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]

-1/8*(-a*d+b*c)^5*(d*x+c)^8/d^6+5/9*b*(-a*d+b*c)^4*(d*x+c)^9/d^6-b^2*(-a*d+b*c)^3*(d*x+c)^10/d^6+10/11*b^3*(-a
*d+b*c)^2*(d*x+c)^11/d^6-5/12*b^4*(-a*d+b*c)*(d*x+c)^12/d^6+1/13*b^5*(d*x+c)^13/d^6

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Rubi [A]  time = 0.36, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ -\frac {5 b^4 (c+d x)^{12} (b c-a d)}{12 d^6}+\frac {10 b^3 (c+d x)^{11} (b c-a d)^2}{11 d^6}-\frac {b^2 (c+d x)^{10} (b c-a d)^3}{d^6}+\frac {5 b (c+d x)^9 (b c-a d)^4}{9 d^6}-\frac {(c+d x)^8 (b c-a d)^5}{8 d^6}+\frac {b^5 (c+d x)^{13}}{13 d^6} \]

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)

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)^5 (c+d x)^7 \, dx &=\int \left (\frac {(-b c+a d)^5 (c+d x)^7}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^8}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^9}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{10}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{11}}{d^5}+\frac {b^5 (c+d x)^{12}}{d^5}\right ) \, dx\\ &=-\frac {(b c-a d)^5 (c+d x)^8}{8 d^6}+\frac {5 b (b c-a d)^4 (c+d x)^9}{9 d^6}-\frac {b^2 (b c-a d)^3 (c+d x)^{10}}{d^6}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{11}}{11 d^6}-\frac {5 b^4 (b c-a d) (c+d x)^{12}}{12 d^6}+\frac {b^5 (c+d x)^{13}}{13 d^6}\\ \end {align*}

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Mathematica [B]  time = 0.08, 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 + 5
0*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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fricas [B]  time = 0.39, size = 670, normalized size = 4.65 \[ \frac {1}{13} x^{13} d^{7} b^{5} + \frac {7}{12} x^{12} d^{6} c b^{5} + \frac {5}{12} x^{12} d^{7} b^{4} a + \frac {21}{11} x^{11} d^{5} c^{2} b^{5} + \frac {35}{11} x^{11} d^{6} c b^{4} a + \frac {10}{11} x^{11} d^{7} b^{3} a^{2} + \frac {7}{2} x^{10} d^{4} c^{3} b^{5} + \frac {21}{2} x^{10} d^{5} c^{2} b^{4} a + 7 x^{10} d^{6} c b^{3} a^{2} + x^{10} d^{7} b^{2} a^{3} + \frac {35}{9} x^{9} d^{3} c^{4} b^{5} + \frac {175}{9} x^{9} d^{4} c^{3} b^{4} a + \frac {70}{3} x^{9} d^{5} c^{2} b^{3} a^{2} + \frac {70}{9} x^{9} d^{6} c b^{2} a^{3} + \frac {5}{9} x^{9} d^{7} b a^{4} + \frac {21}{8} x^{8} d^{2} c^{5} b^{5} + \frac {175}{8} x^{8} d^{3} c^{4} b^{4} a + \frac {175}{4} x^{8} d^{4} c^{3} b^{3} a^{2} + \frac {105}{4} x^{8} d^{5} c^{2} b^{2} a^{3} + \frac {35}{8} x^{8} d^{6} c b a^{4} + \frac {1}{8} x^{8} d^{7} a^{5} + x^{7} d c^{6} b^{5} + 15 x^{7} d^{2} c^{5} b^{4} a + 50 x^{7} d^{3} c^{4} b^{3} a^{2} + 50 x^{7} d^{4} c^{3} b^{2} a^{3} + 15 x^{7} d^{5} c^{2} b a^{4} + x^{7} d^{6} c a^{5} + \frac {1}{6} x^{6} c^{7} b^{5} + \frac {35}{6} x^{6} d c^{6} b^{4} a + 35 x^{6} d^{2} c^{5} b^{3} a^{2} + \frac {175}{3} x^{6} d^{3} c^{4} b^{2} a^{3} + \frac {175}{6} x^{6} d^{4} c^{3} b a^{4} + \frac {7}{2} x^{6} d^{5} c^{2} a^{5} + x^{5} c^{7} b^{4} a + 14 x^{5} d c^{6} b^{3} a^{2} + 42 x^{5} d^{2} c^{5} b^{2} a^{3} + 35 x^{5} d^{3} c^{4} b a^{4} + 7 x^{5} d^{4} c^{3} a^{5} + \frac {5}{2} x^{4} c^{7} b^{3} a^{2} + \frac {35}{2} x^{4} d c^{6} b^{2} a^{3} + \frac {105}{4} x^{4} d^{2} c^{5} b a^{4} + \frac {35}{4} x^{4} d^{3} c^{4} a^{5} + \frac {10}{3} x^{3} c^{7} b^{2} a^{3} + \frac {35}{3} x^{3} d c^{6} b a^{4} + 7 x^{3} d^{2} c^{5} a^{5} + \frac {5}{2} x^{2} c^{7} b a^{4} + \frac {7}{2} x^{2} d c^{6} a^{5} + x c^{7} a^{5} \]

