3.13.1 \(\int (a+b x)^4 (c+d x)^{10} \, dx\)

Optimal. Leaf size=119 \[ -\frac {2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac {6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac {b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac {(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5} \]

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Rubi [A]  time = 0.44, antiderivative size = 119, 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 {2 b^3 (c+d x)^{14} (b c-a d)}{7 d^5}+\frac {6 b^2 (c+d x)^{13} (b c-a d)^2}{13 d^5}-\frac {b (c+d x)^{12} (b c-a d)^3}{3 d^5}+\frac {(c+d x)^{11} (b c-a d)^4}{11 d^5}+\frac {b^4 (c+d x)^{15}}{15 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^10,x]

[Out]

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

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

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Mathematica [B]  time = 0.08, size = 660, normalized size = 5.55 \begin {gather*} a^4 c^{10} x+a^3 c^9 x^2 (5 a d+2 b c)+\frac {1}{13} b^2 d^8 x^{13} \left (6 a^2 d^2+40 a b c d+45 b^2 c^2\right )+\frac {1}{3} a^2 c^8 x^3 \left (45 a^2 d^2+40 a b c d+6 b^2 c^2\right )+\frac {1}{3} b d^7 x^{12} \left (a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+30 b^3 c^3\right )+a c^7 x^4 \left (30 a^3 d^3+45 a^2 b c d^2+15 a b^2 c^2 d+b^3 c^3\right )+\frac {1}{3} c^2 d^4 x^9 \left (15 a^4 d^4+160 a^3 b c d^3+420 a^2 b^2 c^2 d^2+336 a b^3 c^3 d+70 b^4 c^4\right )+3 c^3 d^3 x^8 \left (5 a^4 d^4+35 a^3 b c d^3+63 a^2 b^2 c^2 d^2+35 a b^3 c^3 d+5 b^4 c^4\right )+\frac {3}{7} c^4 d^2 x^7 \left (70 a^4 d^4+336 a^3 b c d^3+420 a^2 b^2 c^2 d^2+160 a b^3 c^3 d+15 b^4 c^4\right )+\frac {1}{11} d^6 x^{11} \left (a^4 d^4+40 a^3 b c d^3+270 a^2 b^2 c^2 d^2+480 a b^3 c^3 d+210 b^4 c^4\right )+\frac {1}{5} c d^5 x^{10} \left (5 a^4 d^4+90 a^3 b c d^3+360 a^2 b^2 c^2 d^2+420 a b^3 c^3 d+126 b^4 c^4\right )+\frac {1}{5} c^6 x^5 \left (210 a^4 d^4+480 a^3 b c d^3+270 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+b^4 c^4\right )+\frac {1}{3} c^5 d x^6 \left (126 a^4 d^4+420 a^3 b c d^3+360 a^2 b^2 c^2 d^2+90 a b^3 c^3 d+5 b^4 c^4\right )+\frac {1}{7} b^3 d^9 x^{14} (2 a d+5 b c)+\frac {1}{15} b^4 d^{10} x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^10,x]

[Out]

