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

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

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Rubi [A]  time = 0.67, antiderivative size = 170, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [B]  time = 0.12, size = 939, normalized size = 5.52 \begin {gather*} \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 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^6 (c+d x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^6*(c + d*x)^10,x]

[Out]

IntegrateAlgebraic[(a + b*x)^6*(c + d*x)^10, x]

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fricas [B]  time = 1.06, size = 1124, normalized size = 6.61 \begin {gather*} \frac {1}{17} x^{17} d^{10} b^{6} + \frac {5}{8} x^{16} d^{9} c b^{6} + \frac {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} + \frac {60}{7} x^{14} d^{7} c^{3} b^{6} + \frac {135}{7} x^{14} d^{8} c^{2} b^{5} a + \frac {75}{7} x^{14} d^{9} c b^{4} a^{2} + \frac {10}{7} x^{14} d^{10} b^{3} a^{3} + \frac {210}{13} x^{13} d^{6} c^{4} b^{6} + \frac {720}{13} x^{13} d^{7} c^{3} b^{5} a + \frac {675}{13} x^{13} d^{8} c^{2} b^{4} a^{2} + \frac {200}{13} x^{13} d^{9} c b^{3} a^{3} + \frac {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} + \frac {25}{2} x^{12} d^{9} c b^{2} a^{4} + \frac {1}{2} x^{12} d^{10} b a^{5} + \frac {210}{11} x^{11} d^{4} c^{6} b^{6} + \frac {1512}{11} x^{11} d^{5} c^{5} b^{5} a + \frac {3150}{11} x^{11} d^{6} c^{4} b^{4} a^{2} + \frac {2400}{11} x^{11} d^{7} c^{3} b^{3} a^{3} + \frac {675}{11} x^{11} d^{8} c^{2} b^{2} a^{4} + \frac {60}{11} x^{11} d^{9} c b a^{5} + \frac {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} + \frac {5}{4} x^{8} d c^{9} b^{6} + \frac {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} + \frac {945}{2} x^{8} d^{5} c^{5} b^{2} a^{4} + \frac {315}{2} x^{8} d^{6} c^{4} b a^{5} + 15 x^{8} d^{7} c^{3} a^{6} + \frac {1}{7} x^{7} c^{10} b^{6} + \frac {60}{7} x^{7} d c^{9} b^{5} a + \frac {675}{7} x^{7} d^{2} c^{8} b^{4} a^{2} + \frac {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} + \frac {75}{2} x^{4} d c^{9} b^{2} a^{4} + \frac {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} \end {gather*}

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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giac [B]  time = 1.30, size = 1124, normalized size = 6.61 \begin {gather*} \frac {1}{17} \, b^{6} d^{10} x^{17} + \frac {5}{8} \, b^{6} c d^{9} x^{16} + \frac {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} + \frac {60}{7} \, b^{6} c^{3} d^{7} x^{14} + \frac {135}{7} \, a b^{5} c^{2} d^{8} x^{14} + \frac {75}{7} \, a^{2} b^{4} c d^{9} x^{14} + \frac {10}{7} \, a^{3} b^{3} d^{10} x^{14} + \frac {210}{13} \, b^{6} c^{4} d^{6} x^{13} + \frac {720}{13} \, a b^{5} c^{3} d^{7} x^{13} + \frac {675}{13} \, a^{2} b^{4} c^{2} d^{8} x^{13} + \frac {200}{13} \, a^{3} b^{3} c d^{9} x^{13} + \frac {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} + \frac {25}{2} \, a^{4} b^{2} c d^{9} x^{12} + \frac {1}{2} \, a^{5} b d^{10} x^{12} + \frac {210}{11} \, b^{6} c^{6} d^{4} x^{11} + \frac {1512}{11} \, a b^{5} c^{5} d^{5} x^{11} + \frac {3150}{11} \, a^{2} b^{4} c^{4} d^{6} x^{11} + \frac {2400}{11} \, a^{3} b^{3} c^{3} d^{7} x^{11} + \frac {675}{11} \, a^{4} b^{2} c^{2} d^{8} x^{11} + \frac {60}{11} \, a^{5} b c d^{9} x^{11} + \frac {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} + \frac {5}{4} \, b^{6} c^{9} d x^{8} + \frac {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} + \frac {945}{2} \, a^{4} b^{2} c^{5} d^{5} x^{8} + \frac {315}{2} \, a^{5} b c^{4} d^{6} x^{8} + 15 \, a^{6} c^{3} d^{7} x^{8} + \frac {1}{7} \, b^{6} c^{10} x^{7} + \frac {60}{7} \, a b^{5} c^{9} d x^{7} + \frac {675}{7} \, a^{2} b^{4} c^{8} d^{2} x^{7} + \frac {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} + \frac {75}{2} \, a^{4} b^{2} c^{9} d x^{4} + \frac {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 \end {gather*}

