3.1305 \(\int (a+b x)^6 (c+d x)^{10} \, dx\)

Optimal. Leaf size=170 \[ -\frac{3 b^5 (c+d x)^{16} (b c-a d)}{8 d^7}+\frac{b^4 (c+d x)^{15} (b c-a d)^2}{d^7}-\frac{10 b^3 (c+d x)^{14} (b c-a d)^3}{7 d^7}+\frac{15 b^2 (c+d x)^{13} (b c-a d)^4}{13 d^7}-\frac{b (c+d x)^{12} (b c-a d)^5}{2 d^7}+\frac{(c+d x)^{11} (b c-a d)^6}{11 d^7}+\frac{b^6 (c+d x)^{17}}{17 d^7} \]

[Out]

((b*c - a*d)^6*(c + d*x)^11)/(11*d^7) - (b*(b*c - a*d)^5*(c + d*x)^12)/(2*d^7) + (15*b^2*(b*c - a*d)^4*(c + d*
x)^13)/(13*d^7) - (10*b^3*(b*c - a*d)^3*(c + d*x)^14)/(7*d^7) + (b^4*(b*c - a*d)^2*(c + d*x)^15)/d^7 - (3*b^5*
(b*c - a*d)*(c + d*x)^16)/(8*d^7) + (b^6*(c + d*x)^17)/(17*d^7)

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Rubi [A]  time = 0.672839, antiderivative size = 170, 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, Rules used = {43} \[ -\frac{3 b^5 (c+d x)^{16} (b c-a d)}{8 d^7}+\frac{b^4 (c+d x)^{15} (b c-a d)^2}{d^7}-\frac{10 b^3 (c+d x)^{14} (b c-a d)^3}{7 d^7}+\frac{15 b^2 (c+d x)^{13} (b c-a d)^4}{13 d^7}-\frac{b (c+d x)^{12} (b c-a d)^5}{2 d^7}+\frac{(c+d x)^{11} (b c-a d)^6}{11 d^7}+\frac{b^6 (c+d x)^{17}}{17 d^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(c + d*x)^10,x]

[Out]

((b*c - a*d)^6*(c + d*x)^11)/(11*d^7) - (b*(b*c - a*d)^5*(c + d*x)^12)/(2*d^7) + (15*b^2*(b*c - a*d)^4*(c + d*
x)^13)/(13*d^7) - (10*b^3*(b*c - a*d)^3*(c + d*x)^14)/(7*d^7) + (b^4*(b*c - a*d)^2*(c + d*x)^15)/d^7 - (3*b^5*
(b*c - a*d)*(c + d*x)^16)/(8*d^7) + (b^6*(c + d*x)^17)/(17*d^7)

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)^6 (c+d x)^{10} \, dx &=\int \left (\frac{(-b c+a d)^6 (c+d x)^{10}}{d^6}-\frac{6 b (b c-a d)^5 (c+d x)^{11}}{d^6}+\frac{15 b^2 (b c-a d)^4 (c+d x)^{12}}{d^6}-\frac{20 b^3 (b c-a d)^3 (c+d x)^{13}}{d^6}+\frac{15 b^4 (b c-a d)^2 (c+d x)^{14}}{d^6}-\frac{6 b^5 (b c-a d) (c+d x)^{15}}{d^6}+\frac{b^6 (c+d x)^{16}}{d^6}\right ) \, dx\\ &=\frac{(b c-a d)^6 (c+d x)^{11}}{11 d^7}-\frac{b (b c-a d)^5 (c+d x)^{12}}{2 d^7}+\frac{15 b^2 (b c-a d)^4 (c+d x)^{13}}{13 d^7}-\frac{10 b^3 (b c-a d)^3 (c+d x)^{14}}{7 d^7}+\frac{b^4 (b c-a d)^2 (c+d x)^{15}}{d^7}-\frac{3 b^5 (b c-a d) (c+d x)^{16}}{8 d^7}+\frac{b^6 (c+d x)^{17}}{17 d^7}\\ \end{align*}

