d+21 b^5 c^5\right )+c d x^7 \left (a^5 d^5+15 a^4 b c d^4+50 a^3 b^2 c^2 d^3+50 a^2 b^3 c^3 d^2+15 a b^4 c^4 d+b^5 c^5\right )+\frac{1}{6} c^2 x^6 \left (21 a^5 d^5+175 a^4 b c d^4+350 a^3 b^2 c^2 d^3+210 a^2 b^3 c^3 d^2+35 a b^4 c^4 d+b^5 c^5\right )+\frac{1}{12} b^4 d^6 x^{12} (5 a d+7 b c)+\frac{1}{13} b^5 d^7 x^{13} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5*(c + d*x)^7,x]

[Out]

a^5*c^7*x + (a^4*c^6*(5*b*c + 7*a*d)*x^2)/2 + (a^3*c^5*(10*b^2*c^2 + 35*a*b*c*d
+ 21*a^2*d^2)*x^3)/3 + (5*a^2*c^4*(2*b^3*c^3 + 14*a*b^2*c^2*d + 21*a^2*b*c*d^2 +
7*a^3*d^3)*x^4)/4 + a*c^3*(b^4*c^4 + 14*a*b^3*c^3*d + 42*a^2*b^2*c^2*d^2 + 35*a
^3*b*c*d^3 + 7*a^4*d^4)*x^5 + (c^2*(b^5*c^5 + 35*a*b^4*c^4*d + 210*a^2*b^3*c^3*d
^2 + 350*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 + 21*a^5*d^5)*x^6)/6 + c*d*(b^5*c^5 +
15*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 + 50*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 + a^5
*d^5)*x^7 + (d^2*(21*b^5*c^5 + 175*a*b^4*c^4*d + 350*a^2*b^3*c^3*d^2 + 210*a^3*b
^2*c^2*d^3 + 35*a^4*b*c*d^4 + a^5*d^5)*x^8)/8 + (5*b*d^3*(7*b^4*c^4 + 35*a*b^3*c
^3*d + 42*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 + a^4*d^4)*x^9)/9 + (b^2*d^4*(7*b^3*c
^3 + 21*a*b^2*c^2*d + 14*a^2*b*c*d^2 + 2*a^3*d^3)*x^10)/2 + (b^3*d^5*(21*b^2*c^2
+ 35*a*b*c*d + 10*a^2*d^2)*x^11)/11 + (b^4*d^6*(7*b*c + 5*a*d)*x^12)/12 + (b^5*
d^7*x^13)/13