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

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maple [B]  time = 0.00, size = 991, normalized size = 5.83 \begin {gather*} \frac {b^{6} d^{10} x^{17}}{17}+a^{6} c^{10} x +\frac {\left (6 a \,b^{5} d^{10}+10 b^{6} c \,d^{9}\right ) x^{16}}{16}+\frac {\left (15 a^{2} b^{4} d^{10}+60 a \,b^{5} c \,d^{9}+45 b^{6} c^{2} d^{8}\right ) x^{15}}{15}+\frac {\left (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}\right ) x^{14}}{14}+\frac {\left (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}\right ) x^{13}}{13}+\frac {\left (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}\right ) x^{12}}{12}+\frac {\left (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}\right ) x^{11}}{11}+\frac {\left (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}\right ) x^{10}}{10}+\frac {\left (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}+3150 a^{2} b^{4} c^{6} d^{4}+720 a \,b^{5} c^{7} d^{3}+45 b^{6} c^{8} d^{2}\right ) x^{9}}{9}+\frac {\left (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 \right ) x^{8}}{8}+\frac {\left (210 a^{6} c^{4} d^{6}+1512 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}\right ) x^{7}}{7}+\frac {\left (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}\right ) x^{6}}{6}+\frac {\left (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}\right ) x^{5}}{5}+\frac {\left (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}\right ) x^{4}}{4}+\frac {\left (45 a^{6} c^{8} d^{2}+60 a^{5} b \,c^{9} d +15 a^{4} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{6} c^{9} d +6 a^{5} b \,c^{10}\right ) x^{2}}{2} \end {gather*}

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 = 1.44, size = 977, normalized size = 5.75 \begin {gather*} \frac {1}{17} \, b^{6} d^{10} x^{17} + a^{6} c^{10} x + \frac {1}{8} \, {\left (5 \, b^{6} c d^{9} + 3 \, a b^{5} d^{10}\right )} x^{16} + {\left (3 \, b^{6} c^{2} d^{8} + 4 \, a b^{5} c d^{9} + a^{2} b^{4} d^{10}\right )} x^{15} + \frac {5}{7} \, {\left (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}\right )} x^{14} + \frac {5}{13} \, {\left (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}\right )} x^{13} + \frac {1}{2} \, {\left (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}\right )} x^{12} + \frac {1}{11} \, {\left (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}\right )} x^{11} + {\left (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}\right )} x^{10} + 5 \, {\left (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}\right )} x^{9} + \frac {5}{4} \, {\left (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}\right )} x^{8} + \frac {1}{7} \, {\left (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}\right )} x^{7} + {\left (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}\right )} x^{6} + {\left (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}\right )} x^{5} + \frac {5}{2} \, {\left (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}\right )} x^{4} + 5 \, {\left (a^{4} b^{2} c^{10} + 4 \, a^{5} b c^{9} d + 3 \, a^{6} c^{8} d^{2}\right )} x^{3} + {\left (3 \, a^{5} b c^{10} + 5 \, a^{6} c^{9} d\right )} x^{2} \end {gather*}