a^4*c^10*x + a^3*c^9*(2*b*c + 5*a*d)*x^2 + (a^2*c^8*(6*b^2*c^2 + 40*a*b*c*d + 45*a^2*d^2)*x^3)/3 + a*c^7*(b^3*
c^3 + 15*a*b^2*c^2*d + 45*a^2*b*c*d^2 + 30*a^3*d^3)*x^4 + (c^6*(b^4*c^4 + 40*a*b^3*c^3*d + 270*a^2*b^2*c^2*d^2
+ 480*a^3*b*c*d^3 + 210*a^4*d^4)*x^5)/5 + (c^5*d*(5*b^4*c^4 + 90*a*b^3*c^3*d + 360*a^2*b^2*c^2*d^2 + 420*a^3*
b*c*d^3 + 126*a^4*d^4)*x^6)/3 + (3*c^4*d^2*(15*b^4*c^4 + 160*a*b^3*c^3*d + 420*a^2*b^2*c^2*d^2 + 336*a^3*b*c*d
^3 + 70*a^4*d^4)*x^7)/7 + 3*c^3*d^3*(5*b^4*c^4 + 35*a*b^3*c^3*d + 63*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^3 + 5*a^4*
d^4)*x^8 + (c^2*d^4*(70*b^4*c^4 + 336*a*b^3*c^3*d + 420*a^2*b^2*c^2*d^2 + 160*a^3*b*c*d^3 + 15*a^4*d^4)*x^9)/3
+ (c*d^5*(126*b^4*c^4 + 420*a*b^3*c^3*d + 360*a^2*b^2*c^2*d^2 + 90*a^3*b*c*d^3 + 5*a^4*d^4)*x^10)/5 + (d^6*(2
10*b^4*c^4 + 480*a*b^3*c^3*d + 270*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 + a^4*d^4)*x^11)/11 + (b*d^7*(30*b^3*c^3 +
45*a*b^2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*x^12)/3 + (b^2*d^8*(45*b^2*c^2 + 40*a*b*c*d + 6*a^2*d^2)*x^13)/13
+ (b^3*d^9*(5*b*c + 2*a*d)*x^14)/7 + (b^4*d^10*x^15)/15

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^4 (c+d x)^{10} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^4*(c + d*x)^10,x]

[Out]

IntegrateAlgebraic[(a + b*x)^4*(c + d*x)^10, x]