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Maple [B]  time = 0.004, size = 601, normalized size = 4.2 \[{\frac{{b}^{5}{d}^{7}{x}^{13}}{13}}+{\frac{ \left ( 5\,a{b}^{4}{d}^{7}+7\,{b}^{5}c{d}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}{d}^{7}+35\,a{b}^{4}c{d}^{6}+21\,{b}^{5}{c}^{2}{d}^{5} \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}{d}^{7}+70\,{a}^{2}{b}^{3}c{d}^{6}+105\,a{b}^{4}{c}^{2}{d}^{5}+35\,{b}^{5}{c}^{3}{d}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 5\,{a}^{4}b{d}^{7}+70\,{a}^{3}{b}^{2}c{d}^{6}+210\,{a}^{2}{b}^{3}{c}^{2}{d}^{5}+175\,a{b}^{4}{c}^{3}{d}^{4}+35\,{b}^{5}{c}^{4}{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{5}{d}^{7}+35\,{a}^{4}bc{d}^{6}+210\,{a}^{3}{b}^{2}{c}^{2}{d}^{5}+350\,{a}^{2}{b}^{3}{c}^{3}{d}^{4}+175\,a{b}^{4}{c}^{4}{d}^{3}+21\,{b}^{5}{c}^{5}{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{5}c{d}^{6}+105\,{a}^{4}b{c}^{2}{d}^{5}+350\,{a}^{3}{b}^{2}{c}^{3}{d}^{4}+350\,{a}^{2}{b}^{3}{c}^{4}{d}^{3}+105\,a{b}^{4}{c}^{5}{d}^{2}+7\,{b}^{5}{c}^{6}d \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{5}{c}^{2}{d}^{5}+175\,{a}^{4}b{c}^{3}{d}^{4}+350\,{a}^{3}{b}^{2}{c}^{4}{d}^{3}+210\,{a}^{2}{b}^{3}{c}^{5}{d}^{2}+35\,a{b}^{4}{c}^{6}d+{b}^{5}{c}^{7} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{5}{c}^{3}{d}^{4}+175\,{a}^{4}b{c}^{4}{d}^{3}+210\,{a}^{3}{b}^{2}{c}^{5}{d}^{2}+70\,{a}^{2}{b}^{3}{c}^{6}d+5\,a{b}^{4}{c}^{7} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{5}{c}^{4}{d}^{3}+105\,{a}^{4}b{c}^{5}{d}^{2}+70\,{a}^{3}{b}^{2}{c}^{6}d+10\,{a}^{2}{b}^{3}{c}^{7} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{5}{c}^{5}{d}^{2}+35\,{a}^{4}b{c}^{6}d+10\,{a}^{3}{b}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{5}{c}^{6}d+5\,{a}^{4}b{c}^{7} \right ){x}^{2}}{2}}+{a}^{5}{c}^{7}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(d*x+c)^7,x)

[Out]

1/13*b^5*d^7*x^13+1/12*(5*a*b^4*d^7+7*b^5*c*d^6)*x^12+1/11*(10*a^2*b^3*d^7+35*a*
b^4*c*d^6+21*b^5*c^2*d^5)*x^11+1/10*(10*a^3*b^2*d^7+70*a^2*b^3*c*d^6+105*a*b^4*c
^2*d^5+35*b^5*c^3*d^4)*x^10+1/9*(5*a^4*b*d^7+70*a^3*b^2*c*d^6+210*a^2*b^3*c^2*d^
5+175*a*b^4*c^3*d^4+35*b^5*c^4*d^3)*x^9+1/8*(a^5*d^7+35*a^4*b*c*d^6+210*a^3*b^2*
c^2*d^5+350*a^2*b^3*c^3*d^4+175*a*b^4*c^4*d^3+21*b^5*c^5*d^2)*x^8+1/7*(7*a^5*c*d
^6+105*a^4*b*c^2*d^5+350*a^3*b^2*c^3*d^4+350*a^2*b^3*c^4*d^3+105*a*b^4*c^5*d^2+7
*b^5*c^6*d)*x^7+1/6*(21*a^5*c^2*d^5+175*a^4*b*c^3*d^4+350*a^3*b^2*c^4*d^3+210*a^
2*b^3*c^5*d^2+35*a*b^4*c^6*d+b^5*c^7)*x^6+1/5*(35*a^5*c^3*d^4+175*a^4*b*c^4*d^3+
210*a^3*b^2*c^5*d^2+70*a^2*b^3*c^6*d+5*a*b^4*c^7)*x^5+1/4*(35*a^5*c^4*d^3+105*a^
4*b*c^5*d^2+70*a^3*b^2*c^6*d+10*a^2*b^3*c^7)*x^4+1/3*(21*a^5*c^5*d^2+35*a^4*b*c^
6*d+10*a^3*b^2*c^7)*x^3+1/2*(7*a^5*c^6*d+5*a^4*b*c^7)*x^2+a^5*c^7*x

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Maxima [A]  time = 1.33926, size = 802, normalized size = 5.57 \[ \frac{1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac{1}{12} \,{\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac{1}{11} \,{\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac{1}{2} \,{\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac{5}{9} \,{\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} +{\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} +{\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{7} + 7 \, a^{5} c^{6} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^7,x, algorithm="maxima")

[Out]