Mathematica [B]  time = 0.112031, size = 939, normalized size = 5.52 \[ \frac{1}{17} b^6 d^{10} x^{17}+\frac{1}{8} b^5 d^9 (5 b c+3 a d) x^{16}+b^4 d^8 \left (3 b^2 c^2+4 a b d c+a^2 d^2\right ) x^{15}+\frac{5}{7} b^3 d^7 \left (12 b^3 c^3+27 a b^2 d c^2+15 a^2 b d^2 c+2 a^3 d^3\right ) x^{14}+\frac{5}{13} b^2 d^6 \left (42 b^4 c^4+144 a b^3 d c^3+135 a^2 b^2 d^2 c^2+40 a^3 b d^3 c+3 a^4 d^4\right ) x^{13}+\frac{1}{2} b d^5 \left (42 b^5 c^5+210 a b^4 d c^4+300 a^2 b^3 d^2 c^3+150 a^3 b^2 d^3 c^2+25 a^4 b d^4 c+a^5 d^5\right ) x^{12}+\frac{1}{11} d^4 \left (210 b^6 c^6+1512 a b^5 d c^5+3150 a^2 b^4 d^2 c^4+2400 a^3 b^3 d^3 c^3+675 a^4 b^2 d^4 c^2+60 a^5 b d^5 c+a^6 d^6\right ) x^{11}+c d^3 \left (12 b^6 c^6+126 a b^5 d c^5+378 a^2 b^4 d^2 c^4+420 a^3 b^3 d^3 c^3+180 a^4 b^2 d^4 c^2+27 a^5 b d^5 c+a^6 d^6\right ) x^{10}+5 c^2 d^2 \left (b^6 c^6+16 a b^5 d c^5+70 a^2 b^4 d^2 c^4+112 a^3 b^3 d^3 c^3+70 a^4 b^2 d^4 c^2+16 a^5 b d^5 c+a^6 d^6\right ) x^9+\frac{5}{4} c^3 d \left (b^6 c^6+27 a b^5 d c^5+180 a^2 b^4 d^2 c^4+420 a^3 b^3 d^3 c^3+378 a^4 b^2 d^4 c^2+126 a^5 b d^5 c+12 a^6 d^6\right ) x^8+\frac{1}{7} c^4 \left (b^6 c^6+60 a b^5 d c^5+675 a^2 b^4 d^2 c^4+2400 a^3 b^3 d^3 c^3+3150 a^4 b^2 d^4 c^2+1512 a^5 b d^5 c+210 a^6 d^6\right ) x^7+a c^5 \left (b^5 c^5+25 a b^4 d c^4+150 a^2 b^3 d^2 c^3+300 a^3 b^2 d^3 c^2+210 a^4 b d^4 c+42 a^5 d^5\right ) x^6+a^2 c^6 \left (3 b^4 c^4+40 a b^3 d c^3+135 a^2 b^2 d^2 c^2+144 a^3 b d^3 c+42 a^4 d^4\right ) x^5+\frac{5}{2} a^3 c^7 \left (2 b^3 c^3+15 a b^2 d c^2+27 a^2 b d^2 c+12 a^3 d^3\right ) x^4+5 a^4 c^8 \left (b^2 c^2+4 a b d c+3 a^2 d^2\right ) x^3+a^5 c^9 (3 b c+5 a d) x^2+a^6 c^{10} x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(c + d*x)^10,x]

[Out]