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fricas [B]  time = 1.10, size = 771, normalized size = 6.48 \begin {gather*} \frac {1}{15} x^{15} d^{10} b^{4} + \frac {5}{7} x^{14} d^{9} c b^{4} + \frac {2}{7} x^{14} d^{10} b^{3} a + \frac {45}{13} x^{13} d^{8} c^{2} b^{4} + \frac {40}{13} x^{13} d^{9} c b^{3} a + \frac {6}{13} x^{13} d^{10} b^{2} a^{2} + 10 x^{12} d^{7} c^{3} b^{4} + 15 x^{12} d^{8} c^{2} b^{3} a + 5 x^{12} d^{9} c b^{2} a^{2} + \frac {1}{3} x^{12} d^{10} b a^{3} + \frac {210}{11} x^{11} d^{6} c^{4} b^{4} + \frac {480}{11} x^{11} d^{7} c^{3} b^{3} a + \frac {270}{11} x^{11} d^{8} c^{2} b^{2} a^{2} + \frac {40}{11} x^{11} d^{9} c b a^{3} + \frac {1}{11} x^{11} d^{10} a^{4} + \frac {126}{5} x^{10} d^{5} c^{5} b^{4} + 84 x^{10} d^{6} c^{4} b^{3} a + 72 x^{10} d^{7} c^{3} b^{2} a^{2} + 18 x^{10} d^{8} c^{2} b a^{3} + x^{10} d^{9} c a^{4} + \frac {70}{3} x^{9} d^{4} c^{6} b^{4} + 112 x^{9} d^{5} c^{5} b^{3} a + 140 x^{9} d^{6} c^{4} b^{2} a^{2} + \frac {160}{3} x^{9} d^{7} c^{3} b a^{3} + 5 x^{9} d^{8} c^{2} a^{4} + 15 x^{8} d^{3} c^{7} b^{4} + 105 x^{8} d^{4} c^{6} b^{3} a + 189 x^{8} d^{5} c^{5} b^{2} a^{2} + 105 x^{8} d^{6} c^{4} b a^{3} + 15 x^{8} d^{7} c^{3} a^{4} + \frac {45}{7} x^{7} d^{2} c^{8} b^{4} + \frac {480}{7} x^{7} d^{3} c^{7} b^{3} a + 180 x^{7} d^{4} c^{6} b^{2} a^{2} + 144 x^{7} d^{5} c^{5} b a^{3} + 30 x^{7} d^{6} c^{4} a^{4} + \frac {5}{3} x^{6} d c^{9} b^{4} + 30 x^{6} d^{2} c^{8} b^{3} a + 120 x^{6} d^{3} c^{7} b^{2} a^{2} + 140 x^{6} d^{4} c^{6} b a^{3} + 42 x^{6} d^{5} c^{5} a^{4} + \frac {1}{5} x^{5} c^{10} b^{4} + 8 x^{5} d c^{9} b^{3} a + 54 x^{5} d^{2} c^{8} b^{2} a^{2} + 96 x^{5} d^{3} c^{7} b a^{3} + 42 x^{5} d^{4} c^{6} a^{4} + x^{4} c^{10} b^{3} a + 15 x^{4} d c^{9} b^{2} a^{2} + 45 x^{4} d^{2} c^{8} b a^{3} + 30 x^{4} d^{3} c^{7} a^{4} + 2 x^{3} c^{10} b^{2} a^{2} + \frac {40}{3} x^{3} d c^{9} b a^{3} + 15 x^{3} d^{2} c^{8} a^{4} + 2 x^{2} c^{10} b a^{3} + 5 x^{2} d c^{9} a^{4} + x c^{10} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/15*x^15*d^10*b^4 + 5/7*x^14*d^9*c*b^4 + 2/7*x^14*d^10*b^3*a + 45/13*x^13*d^8*c^2*b^4 + 40/13*x^13*d^9*c*b^3*
a + 6/13*x^13*d^10*b^2*a^2 + 10*x^12*d^7*c^3*b^4 + 15*x^12*d^8*c^2*b^3*a + 5*x^12*d^9*c*b^2*a^2 + 1/3*x^12*d^1
0*b*a^3 + 210/11*x^11*d^6*c^4*b^4 + 480/11*x^11*d^7*c^3*b^3*a + 270/11*x^11*d^8*c^2*b^2*a^2 + 40/11*x^11*d^9*c
*b*a^3 + 1/11*x^11*d^10*a^4 + 126/5*x^10*d^5*c^5*b^4 + 84*x^10*d^6*c^4*b^3*a + 72*x^10*d^7*c^3*b^2*a^2 + 18*x^
10*d^8*c^2*b*a^3 + x^10*d^9*c*a^4 + 70/3*x^9*d^4*c^6*b^4 + 112*x^9*d^5*c^5*b^3*a + 140*x^9*d^6*c^4*b^2*a^2 + 1
60/3*x^9*d^7*c^3*b*a^3 + 5*x^9*d^8*c^2*a^4 + 15*x^8*d^3*c^7*b^4 + 105*x^8*d^4*c^6*b^3*a + 189*x^8*d^5*c^5*b^2*
a^2 + 105*x^8*d^6*c^4*b*a^3 + 15*x^8*d^7*c^3*a^4 + 45/7*x^7*d^2*c^8*b^4 + 480/7*x^7*d^3*c^7*b^3*a + 180*x^7*d^
4*c^6*b^2*a^2 + 144*x^7*d^5*c^5*b*a^3 + 30*x^7*d^6*c^4*a^4 + 5/3*x^6*d*c^9*b^4 + 30*x^6*d^2*c^8*b^3*a + 120*x^
6*d^3*c^7*b^2*a^2 + 140*x^6*d^4*c^6*b*a^3 + 42*x^6*d^5*c^5*a^4 + 1/5*x^5*c^10*b^4 + 8*x^5*d*c^9*b^3*a + 54*x^5
*d^2*c^8*b^2*a^2 + 96*x^5*d^3*c^7*b*a^3 + 42*x^5*d^4*c^6*a^4 + x^4*c^10*b^3*a + 15*x^4*d*c^9*b^2*a^2 + 45*x^4*
d^2*c^8*b*a^3 + 30*x^4*d^3*c^7*a^4 + 2*x^3*c^10*b^2*a^2 + 40/3*x^3*d*c^9*b*a^3 + 15*x^3*d^2*c^8*a^4 + 2*x^2*c^
10*b*a^3 + 5*x^2*d*c^9*a^4 + x*c^10*a^4