1/13*b^5*d^7*x^13 + a^5*c^7*x + 1/12*(7*b^5*c*d^6 + 5*a*b^4*d^7)*x^12 + 1/11*(21
*b^5*c^2*d^5 + 35*a*b^4*c*d^6 + 10*a^2*b^3*d^7)*x^11 + 1/2*(7*b^5*c^3*d^4 + 21*a
*b^4*c^2*d^5 + 14*a^2*b^3*c*d^6 + 2*a^3*b^2*d^7)*x^10 + 5/9*(7*b^5*c^4*d^3 + 35*
a*b^4*c^3*d^4 + 42*a^2*b^3*c^2*d^5 + 14*a^3*b^2*c*d^6 + a^4*b*d^7)*x^9 + 1/8*(21
*b^5*c^5*d^2 + 175*a*b^4*c^4*d^3 + 350*a^2*b^3*c^3*d^4 + 210*a^3*b^2*c^2*d^5 + 3
5*a^4*b*c*d^6 + a^5*d^7)*x^8 + (b^5*c^6*d + 15*a*b^4*c^5*d^2 + 50*a^2*b^3*c^4*d^
3 + 50*a^3*b^2*c^3*d^4 + 15*a^4*b*c^2*d^5 + a^5*c*d^6)*x^7 + 1/6*(b^5*c^7 + 35*a
*b^4*c^6*d + 210*a^2*b^3*c^5*d^2 + 350*a^3*b^2*c^4*d^3 + 175*a^4*b*c^3*d^4 + 21*
a^5*c^2*d^5)*x^6 + (a*b^4*c^7 + 14*a^2*b^3*c^6*d + 42*a^3*b^2*c^5*d^2 + 35*a^4*b
*c^4*d^3 + 7*a^5*c^3*d^4)*x^5 + 5/4*(2*a^2*b^3*c^7 + 14*a^3*b^2*c^6*d + 21*a^4*b
*c^5*d^2 + 7*a^5*c^4*d^3)*x^4 + 1/3*(10*a^3*b^2*c^7 + 35*a^4*b*c^6*d + 21*a^5*c^
5*d^2)*x^3 + 1/2*(5*a^4*b*c^7 + 7*a^5*c^6*d)*x^2

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Fricas [A]  time = 0.179958, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} d^{7} b^{5} + \frac{7}{12} x^{12} d^{6} c b^{5} + \frac{5}{12} x^{12} d^{7} b^{4} a + \frac{21}{11} x^{11} d^{5} c^{2} b^{5} + \frac{35}{11} x^{11} d^{6} c b^{4} a + \frac{10}{11} x^{11} d^{7} b^{3} a^{2} + \frac{7}{2} x^{10} d^{4} c^{3} b^{5} + \frac{21}{2} x^{10} d^{5} c^{2} b^{4} a + 7 x^{10} d^{6} c b^{3} a^{2} + x^{10} d^{7} b^{2} a^{3} + \frac{35}{9} x^{9} d^{3} c^{4} b^{5} + \frac{175}{9} x^{9} d^{4} c^{3} b^{4} a + \frac{70}{3} x^{9} d^{5} c^{2} b^{3} a^{2} + \frac{70}{9} x^{9} d^{6} c b^{2} a^{3} + \frac{5}{9} x^{9} d^{7} b a^{4} + \frac{21}{8} x^{8} d^{2} c^{5} b^{5} + \frac{175}{8} x^{8} d^{3} c^{4} b^{4} a + \frac{175}{4} x^{8} d^{4} c^{3} b^{3} a^{2} + \frac{105}{4} x^{8} d^{5} c^{2} b^{2} a^{3} + \frac{35}{8} x^{8} d^{6} c b a^{4} + \frac{1}{8} x^{8} d^{7} a^{5} + x^{7} d c^{6} b^{5} + 15 x^{7} d^{2} c^{5} b^{4} a + 50 x^{7} d^{3} c^{4} b^{3} a^{2} + 50 x^{7} d^{4} c^{3} b^{2} a^{3} + 15 x^{7} d^{5} c^{2} b a^{4} + x^{7} d^{6} c a^{5} + \frac{1}{6} x^{6} c^{7} b^{5} + \frac{35}{6} x^{6} d c^{6} b^{4} a + 35 x^{6} d^{2} c^{5} b^{3} a^{2} + \frac{175}{3} x^{6} d^{3} c^{4} b^{2} a^{3} + \frac{175}{6} x^{6} d^{4} c^{3} b a^{4} + \frac{7}{2} x^{6} d^{5} c^{2} a^{5} + x^{5} c^{7} b^{4} a + 14 x^{5} d c^{6} b^{3} a^{2} + 42 x^{5} d^{2} c^{5} b^{2} a^{3} + 35 x^{5} d^{3} c^{4} b a^{4} + 7 x^{5} d^{4} c^{3} a^{5} + \frac{5}{2} x^{4} c^{7} b^{3} a^{2} + \frac{35}{2} x^{4} d c^{6} b^{2} a^{3} + \frac{105}{4} x^{4} d^{2} c^{5} b a^{4} + \frac{35}{4} x^{4} d^{3} c^{4} a^{5} + \frac{10}{3} x^{3} c^{7} b^{2} a^{3} + \frac{35}{3} x^{3} d c^{6} b a^{4} + 7 x^{3} d^{2} c^{5} a^{5} + \frac{5}{2} x^{2} c^{7} b a^{4} + \frac{7}{2} x^{2} d c^{6} a^{5} + x c^{7} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^7,x, algorithm="fricas")