a^6*c^10*x + a^5*c^9*(3*b*c + 5*a*d)*x^2 + 5*a^4*c^8*(b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*x^3 + (5*a^3*c^7*(2*b^3
*c^3 + 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 12*a^3*d^3)*x^4)/2 + a^2*c^6*(3*b^4*c^4 + 40*a*b^3*c^3*d + 135*a^2*b^
2*c^2*d^2 + 144*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + a*c^5*(b^5*c^5 + 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 300*a^
3*b^2*c^2*d^3 + 210*a^4*b*c*d^4 + 42*a^5*d^5)*x^6 + (c^4*(b^6*c^6 + 60*a*b^5*c^5*d + 675*a^2*b^4*c^4*d^2 + 240
0*a^3*b^3*c^3*d^3 + 3150*a^4*b^2*c^2*d^4 + 1512*a^5*b*c*d^5 + 210*a^6*d^6)*x^7)/7 + (5*c^3*d*(b^6*c^6 + 27*a*b
^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 378*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 + 12*a^6*d^6)*x^8
)/4 + 5*c^2*d^2*(b^6*c^6 + 16*a*b^5*c^5*d + 70*a^2*b^4*c^4*d^2 + 112*a^3*b^3*c^3*d^3 + 70*a^4*b^2*c^2*d^4 + 16
*a^5*b*c*d^5 + a^6*d^6)*x^9 + c*d^3*(12*b^6*c^6 + 126*a*b^5*c^5*d + 378*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3
+ 180*a^4*b^2*c^2*d^4 + 27*a^5*b*c*d^5 + a^6*d^6)*x^10 + (d^4*(210*b^6*c^6 + 1512*a*b^5*c^5*d + 3150*a^2*b^4*c
^4*d^2 + 2400*a^3*b^3*c^3*d^3 + 675*a^4*b^2*c^2*d^4 + 60*a^5*b*c*d^5 + a^6*d^6)*x^11)/11 + (b*d^5*(42*b^5*c^5
+ 210*a*b^4*c^4*d + 300*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 + 25*a^4*b*c*d^4 + a^5*d^5)*x^12)/2 + (5*b^2*d^6
*(42*b^4*c^4 + 144*a*b^3*c^3*d + 135*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 + 3*a^4*d^4)*x^13)/13 + (5*b^3*d^7*(12*b
^3*c^3 + 27*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 2*a^3*d^3)*x^14)/7 + b^4*d^8*(3*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^15
+ (b^5*d^9*(5*b*c + 3*a*d)*x^16)/8 + (b^6*d^10*x^17)/17

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Maple [B]  time = 0.003, size = 991, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(d*x+c)^10,x)

[Out]

1/17*b^6*d^10*x^17+1/16*(6*a*b^5*d^10+10*b^6*c*d^9)*x^16+1/15*(15*a^2*b^4*d^10+60*a*b^5*c*d^9+45*b^6*c^2*d^8)*
x^15+1/14*(20*a^3*b^3*d^10+150*a^2*b^4*c*d^9+270*a*b^5*c^2*d^8+120*b^6*c^3*d^7)*x^14+1/13*(15*a^4*b^2*d^10+200
*a^3*b^3*c*d^9+675*a^2*b^4*c^2*d^8+720*a*b^5*c^3*d^7+210*b^6*c^4*d^6)*x^13+1/12*(6*a^5*b*d^10+150*a^4*b^2*c*d^
9+900*a^3*b^3*c^2*d^8+1800*a^2*b^4*c^3*d^7+1260*a*b^5*c^4*d^6+252*b^6*c^5*d^5)*x^12+1/11*(a^6*d^10+60*a^5*b*c*
d^9+675*a^4*b^2*c^2*d^8+2400*a^3*b^3*c^3*d^7+3150*a^2*b^4*c^4*d^6+1512*a*b^5*c^5*d^5+210*b^6*c^6*d^4)*x^11+1/1
0*(10*a^6*c*d^9+270*a^5*b*c^2*d^8+1800*a^4*b^2*c^3*d^7+4200*a^3*b^3*c^4*d^6+3780*a^2*b^4*c^5*d^5+1260*a*b^5*c^
6*d^4+120*b^6*c^7*d^3)*x^10+1/9*(45*a^6*c^2*d^8+720*a^5*b*c^3*d^7+3150*a^4*b^2*c^4*d^6+5040*a^3*b^3*c^5*d^5+31
50*a^2*b^4*c^6*d^4+720*a*b^5*c^7*d^3+45*b^6*c^8*d^2)*x^9+1/8*(120*a^6*c^3*d^7+1260*a^5*b*c^4*d^6+3780*a^4*b^2*
c^5*d^5+4200*a^3*b^3*c^6*d^4+1800*a^2*b^4*c^7*d^3+270*a*b^5*c^8*d^2+10*b^6*c^9*d)*x^8+1/7*(210*a^6*c^4*d^6+151
2*a^5*b*c^5*d^5+3150*a^4*b^2*c^6*d^4+2400*a^3*b^3*c^7*d^3+675*a^2*b^4*c^8*d^2+60*a*b^5*c^9*d+b^6*c^10)*x^7+1/6
*(252*a^6*c^5*d^5+1260*a^5*b*c^6*d^4+1800*a^4*b^2*c^7*d^3+900*a^3*b^3*c^8*d^2+150*a^2*b^4*c^9*d+6*a*b^5*c^10)*
x^6+1/5*(210*a^6*c^6*d^4+720*a^5*b*c^7*d^3+675*a^4*b^2*c^8*d^2+200*a^3*b^3*c^9*d+15*a^2*b^4*c^10)*x^5+1/4*(120
*a^6*c^7*d^3+270*a^5*b*c^8*d^2+150*a^4*b^2*c^9*d+20*a^3*b^3*c^10)*x^4+1/3*(45*a^6*c^8*d^2+60*a^5*b*c^9*d+15*a^
4*b^2*c^10)*x^3+1/2*(10*a^6*c^9*d+6*a^5*b*c^10)*x^2+a^6*c^10*x