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giac [B]  time = 1.27, size = 771, normalized size = 6.48 \begin {gather*} \frac {1}{15} \, b^{4} d^{10} x^{15} + \frac {5}{7} \, b^{4} c d^{9} x^{14} + \frac {2}{7} \, a b^{3} d^{10} x^{14} + \frac {45}{13} \, b^{4} c^{2} d^{8} x^{13} + \frac {40}{13} \, a b^{3} c d^{9} x^{13} + \frac {6}{13} \, a^{2} b^{2} d^{10} x^{13} + 10 \, b^{4} c^{3} d^{7} x^{12} + 15 \, a b^{3} c^{2} d^{8} x^{12} + 5 \, a^{2} b^{2} c d^{9} x^{12} + \frac {1}{3} \, a^{3} b d^{10} x^{12} + \frac {210}{11} \, b^{4} c^{4} d^{6} x^{11} + \frac {480}{11} \, a b^{3} c^{3} d^{7} x^{11} + \frac {270}{11} \, a^{2} b^{2} c^{2} d^{8} x^{11} + \frac {40}{11} \, a^{3} b c d^{9} x^{11} + \frac {1}{11} \, a^{4} d^{10} x^{11} + \frac {126}{5} \, b^{4} c^{5} d^{5} x^{10} + 84 \, a b^{3} c^{4} d^{6} x^{10} + 72 \, a^{2} b^{2} c^{3} d^{7} x^{10} + 18 \, a^{3} b c^{2} d^{8} x^{10} + a^{4} c d^{9} x^{10} + \frac {70}{3} \, b^{4} c^{6} d^{4} x^{9} + 112 \, a b^{3} c^{5} d^{5} x^{9} + 140 \, a^{2} b^{2} c^{4} d^{6} x^{9} + \frac {160}{3} \, a^{3} b c^{3} d^{7} x^{9} + 5 \, a^{4} c^{2} d^{8} x^{9} + 15 \, b^{4} c^{7} d^{3} x^{8} + 105 \, a b^{3} c^{6} d^{4} x^{8} + 189 \, a^{2} b^{2} c^{5} d^{5} x^{8} + 105 \, a^{3} b c^{4} d^{6} x^{8} + 15 \, a^{4} c^{3} d^{7} x^{8} + \frac {45}{7} \, b^{4} c^{8} d^{2} x^{7} + \frac {480}{7} \, a b^{3} c^{7} d^{3} x^{7} + 180 \, a^{2} b^{2} c^{6} d^{4} x^{7} + 144 \, a^{3} b c^{5} d^{5} x^{7} + 30 \, a^{4} c^{4} d^{6} x^{7} + \frac {5}{3} \, b^{4} c^{9} d x^{6} + 30 \, a b^{3} c^{8} d^{2} x^{6} + 120 \, a^{2} b^{2} c^{7} d^{3} x^{6} + 140 \, a^{3} b c^{6} d^{4} x^{6} + 42 \, a^{4} c^{5} d^{5} x^{6} + \frac {1}{5} \, b^{4} c^{10} x^{5} + 8 \, a b^{3} c^{9} d x^{5} + 54 \, a^{2} b^{2} c^{8} d^{2} x^{5} + 96 \, a^{3} b c^{7} d^{3} x^{5} + 42 \, a^{4} c^{6} d^{4} x^{5} + a b^{3} c^{10} x^{4} + 15 \, a^{2} b^{2} c^{9} d x^{4} + 45 \, a^{3} b c^{8} d^{2} x^{4} + 30 \, a^{4} c^{7} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{10} x^{3} + \frac {40}{3} \, a^{3} b c^{9} d x^{3} + 15 \, a^{4} c^{8} d^{2} x^{3} + 2 \, a^{3} b c^{10} x^{2} + 5 \, a^{4} c^{9} d x^{2} + a^{4} c^{10} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="giac")

[Out]