[Out]

1/13*x^13*d^7*b^5 + 7/12*x^12*d^6*c*b^5 + 5/12*x^12*d^7*b^4*a + 21/11*x^11*d^5*c
^2*b^5 + 35/11*x^11*d^6*c*b^4*a + 10/11*x^11*d^7*b^3*a^2 + 7/2*x^10*d^4*c^3*b^5
+ 21/2*x^10*d^5*c^2*b^4*a + 7*x^10*d^6*c*b^3*a^2 + x^10*d^7*b^2*a^3 + 35/9*x^9*d
^3*c^4*b^5 + 175/9*x^9*d^4*c^3*b^4*a + 70/3*x^9*d^5*c^2*b^3*a^2 + 70/9*x^9*d^6*c
*b^2*a^3 + 5/9*x^9*d^7*b*a^4 + 21/8*x^8*d^2*c^5*b^5 + 175/8*x^8*d^3*c^4*b^4*a +
175/4*x^8*d^4*c^3*b^3*a^2 + 105/4*x^8*d^5*c^2*b^2*a^3 + 35/8*x^8*d^6*c*b*a^4 + 1
/8*x^8*d^7*a^5 + x^7*d*c^6*b^5 + 15*x^7*d^2*c^5*b^4*a + 50*x^7*d^3*c^4*b^3*a^2 +
50*x^7*d^4*c^3*b^2*a^3 + 15*x^7*d^5*c^2*b*a^4 + x^7*d^6*c*a^5 + 1/6*x^6*c^7*b^5
+ 35/6*x^6*d*c^6*b^4*a + 35*x^6*d^2*c^5*b^3*a^2 + 175/3*x^6*d^3*c^4*b^2*a^3 + 1
75/6*x^6*d^4*c^3*b*a^4 + 7/2*x^6*d^5*c^2*a^5 + x^5*c^7*b^4*a + 14*x^5*d*c^6*b^3*
a^2 + 42*x^5*d^2*c^5*b^2*a^3 + 35*x^5*d^3*c^4*b*a^4 + 7*x^5*d^4*c^3*a^5 + 5/2*x^
4*c^7*b^3*a^2 + 35/2*x^4*d*c^6*b^2*a^3 + 105/4*x^4*d^2*c^5*b*a^4 + 35/4*x^4*d^3*
c^4*a^5 + 10/3*x^3*c^7*b^2*a^3 + 35/3*x^3*d*c^6*b*a^4 + 7*x^3*d^2*c^5*a^5 + 5/2*
x^2*c^7*b*a^4 + 7/2*x^2*d*c^6*a^5 + x*c^7*a^5