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Maxima [B]  time = 1.00393, size = 1319, normalized size = 7.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/17*b^6*d^10*x^17 + a^6*c^10*x + 1/8*(5*b^6*c*d^9 + 3*a*b^5*d^10)*x^16 + (3*b^6*c^2*d^8 + 4*a*b^5*c*d^9 + a^2
*b^4*d^10)*x^15 + 5/7*(12*b^6*c^3*d^7 + 27*a*b^5*c^2*d^8 + 15*a^2*b^4*c*d^9 + 2*a^3*b^3*d^10)*x^14 + 5/13*(42*
b^6*c^4*d^6 + 144*a*b^5*c^3*d^7 + 135*a^2*b^4*c^2*d^8 + 40*a^3*b^3*c*d^9 + 3*a^4*b^2*d^10)*x^13 + 1/2*(42*b^6*
c^5*d^5 + 210*a*b^5*c^4*d^6 + 300*a^2*b^4*c^3*d^7 + 150*a^3*b^3*c^2*d^8 + 25*a^4*b^2*c*d^9 + a^5*b*d^10)*x^12
+ 1/11*(210*b^6*c^6*d^4 + 1512*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 + 2400*a^3*b^3*c^3*d^7 + 675*a^4*b^2*c^2*d
^8 + 60*a^5*b*c*d^9 + a^6*d^10)*x^11 + (12*b^6*c^7*d^3 + 126*a*b^5*c^6*d^4 + 378*a^2*b^4*c^5*d^5 + 420*a^3*b^3
*c^4*d^6 + 180*a^4*b^2*c^3*d^7 + 27*a^5*b*c^2*d^8 + a^6*c*d^9)*x^10 + 5*(b^6*c^8*d^2 + 16*a*b^5*c^7*d^3 + 70*a
^2*b^4*c^6*d^4 + 112*a^3*b^3*c^5*d^5 + 70*a^4*b^2*c^4*d^6 + 16*a^5*b*c^3*d^7 + a^6*c^2*d^8)*x^9 + 5/4*(b^6*c^9
*d + 27*a*b^5*c^8*d^2 + 180*a^2*b^4*c^7*d^3 + 420*a^3*b^3*c^6*d^4 + 378*a^4*b^2*c^5*d^5 + 126*a^5*b*c^4*d^6 +
12*a^6*c^3*d^7)*x^8 + 1/7*(b^6*c^10 + 60*a*b^5*c^9*d + 675*a^2*b^4*c^8*d^2 + 2400*a^3*b^3*c^7*d^3 + 3150*a^4*b
^2*c^6*d^4 + 1512*a^5*b*c^5*d^5 + 210*a^6*c^4*d^6)*x^7 + (a*b^5*c^10 + 25*a^2*b^4*c^9*d + 150*a^3*b^3*c^8*d^2
+ 300*a^4*b^2*c^7*d^3 + 210*a^5*b*c^6*d^4 + 42*a^6*c^5*d^5)*x^6 + (3*a^2*b^4*c^10 + 40*a^3*b^3*c^9*d + 135*a^4
*b^2*c^8*d^2 + 144*a^5*b*c^7*d^3 + 42*a^6*c^6*d^4)*x^5 + 5/2*(2*a^3*b^3*c^10 + 15*a^4*b^2*c^9*d + 27*a^5*b*c^8
*d^2 + 12*a^6*c^7*d^3)*x^4 + 5*(a^4*b^2*c^10 + 4*a^5*b*c^9*d + 3*a^6*c^8*d^2)*x^3 + (3*a^5*b*c^10 + 5*a^6*c^9*
d)*x^2