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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mupad [B]  time = 0.53, size = 953, normalized size = 5.61 \begin {gather*} x^7\,\left (30\,a^6\,c^4\,d^6+216\,a^5\,b\,c^5\,d^5+450\,a^4\,b^2\,c^6\,d^4+\frac {2400\,a^3\,b^3\,c^7\,d^3}{7}+\frac {675\,a^2\,b^4\,c^8\,d^2}{7}+\frac {60\,a\,b^5\,c^9\,d}{7}+\frac {b^6\,c^{10}}{7}\right )+x^{11}\,\left (\frac {a^6\,d^{10}}{11}+\frac {60\,a^5\,b\,c\,d^9}{11}+\frac {675\,a^4\,b^2\,c^2\,d^8}{11}+\frac {2400\,a^3\,b^3\,c^3\,d^7}{11}+\frac {3150\,a^2\,b^4\,c^4\,d^6}{11}+\frac {1512\,a\,b^5\,c^5\,d^5}{11}+\frac {210\,b^6\,c^6\,d^4}{11}\right )+x^9\,\left (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\right )+x^5\,\left (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}\right )+x^{13}\,\left (\frac {15\,a^4\,b^2\,d^{10}}{13}+\frac {200\,a^3\,b^3\,c\,d^9}{13}+\frac {675\,a^2\,b^4\,c^2\,d^8}{13}+\frac {720\,a\,b^5\,c^3\,d^7}{13}+\frac {210\,b^6\,c^4\,d^6}{13}\right )+x^6\,\left (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}\right )+x^{12}\,\left (\frac {a^5\,b\,d^{10}}{2}+\frac {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\right )+x^{10}\,\left (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\right )+x^8\,\left (15\,a^6\,c^3\,d^7+\frac {315\,a^5\,b\,c^4\,d^6}{2}+\frac {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+\frac {135\,a\,b^5\,c^8\,d^2}{4}+\frac {5\,b^6\,c^9\,d}{4}\right )+a^6\,c^{10}\,x+\frac {b^6\,d^{10}\,x^{17}}{17}+\frac {5\,a^3\,c^7\,x^4\,\left (12\,a^3\,d^3+27\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2}+\frac {5\,b^3\,d^7\,x^{14}\,\left (2\,a^3\,d^3+15\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{7}+a^5\,c^9\,x^2\,\left (5\,a\,d+3\,b\,c\right )+\frac {b^5\,d^9\,x^{16}\,\left (3\,a\,d+5\,b\,c\right )}{8}+5\,a^4\,c^8\,x^3\,\left (3\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+b^4\,d^8\,x^{15}\,\left (a^2\,d^2+4\,a\,b\,c\,d+3\,b^2\,c^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6*(c + d*x)^10,x)

[Out]

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

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sympy [B]  time = 0.23, size = 1088, normalized size = 6.40 \begin {gather*} a^{6} c^{10} x + \frac {b^{6} d^{10} x^{17}}{17} + x^{16} \left (\frac {3 a b^{5} d^{10}}{8} + \frac {5 b^{6} c d^{9}}{8}\right ) + x^{15} \left (a^{2} b^{4} d^{10} + 4 a b^{5} c d^{9} + 3 b^{6} c^{2} d^{8}\right ) + x^{14} \left (\frac {10 a^{3} b^{3} d^{10}}{7} + \frac {75 a^{2} b^{4} c d^{9}}{7} + \frac {135 a b^{5} c^{2} d^{8}}{7} + \frac {60 b^{6} c^{3} d^{7}}{7}\right ) + x^{13} \left (\frac {15 a^{4} b^{2} d^{10}}{13} + \frac {200 a^{3} b^{3} c d^{9}}{13} + \frac {675 a^{2} b^{4} c^{2} d^{8}}{13} + \frac {720 a b^{5} c^{3} d^{7}}{13} + \frac {210 b^{6} c^{4} d^{6}}{13}\right ) + x^{12} \left (\frac {a^{5} b d^{10}}{2} + \frac {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}\right ) + x^{11} \left (\frac {a^{6} d^{10}}{11} + \frac {60 a^{5} b c d^{9}}{11} + \frac {675 a^{4} b^{2} c^{2} d^{8}}{11} + \frac {2400 a^{3} b^{3} c^{3} d^{7}}{11} + \frac {3150 a^{2} b^{4} c^{4} d^{6}}{11} + \frac {1512 a b^{5} c^{5} d^{5}}{11} + \frac {210 b^{6} c^{6} d^{4}}{11}\right ) + x^{10} \left (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}\right ) + x^{9} \left (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}\right ) + x^{8} \left (15 a^{6} c^{3} d^{7} + \frac {315 a^{5} b c^{4} d^{6}}{2} + \frac {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} + \frac {135 a b^{5} c^{8} d^{2}}{4} + \frac {5 b^{6} c^{9} d}{4}\right ) + x^{7} \left (30 a^{6} c^{4} d^{6} + 216 a^{5} b c^{5} d^{5} + 450 a^{4} b^{2} c^{6} d^{4} + \frac {2400 a^{3} b^{3} c^{7} d^{3}}{7} + \frac {675 a^{2} b^{4} c^{8} d^{2}}{7} + \frac {60 a b^{5} c^{9} d}{7} + \frac {b^{6} c^{10}}{7}\right ) + x^{6} \left (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}\right ) + x^{5} \left (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}\right ) + x^{4} \left (30 a^{6} c^{7} d^{3} + \frac {135 a^{5} b c^{8} d^{2}}{2} + \frac {75 a^{4} b^{2} c^{9} d}{2} + 5 a^{3} b^{3} c^{10}\right ) + x^{3} \left (15 a^{6} c^{8} d^{2} + 20 a^{5} b c^{9} d + 5 a^{4} b^{2} c^{10}\right ) + x^{2} \left (5 a^{6} c^{9} d + 3 a^{5} b c^{10}\right ) \end {gather*}

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)

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