1/15*b^4*d^10*x^15 + 5/7*b^4*c*d^9*x^14 + 2/7*a*b^3*d^10*x^14 + 45/13*b^4*c^2*d^8*x^13 + 40/13*a*b^3*c*d^9*x^1
3 + 6/13*a^2*b^2*d^10*x^13 + 10*b^4*c^3*d^7*x^12 + 15*a*b^3*c^2*d^8*x^12 + 5*a^2*b^2*c*d^9*x^12 + 1/3*a^3*b*d^
10*x^12 + 210/11*b^4*c^4*d^6*x^11 + 480/11*a*b^3*c^3*d^7*x^11 + 270/11*a^2*b^2*c^2*d^8*x^11 + 40/11*a^3*b*c*d^
9*x^11 + 1/11*a^4*d^10*x^11 + 126/5*b^4*c^5*d^5*x^10 + 84*a*b^3*c^4*d^6*x^10 + 72*a^2*b^2*c^3*d^7*x^10 + 18*a^
3*b*c^2*d^8*x^10 + a^4*c*d^9*x^10 + 70/3*b^4*c^6*d^4*x^9 + 112*a*b^3*c^5*d^5*x^9 + 140*a^2*b^2*c^4*d^6*x^9 + 1
60/3*a^3*b*c^3*d^7*x^9 + 5*a^4*c^2*d^8*x^9 + 15*b^4*c^7*d^3*x^8 + 105*a*b^3*c^6*d^4*x^8 + 189*a^2*b^2*c^5*d^5*
x^8 + 105*a^3*b*c^4*d^6*x^8 + 15*a^4*c^3*d^7*x^8 + 45/7*b^4*c^8*d^2*x^7 + 480/7*a*b^3*c^7*d^3*x^7 + 180*a^2*b^
2*c^6*d^4*x^7 + 144*a^3*b*c^5*d^5*x^7 + 30*a^4*c^4*d^6*x^7 + 5/3*b^4*c^9*d*x^6 + 30*a*b^3*c^8*d^2*x^6 + 120*a^
2*b^2*c^7*d^3*x^6 + 140*a^3*b*c^6*d^4*x^6 + 42*a^4*c^5*d^5*x^6 + 1/5*b^4*c^10*x^5 + 8*a*b^3*c^9*d*x^5 + 54*a^2
*b^2*c^8*d^2*x^5 + 96*a^3*b*c^7*d^3*x^5 + 42*a^4*c^6*d^4*x^5 + a*b^3*c^10*x^4 + 15*a^2*b^2*c^9*d*x^4 + 45*a^3*
b*c^8*d^2*x^4 + 30*a^4*c^7*d^3*x^4 + 2*a^2*b^2*c^10*x^3 + 40/3*a^3*b*c^9*d*x^3 + 15*a^4*c^8*d^2*x^3 + 2*a^3*b*
c^10*x^2 + 5*a^4*c^9*d*x^2 + a^4*c^10*x