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Fricas [B]  time = 1.54228, size = 2484, normalized size = 14.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/17*x^17*d^10*b^6 + 5/8*x^16*d^9*c*b^6 + 3/8*x^16*d^10*b^5*a + 3*x^15*d^8*c^2*b^6 + 4*x^15*d^9*c*b^5*a + x^15
*d^10*b^4*a^2 + 60/7*x^14*d^7*c^3*b^6 + 135/7*x^14*d^8*c^2*b^5*a + 75/7*x^14*d^9*c*b^4*a^2 + 10/7*x^14*d^10*b^
3*a^3 + 210/13*x^13*d^6*c^4*b^6 + 720/13*x^13*d^7*c^3*b^5*a + 675/13*x^13*d^8*c^2*b^4*a^2 + 200/13*x^13*d^9*c*
b^3*a^3 + 15/13*x^13*d^10*b^2*a^4 + 21*x^12*d^5*c^5*b^6 + 105*x^12*d^6*c^4*b^5*a + 150*x^12*d^7*c^3*b^4*a^2 +
75*x^12*d^8*c^2*b^3*a^3 + 25/2*x^12*d^9*c*b^2*a^4 + 1/2*x^12*d^10*b*a^5 + 210/11*x^11*d^4*c^6*b^6 + 1512/11*x^
11*d^5*c^5*b^5*a + 3150/11*x^11*d^6*c^4*b^4*a^2 + 2400/11*x^11*d^7*c^3*b^3*a^3 + 675/11*x^11*d^8*c^2*b^2*a^4 +
60/11*x^11*d^9*c*b*a^5 + 1/11*x^11*d^10*a^6 + 12*x^10*d^3*c^7*b^6 + 126*x^10*d^4*c^6*b^5*a + 378*x^10*d^5*c^5
*b^4*a^2 + 420*x^10*d^6*c^4*b^3*a^3 + 180*x^10*d^7*c^3*b^2*a^4 + 27*x^10*d^8*c^2*b*a^5 + x^10*d^9*c*a^6 + 5*x^
9*d^2*c^8*b^6 + 80*x^9*d^3*c^7*b^5*a + 350*x^9*d^4*c^6*b^4*a^2 + 560*x^9*d^5*c^5*b^3*a^3 + 350*x^9*d^6*c^4*b^2
*a^4 + 80*x^9*d^7*c^3*b*a^5 + 5*x^9*d^8*c^2*a^6 + 5/4*x^8*d*c^9*b^6 + 135/4*x^8*d^2*c^8*b^5*a + 225*x^8*d^3*c^
7*b^4*a^2 + 525*x^8*d^4*c^6*b^3*a^3 + 945/2*x^8*d^5*c^5*b^2*a^4 + 315/2*x^8*d^6*c^4*b*a^5 + 15*x^8*d^7*c^3*a^6
+ 1/7*x^7*c^10*b^6 + 60/7*x^7*d*c^9*b^5*a + 675/7*x^7*d^2*c^8*b^4*a^2 + 2400/7*x^7*d^3*c^7*b^3*a^3 + 450*x^7*
d^4*c^6*b^2*a^4 + 216*x^7*d^5*c^5*b*a^5 + 30*x^7*d^6*c^4*a^6 + x^6*c^10*b^5*a + 25*x^6*d*c^9*b^4*a^2 + 150*x^6
*d^2*c^8*b^3*a^3 + 300*x^6*d^3*c^7*b^2*a^4 + 210*x^6*d^4*c^6*b*a^5 + 42*x^6*d^5*c^5*a^6 + 3*x^5*c^10*b^4*a^2 +
40*x^5*d*c^9*b^3*a^3 + 135*x^5*d^2*c^8*b^2*a^4 + 144*x^5*d^3*c^7*b*a^5 + 42*x^5*d^4*c^6*a^6 + 5*x^4*c^10*b^3*
a^3 + 75/2*x^4*d*c^9*b^2*a^4 + 135/2*x^4*d^2*c^8*b*a^5 + 30*x^4*d^3*c^7*a^6 + 5*x^3*c^10*b^2*a^4 + 20*x^3*d*c^
9*b*a^5 + 15*x^3*d^2*c^8*a^6 + 3*x^2*c^10*b*a^5 + 5*x^2*d*c^9*a^6 + x*c^10*a^6