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Sympy [A]  time = 0.368807, size = 673, normalized size = 4.67 \[ a^{5} c^{7} x + \frac{b^{5} d^{7} x^{13}}{13} + x^{12} \left (\frac{5 a b^{4} d^{7}}{12} + \frac{7 b^{5} c d^{6}}{12}\right ) + x^{11} \left (\frac{10 a^{2} b^{3} d^{7}}{11} + \frac{35 a b^{4} c d^{6}}{11} + \frac{21 b^{5} c^{2} d^{5}}{11}\right ) + x^{10} \left (a^{3} b^{2} d^{7} + 7 a^{2} b^{3} c d^{6} + \frac{21 a b^{4} c^{2} d^{5}}{2} + \frac{7 b^{5} c^{3} d^{4}}{2}\right ) + x^{9} \left (\frac{5 a^{4} b d^{7}}{9} + \frac{70 a^{3} b^{2} c d^{6}}{9} + \frac{70 a^{2} b^{3} c^{2} d^{5}}{3} + \frac{175 a b^{4} c^{3} d^{4}}{9} + \frac{35 b^{5} c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac{a^{5} d^{7}}{8} + \frac{35 a^{4} b c d^{6}}{8} + \frac{105 a^{3} b^{2} c^{2} d^{5}}{4} + \frac{175 a^{2} b^{3} c^{3} d^{4}}{4} + \frac{175 a b^{4} c^{4} d^{3}}{8} + \frac{21 b^{5} c^{5} d^{2}}{8}\right ) + x^{7} \left (a^{5} c d^{6} + 15 a^{4} b c^{2} d^{5} + 50 a^{3} b^{2} c^{3} d^{4} + 50 a^{2} b^{3} c^{4} d^{3} + 15 a b^{4} c^{5} d^{2} + b^{5} c^{6} d\right ) + x^{6} \left (\frac{7 a^{5} c^{2} d^{5}}{2} + \frac{175 a^{4} b c^{3} d^{4}}{6} + \frac{175 a^{3} b^{2} c^{4} d^{3}}{3} + 35 a^{2} b^{3} c^{5} d^{2} + \frac{35 a b^{4} c^{6} d}{6} + \frac{b^{5} c^{7}}{6}\right ) + x^{5} \left (7 a^{5} c^{3} d^{4} + 35 a^{4} b c^{4} d^{3} + 42 a^{3} b^{2} c^{5} d^{2} + 14 a^{2} b^{3} c^{6} d + a b^{4} c^{7}\right ) + x^{4} \left (\frac{35 a^{5} c^{4} d^{3}}{4} + \frac{105 a^{4} b c^{5} d^{2}}{4} + \frac{35 a^{3} b^{2} c^{6} d}{2} + \frac{5 a^{2} b^{3} c^{7}}{2}\right ) + x^{3} \left (7 a^{5} c^{5} d^{2} + \frac{35 a^{4} b c^{6} d}{3} + \frac{10 a^{3} b^{2} c^{7}}{3}\right ) + x^{2} \left (\frac{7 a^{5} c^{6} d}{2} + \frac{5 a^{4} b c^{7}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(d*x+c)**7,x)