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 + 175/6*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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giac [B]  time = 1.28, size = 670, normalized size = 4.65 \[ \frac {1}{13} \, b^{5} d^{7} x^{13} + \frac {7}{12} \, b^{5} c d^{6} x^{12} + \frac {5}{12} \, a b^{4} d^{7} x^{12} + \frac {21}{11} \, b^{5} c^{2} d^{5} x^{11} + \frac {35}{11} \, a b^{4} c d^{6} x^{11} + \frac {10}{11} \, a^{2} b^{3} d^{7} x^{11} + \frac {7}{2} \, b^{5} c^{3} d^{4} x^{10} + \frac {21}{2} \, a b^{4} c^{2} d^{5} x^{10} + 7 \, a^{2} b^{3} c d^{6} x^{10} + a^{3} b^{2} d^{7} x^{10} + \frac {35}{9} \, b^{5} c^{4} d^{3} x^{9} + \frac {175}{9} \, a b^{4} c^{3} d^{4} x^{9} + \frac {70}{3} \, a^{2} b^{3} c^{2} d^{5} x^{9} + \frac {70}{9} \, a^{3} b^{2} c d^{6} x^{9} + \frac {5}{9} \, a^{4} b d^{7} x^{9} + \frac {21}{8} \, b^{5} c^{5} d^{2} x^{8} + \frac {175}{8} \, a b^{4} c^{4} d^{3} x^{8} + \frac {175}{4} \, a^{2} b^{3} c^{3} d^{4} x^{8} + \frac {105}{4} \, a^{3} b^{2} c^{2} d^{5} x^{8} + \frac {35}{8} \, a^{4} b c d^{6} x^{8} + \frac {1}{8} \, a^{5} d^{7} x^{8} + b^{5} c^{6} d x^{7} + 15 \, a b^{4} c^{5} d^{2} x^{7} + 50 \, a^{2} b^{3} c^{4} d^{3} x^{7} + 50 \, a^{3} b^{2} c^{3} d^{4} x^{7} + 15 \, a^{4} b c^{2} d^{5} x^{7} + a^{5} c d^{6} x^{7} + \frac {1}{6} \, b^{5} c^{7} x^{6} + \frac {35}{6} \, a b^{4} c^{6} d x^{6} + 35 \, a^{2} b^{3} c^{5} d^{2} x^{6} + \frac {175}{3} \, a^{3} b^{2} c^{4} d^{3} x^{6} + \frac {175}{6} \, a^{4} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{5} c^{2} d^{5} x^{6} + a b^{4} c^{7} x^{5} + 14 \, a^{2} b^{3} c^{6} d x^{5} + 42 \, a^{3} b^{2} c^{5} d^{2} x^{5} + 35 \, a^{4} b c^{4} d^{3} x^{5} + 7 \, a^{5} c^{3} d^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{7} x^{4} + \frac {35}{2} \, a^{3} b^{2} c^{6} d x^{4} + \frac {105}{4} \, a^{4} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{5} c^{4} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{7} x^{3} + \frac {35}{3} \, a^{4} b c^{6} d x^{3} + 7 \, a^{5} c^{5} d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{7} x^{2} + \frac {7}{2} \, a^{5} c^{6} d x^{2} + a^{5} c^{7} x \]