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Sympy [B]  time = 0.187351, size = 1088, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(d*x+c)**10,x)

[Out]

a**6*c**10*x + b**6*d**10*x**17/17 + x**16*(3*a*b**5*d**10/8 + 5*b**6*c*d**9/8) + x**15*(a**2*b**4*d**10 + 4*a
*b**5*c*d**9 + 3*b**6*c**2*d**8) + x**14*(10*a**3*b**3*d**10/7 + 75*a**2*b**4*c*d**9/7 + 135*a*b**5*c**2*d**8/
7 + 60*b**6*c**3*d**7/7) + x**13*(15*a**4*b**2*d**10/13 + 200*a**3*b**3*c*d**9/13 + 675*a**2*b**4*c**2*d**8/13
+ 720*a*b**5*c**3*d**7/13 + 210*b**6*c**4*d**6/13) + x**12*(a**5*b*d**10/2 + 25*a**4*b**2*c*d**9/2 + 75*a**3*
b**3*c**2*d**8 + 150*a**2*b**4*c**3*d**7 + 105*a*b**5*c**4*d**6 + 21*b**6*c**5*d**5) + x**11*(a**6*d**10/11 +
60*a**5*b*c*d**9/11 + 675*a**4*b**2*c**2*d**8/11 + 2400*a**3*b**3*c**3*d**7/11 + 3150*a**2*b**4*c**4*d**6/11 +
1512*a*b**5*c**5*d**5/11 + 210*b**6*c**6*d**4/11) + x**10*(a**6*c*d**9 + 27*a**5*b*c**2*d**8 + 180*a**4*b**2*
c**3*d**7 + 420*a**3*b**3*c**4*d**6 + 378*a**2*b**4*c**5*d**5 + 126*a*b**5*c**6*d**4 + 12*b**6*c**7*d**3) + x*
*9*(5*a**6*c**2*d**8 + 80*a**5*b*c**3*d**7 + 350*a**4*b**2*c**4*d**6 + 560*a**3*b**3*c**5*d**5 + 350*a**2*b**4
*c**6*d**4 + 80*a*b**5*c**7*d**3 + 5*b**6*c**8*d**2) + x**8*(15*a**6*c**3*d**7 + 315*a**5*b*c**4*d**6/2 + 945*
a**4*b**2*c**5*d**5/2 + 525*a**3*b**3*c**6*d**4 + 225*a**2*b**4*c**7*d**3 + 135*a*b**5*c**8*d**2/4 + 5*b**6*c*
*9*d/4) + x**7*(30*a**6*c**4*d**6 + 216*a**5*b*c**5*d**5 + 450*a**4*b**2*c**6*d**4 + 2400*a**3*b**3*c**7*d**3/
7 + 675*a**2*b**4*c**8*d**2/7 + 60*a*b**5*c**9*d/7 + b**6*c**10/7) + x**6*(42*a**6*c**5*d**5 + 210*a**5*b*c**6
*d**4 + 300*a**4*b**2*c**7*d**3 + 150*a**3*b**3*c**8*d**2 + 25*a**2*b**4*c**9*d + a*b**5*c**10) + x**5*(42*a**
6*c**6*d**4 + 144*a**5*b*c**7*d**3 + 135*a**4*b**2*c**8*d**2 + 40*a**3*b**3*c**9*d + 3*a**2*b**4*c**10) + x**4
*(30*a**6*c**7*d**3 + 135*a**5*b*c**8*d**2/2 + 75*a**4*b**2*c**9*d/2 + 5*a**3*b**3*c**10) + x**3*(15*a**6*c**8
*d**2 + 20*a**5*b*c**9*d + 5*a**4*b**2*c**10) + x**2*(5*a**6*c**9*d + 3*a**5*b*c**10)

________________________________________________________________________________________

Giac [B]  time = 1.05535, size = 1517, normalized size = 8.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(d*x+c)^10,x, algorithm="giac")

[Out]