[Out]

a**5*c**7*x + b**5*d**7*x**13/13 + x**12*(5*a*b**4*d**7/12 + 7*b**5*c*d**6/12) +
x**11*(10*a**2*b**3*d**7/11 + 35*a*b**4*c*d**6/11 + 21*b**5*c**2*d**5/11) + x**
10*(a**3*b**2*d**7 + 7*a**2*b**3*c*d**6 + 21*a*b**4*c**2*d**5/2 + 7*b**5*c**3*d*
*4/2) + x**9*(5*a**4*b*d**7/9 + 70*a**3*b**2*c*d**6/9 + 70*a**2*b**3*c**2*d**5/3
+ 175*a*b**4*c**3*d**4/9 + 35*b**5*c**4*d**3/9) + x**8*(a**5*d**7/8 + 35*a**4*b
*c*d**6/8 + 105*a**3*b**2*c**2*d**5/4 + 175*a**2*b**3*c**3*d**4/4 + 175*a*b**4*c
**4*d**3/8 + 21*b**5*c**5*d**2/8) + x**7*(a**5*c*d**6 + 15*a**4*b*c**2*d**5 + 50
*a**3*b**2*c**3*d**4 + 50*a**2*b**3*c**4*d**3 + 15*a*b**4*c**5*d**2 + b**5*c**6*
d) + x**6*(7*a**5*c**2*d**5/2 + 175*a**4*b*c**3*d**4/6 + 175*a**3*b**2*c**4*d**3
/3 + 35*a**2*b**3*c**5*d**2 + 35*a*b**4*c**6*d/6 + b**5*c**7/6) + x**5*(7*a**5*c
**3*d**4 + 35*a**4*b*c**4*d**3 + 42*a**3*b**2*c**5*d**2 + 14*a**2*b**3*c**6*d +
a*b**4*c**7) + x**4*(35*a**5*c**4*d**3/4 + 105*a**4*b*c**5*d**2/4 + 35*a**3*b**2
*c**6*d/2 + 5*a**2*b**3*c**7/2) + x**3*(7*a**5*c**5*d**2 + 35*a**4*b*c**6*d/3 +
10*a**3*b**2*c**7/3) + x**2*(7*a**5*c**6*d/2 + 5*a**4*b*c**7/2)

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GIAC/XCAS [A]  time = 0.215069, size = 905, normalized size = 6.28 \[ \frac{1}{13} \, b^{5} d^{7} x^{13} + \frac{7}{12} \, b^{5} c d^{6} x^{12} + \frac{5}{12} \, a b^{4} d^{7} x^{12} + \frac{21}{11} \, b^{5} c^{2} d^{5} x^{11} + \frac{35}{11} \, a b^{4} c d^{6} x^{11} + \frac{10}{11} \, a^{2} b^{3} d^{7} x^{11} + \frac{7}{2} \, b^{5} c^{3} d^{4} x^{10} + \frac{21}{2} \, a b^{4} c^{2} d^{5} x^{10} + 7 \, a^{2} b^{3} c d^{6} x^{10} + a^{3} b^{2} d^{7} x^{10} + \frac{35}{9} \, b^{5} c^{4} d^{3} x^{9} + \frac{175}{9} \, a b^{4} c^{3} d^{4} x^{9} + \frac{70}{3} \, a^{2} b^{3} c^{2} d^{5} x^{9} + \frac{70}{9} \, a^{3} b^{2} c d^{6} x^{9} + \frac{5}{9} \, a^{4} b d^{7} x^{9} + \frac{21}{8} \, b^{5} c^{5} d^{2} x^{8} + \frac{175}{8} \, a b^{4} c^{4} d^{3} x^{8} + \frac{175}{4} \, a^{2} b^{3} c^{3} d^{4} x^{8} + \frac{105}{4} \, a^{3} b^{2} c^{2} d^{5} x^{8} + \frac{35}{8} \, a^{4} b c d^{6} x^{8} + \frac{1}{8} \, a^{5} d^{7} x^{8} + b^{5} c^{6} d x^{7} + 15 \, a b^{4} c^{5} d^{2} x^{7} + 50 \, a^{2} b^{3} c^{4} d^{3} x^{7} + 50 \, a^{3} b^{2} c^{3} d^{4} x^{7} + 15 \, a^{4} b c^{2} d^{5} x^{7} + a^{5} c d^{6} x^{7} + \frac{1}{6} \, b^{5} c^{7} x^{6} + \frac{35}{6} \, a b^{4} c^{6} d x^{6} + 35 \, a^{2} b^{3} c^{5} d^{2} x^{6} + \frac{175}{3} \, a^{3} b^{2} c^{4} d^{3} x^{6} + \frac{175}{6} \, a^{4} b c^{3} d^{4} x^{6} + \frac{7}{2} \, a^{5} c^{2} d^{5} x^{6} + a b^{4} c^{7} x^{5} + 14 \, a^{2} b^{3} c^{6} d x^{5} + 42 \, a^{3} b^{2} c^{5} d^{2} x^{5} + 35 \, a^{4} b c^{4} d^{3} x^{5} + 7 \, a^{5} c^{3} d^{4} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{7} x^{4} + \frac{35}{2} \, a^{3} b^{2} c^{6} d x^{4} + \frac{105}{4} \, a^{4} b c^{5} d^{2} x^{4} + \frac{35}{4} \, a^{5} c^{4} d^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{7} x^{3} + \frac{35}{3} \, a^{4} b c^{6} d x^{3} + 7 \, a^{5} c^{5} d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{7} x^{2} + \frac{7}{2} \, a^{5} c^{6} d x^{2} + a^{5} c^{7} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^7,x, algorithm="giac")