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^1
1 + 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 + 175/6*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

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maple [B]  time = 0.00, size = 601, normalized size = 4.17 \[ \frac {b^{5} d^{7} x^{13}}{13}+a^{5} c^{7} x +\frac {\left (5 a \,b^{4} d^{7}+7 b^{5} c \,d^{6}\right ) x^{12}}{12}+\frac {\left (10 a^{2} b^{3} d^{7}+35 a \,b^{4} c \,d^{6}+21 b^{5} c^{2} d^{5}\right ) x^{11}}{11}+\frac {\left (10 a^{3} b^{2} d^{7}+70 a^{2} b^{3} c \,d^{6}+105 a \,b^{4} c^{2} d^{5}+35 b^{5} c^{3} d^{4}\right ) x^{10}}{10}+\frac {\left (5 a^{4} b \,d^{7}+70 a^{3} b^{2} c \,d^{6}+210 a^{2} b^{3} c^{2} d^{5}+175 a \,b^{4} c^{3} d^{4}+35 b^{5} c^{4} d^{3}\right ) x^{9}}{9}+\frac {\left (a^{5} d^{7}+35 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+350 a^{2} b^{3} c^{3} d^{4}+175 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{8}}{8}+\frac {\left (7 a^{5} c \,d^{6}+105 a^{4} b \,c^{2} d^{5}+350 a^{3} b^{2} c^{3} d^{4}+350 a^{2} b^{3} c^{4} d^{3}+105 a \,b^{4} c^{5} d^{2}+7 b^{5} c^{6} d \right ) x^{7}}{7}+\frac {\left (21 a^{5} c^{2} d^{5}+175 a^{4} b \,c^{3} d^{4}+350 a^{3} b^{2} c^{4} d^{3}+210 a^{2} b^{3} c^{5} d^{2}+35 a \,b^{4} c^{6} d +b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{5} c^{3} d^{4}+175 a^{4} b \,c^{4} d^{3}+210 a^{3} b^{2} c^{5} d^{2}+70 a^{2} b^{3} c^{6} d +5 a \,b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{5} c^{4} d^{3}+105 a^{4} b \,c^{5} d^{2}+70 a^{3} b^{2} c^{6} d +10 a^{2} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{5} c^{5} d^{2}+35 a^{4} b \,c^{6} d +10 a^{3} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{5} c^{6} d +5 a^{4} b \,c^{7}\right ) x^{2}}{2} \]