1/17*b^6*d^10*x^17 + 5/8*b^6*c*d^9*x^16 + 3/8*a*b^5*d^10*x^16 + 3*b^6*c^2*d^8*x^15 + 4*a*b^5*c*d^9*x^15 + a^2*
b^4*d^10*x^15 + 60/7*b^6*c^3*d^7*x^14 + 135/7*a*b^5*c^2*d^8*x^14 + 75/7*a^2*b^4*c*d^9*x^14 + 10/7*a^3*b^3*d^10
*x^14 + 210/13*b^6*c^4*d^6*x^13 + 720/13*a*b^5*c^3*d^7*x^13 + 675/13*a^2*b^4*c^2*d^8*x^13 + 200/13*a^3*b^3*c*d
^9*x^13 + 15/13*a^4*b^2*d^10*x^13 + 21*b^6*c^5*d^5*x^12 + 105*a*b^5*c^4*d^6*x^12 + 150*a^2*b^4*c^3*d^7*x^12 +
75*a^3*b^3*c^2*d^8*x^12 + 25/2*a^4*b^2*c*d^9*x^12 + 1/2*a^5*b*d^10*x^12 + 210/11*b^6*c^6*d^4*x^11 + 1512/11*a*
b^5*c^5*d^5*x^11 + 3150/11*a^2*b^4*c^4*d^6*x^11 + 2400/11*a^3*b^3*c^3*d^7*x^11 + 675/11*a^4*b^2*c^2*d^8*x^11 +
60/11*a^5*b*c*d^9*x^11 + 1/11*a^6*d^10*x^11 + 12*b^6*c^7*d^3*x^10 + 126*a*b^5*c^6*d^4*x^10 + 378*a^2*b^4*c^5*
d^5*x^10 + 420*a^3*b^3*c^4*d^6*x^10 + 180*a^4*b^2*c^3*d^7*x^10 + 27*a^5*b*c^2*d^8*x^10 + a^6*c*d^9*x^10 + 5*b^
6*c^8*d^2*x^9 + 80*a*b^5*c^7*d^3*x^9 + 350*a^2*b^4*c^6*d^4*x^9 + 560*a^3*b^3*c^5*d^5*x^9 + 350*a^4*b^2*c^4*d^6
*x^9 + 80*a^5*b*c^3*d^7*x^9 + 5*a^6*c^2*d^8*x^9 + 5/4*b^6*c^9*d*x^8 + 135/4*a*b^5*c^8*d^2*x^8 + 225*a^2*b^4*c^
7*d^3*x^8 + 525*a^3*b^3*c^6*d^4*x^8 + 945/2*a^4*b^2*c^5*d^5*x^8 + 315/2*a^5*b*c^4*d^6*x^8 + 15*a^6*c^3*d^7*x^8
+ 1/7*b^6*c^10*x^7 + 60/7*a*b^5*c^9*d*x^7 + 675/7*a^2*b^4*c^8*d^2*x^7 + 2400/7*a^3*b^3*c^7*d^3*x^7 + 450*a^4*
b^2*c^6*d^4*x^7 + 216*a^5*b*c^5*d^5*x^7 + 30*a^6*c^4*d^6*x^7 + a*b^5*c^10*x^6 + 25*a^2*b^4*c^9*d*x^6 + 150*a^3
*b^3*c^8*d^2*x^6 + 300*a^4*b^2*c^7*d^3*x^6 + 210*a^5*b*c^6*d^4*x^6 + 42*a^6*c^5*d^5*x^6 + 3*a^2*b^4*c^10*x^5 +
40*a^3*b^3*c^9*d*x^5 + 135*a^4*b^2*c^8*d^2*x^5 + 144*a^5*b*c^7*d^3*x^5 + 42*a^6*c^6*d^4*x^5 + 5*a^3*b^3*c^10*
x^4 + 75/2*a^4*b^2*c^9*d*x^4 + 135/2*a^5*b*c^8*d^2*x^4 + 30*a^6*c^7*d^3*x^4 + 5*a^4*b^2*c^10*x^3 + 20*a^5*b*c^
9*d*x^3 + 15*a^6*c^8*d^2*x^3 + 3*a^5*b*c^10*x^2 + 5*a^6*c^9*d*x^2 + a^6*c^10*x