[Out]

1/13*b^5*d^7*x^13 + 7/12*b^5*c*d^6*x^12 + 5/12*a*b^4*d^7*x^12 + 21/11*b^5*c^2*d^
5*x^11 + 35/11*a*b^4*c*d^6*x^11 + 10/11*a^2*b^3*d^7*x^11 + 7/2*b^5*c^3*d^4*x^10
+ 21/2*a*b^4*c^2*d^5*x^10 + 7*a^2*b^3*c*d^6*x^10 + a^3*b^2*d^7*x^10 + 35/9*b^5*c
^4*d^3*x^9 + 175/9*a*b^4*c^3*d^4*x^9 + 70/3*a^2*b^3*c^2*d^5*x^9 + 70/9*a^3*b^2*c
*d^6*x^9 + 5/9*a^4*b*d^7*x^9 + 21/8*b^5*c^5*d^2*x^8 + 175/8*a*b^4*c^4*d^3*x^8 +
175/4*a^2*b^3*c^3*d^4*x^8 + 105/4*a^3*b^2*c^2*d^5*x^8 + 35/8*a^4*b*c*d^6*x^8 + 1
/8*a^5*d^7*x^8 + b^5*c^6*d*x^7 + 15*a*b^4*c^5*d^2*x^7 + 50*a^2*b^3*c^4*d^3*x^7 +
50*a^3*b^2*c^3*d^4*x^7 + 15*a^4*b*c^2*d^5*x^7 + a^5*c*d^6*x^7 + 1/6*b^5*c^7*x^6
+ 35/6*a*b^4*c^6*d*x^6 + 35*a^2*b^3*c^5*d^2*x^6 + 175/3*a^3*b^2*c^4*d^3*x^6 + 1
75/6*a^4*b*c^3*d^4*x^6 + 7/2*a^5*c^2*d^5*x^6 + a*b^4*c^7*x^5 + 14*a^2*b^3*c^6*d*
x^5 + 42*a^3*b^2*c^5*d^2*x^5 + 35*a^4*b*c^4*d^3*x^5 + 7*a^5*c^3*d^4*x^5 + 5/2*a^
2*b^3*c^7*x^4 + 35/2*a^3*b^2*c^6*d*x^4 + 105/4*a^4*b*c^5*d^2*x^4 + 35/4*a^5*c^4*
d^3*x^4 + 10/3*a^3*b^2*c^7*x^3 + 35/3*a^4*b*c^6*d*x^3 + 7*a^5*c^5*d^2*x^3 + 5/2*
a^4*b*c^7*x^2 + 7/2*a^5*c^6*d*x^2 + a^5*c^7*x