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+35
0*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 [B]  time = 1.38, size = 594, normalized size = 4.12 \[ \frac {1}{13} \, b^{5} d^{7} x^{13} + a^{5} c^{7} x + \frac {1}{12} \, {\left (7 \, b^{5} c d^{6} + 5 \, a b^{4} d^{7}\right )} x^{12} + \frac {1}{11} \, {\left (21 \, b^{5} c^{2} d^{5} + 35 \, a b^{4} c d^{6} + 10 \, a^{2} b^{3} d^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{5} c^{3} d^{4} + 21 \, a b^{4} c^{2} d^{5} + 14 \, a^{2} b^{3} c d^{6} + 2 \, a^{3} b^{2} d^{7}\right )} x^{10} + \frac {5}{9} \, {\left (7 \, b^{5} c^{4} d^{3} + 35 \, a b^{4} c^{3} d^{4} + 42 \, a^{2} b^{3} c^{2} d^{5} + 14 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (21 \, b^{5} c^{5} d^{2} + 175 \, a b^{4} c^{4} d^{3} + 350 \, a^{2} b^{3} c^{3} d^{4} + 210 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{8} + {\left (b^{5} c^{6} d + 15 \, a b^{4} c^{5} d^{2} + 50 \, a^{2} b^{3} c^{4} d^{3} + 50 \, a^{3} b^{2} c^{3} d^{4} + 15 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{7} + 35 \, a b^{4} c^{6} d + 210 \, a^{2} b^{3} c^{5} d^{2} + 350 \, a^{3} b^{2} c^{4} d^{3} + 175 \, a^{4} b c^{3} d^{4} + 21 \, a^{5} c^{2} d^{5}\right )} x^{6} + {\left (a b^{4} c^{7} + 14 \, a^{2} b^{3} c^{6} d + 42 \, a^{3} b^{2} c^{5} d^{2} + 35 \, a^{4} b c^{4} d^{3} + 7 \, a^{5} c^{3} d^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} c^{7} + 14 \, a^{3} b^{2} c^{6} d + 21 \, a^{4} b c^{5} d^{2} + 7 \, a^{5} c^{4} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{7} + 35 \, a^{4} b c^{6} d + 21 \, a^{5} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{7} + 7 \, a^{5} c^{6} d\right )} x^{2} \]

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 + 35*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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mupad [B]  time = 0.21, size = 570, normalized size = 3.96 \[ x^7\,\left (a^5\,c\,d^6+15\,a^4\,b\,c^2\,d^5+50\,a^3\,b^2\,c^3\,d^4+50\,a^2\,b^3\,c^4\,d^3+15\,a\,b^4\,c^5\,d^2+b^5\,c^6\,d\right )+x^6\,\left (\frac {7\,a^5\,c^2\,d^5}{2}+\frac {175\,a^4\,b\,c^3\,d^4}{6}+\frac {175\,a^3\,b^2\,c^4\,d^3}{3}+35\,a^2\,b^3\,c^5\,d^2+\frac {35\,a\,b^4\,c^6\,d}{6}+\frac {b^5\,c^7}{6}\right )+x^8\,\left (\frac {a^5\,d^7}{8}+\frac {35\,a^4\,b\,c\,d^6}{8}+\frac {105\,a^3\,b^2\,c^2\,d^5}{4}+\frac {175\,a^2\,b^3\,c^3\,d^4}{4}+\frac {175\,a\,b^4\,c^4\,d^3}{8}+\frac {21\,b^5\,c^5\,d^2}{8}\right )+x^5\,\left (7\,a^5\,c^3\,d^4+35\,a^4\,b\,c^4\,d^3+42\,a^3\,b^2\,c^5\,d^2+14\,a^2\,b^3\,c^6\,d+a\,b^4\,c^7\right )+x^9\,\left (\frac {5\,a^4\,b\,d^7}{9}+\frac {70\,a^3\,b^2\,c\,d^6}{9}+\frac {70\,a^2\,b^3\,c^2\,d^5}{3}+\frac {175\,a\,b^4\,c^3\,d^4}{9}+\frac {35\,b^5\,c^4\,d^3}{9}\right )+a^5\,c^7\,x+\frac {b^5\,d^7\,x^{13}}{13}+\frac {5\,a^2\,c^4\,x^4\,\left (7\,a^3\,d^3+21\,a^2\,b\,c\,d^2+14\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{4}+\frac {b^2\,d^4\,x^{10}\,\left (2\,a^3\,d^3+14\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )}{2}+\frac {a^4\,c^6\,x^2\,\left (7\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d^6\,x^{12}\,\left (5\,a\,d+7\,b\,c\right )}{12}+\frac {a^3\,c^5\,x^3\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3}+\frac {b^3\,d^5\,x^{11}\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5*(c + d*x)^7,x)

[Out]

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

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sympy [B]  time = 0.16, 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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