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maple [B]  time = 0.00, size = 691, normalized size = 5.81 \begin {gather*} \frac {b^{4} d^{10} x^{15}}{15}+a^{4} c^{10} x +\frac {\left (4 a \,b^{3} d^{10}+10 b^{4} c \,d^{9}\right ) x^{14}}{14}+\frac {\left (6 a^{2} b^{2} d^{10}+40 a \,b^{3} c \,d^{9}+45 b^{4} c^{2} d^{8}\right ) x^{13}}{13}+\frac {\left (4 a^{3} b \,d^{10}+60 a^{2} b^{2} c \,d^{9}+180 a \,b^{3} c^{2} d^{8}+120 b^{4} c^{3} d^{7}\right ) x^{12}}{12}+\frac {\left (a^{4} d^{10}+40 a^{3} b c \,d^{9}+270 a^{2} b^{2} c^{2} d^{8}+480 a \,b^{3} c^{3} d^{7}+210 b^{4} c^{4} d^{6}\right ) x^{11}}{11}+\frac {\left (10 a^{4} c \,d^{9}+180 a^{3} b \,c^{2} d^{8}+720 a^{2} b^{2} c^{3} d^{7}+840 a \,b^{3} c^{4} d^{6}+252 b^{4} c^{5} d^{5}\right ) x^{10}}{10}+\frac {\left (45 a^{4} c^{2} d^{8}+480 a^{3} b \,c^{3} d^{7}+1260 a^{2} b^{2} c^{4} d^{6}+1008 a \,b^{3} c^{5} d^{5}+210 b^{4} c^{6} d^{4}\right ) x^{9}}{9}+\frac {\left (120 a^{4} c^{3} d^{7}+840 a^{3} b \,c^{4} d^{6}+1512 a^{2} b^{2} c^{5} d^{5}+840 a \,b^{3} c^{6} d^{4}+120 b^{4} c^{7} d^{3}\right ) x^{8}}{8}+\frac {\left (210 a^{4} c^{4} d^{6}+1008 a^{3} b \,c^{5} d^{5}+1260 a^{2} b^{2} c^{6} d^{4}+480 a \,b^{3} c^{7} d^{3}+45 b^{4} c^{8} d^{2}\right ) x^{7}}{7}+\frac {\left (252 a^{4} c^{5} d^{5}+840 a^{3} b \,c^{6} d^{4}+720 a^{2} b^{2} c^{7} d^{3}+180 a \,b^{3} c^{8} d^{2}+10 b^{4} c^{9} d \right ) x^{6}}{6}+\frac {\left (210 a^{4} c^{6} d^{4}+480 a^{3} b \,c^{7} d^{3}+270 a^{2} b^{2} c^{8} d^{2}+40 a \,b^{3} c^{9} d +b^{4} c^{10}\right ) x^{5}}{5}+\frac {\left (120 a^{4} c^{7} d^{3}+180 a^{3} b \,c^{8} d^{2}+60 a^{2} b^{2} c^{9} d +4 a \,b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{4} c^{8} d^{2}+40 a^{3} b \,c^{9} d +6 a^{2} b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{4} c^{9} d +4 a^{3} b \,c^{10}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^10,x)

[Out]

1/15*b^4*d^10*x^15+1/14*(4*a*b^3*d^10+10*b^4*c*d^9)*x^14+1/13*(6*a^2*b^2*d^10+40*a*b^3*c*d^9+45*b^4*c^2*d^8)*x
^13+1/12*(4*a^3*b*d^10+60*a^2*b^2*c*d^9+180*a*b^3*c^2*d^8+120*b^4*c^3*d^7)*x^12+1/11*(a^4*d^10+40*a^3*b*c*d^9+
270*a^2*b^2*c^2*d^8+480*a*b^3*c^3*d^7+210*b^4*c^4*d^6)*x^11+1/10*(10*a^4*c*d^9+180*a^3*b*c^2*d^8+720*a^2*b^2*c
^3*d^7+840*a*b^3*c^4*d^6+252*b^4*c^5*d^5)*x^10+1/9*(45*a^4*c^2*d^8+480*a^3*b*c^3*d^7+1260*a^2*b^2*c^4*d^6+1008
*a*b^3*c^5*d^5+210*b^4*c^6*d^4)*x^9+1/8*(120*a^4*c^3*d^7+840*a^3*b*c^4*d^6+1512*a^2*b^2*c^5*d^5+840*a*b^3*c^6*
d^4+120*b^4*c^7*d^3)*x^8+1/7*(210*a^4*c^4*d^6+1008*a^3*b*c^5*d^5+1260*a^2*b^2*c^6*d^4+480*a*b^3*c^7*d^3+45*b^4
*c^8*d^2)*x^7+1/6*(252*a^4*c^5*d^5+840*a^3*b*c^6*d^4+720*a^2*b^2*c^7*d^3+180*a*b^3*c^8*d^2+10*b^4*c^9*d)*x^6+1
/5*(210*a^4*c^6*d^4+480*a^3*b*c^7*d^3+270*a^2*b^2*c^8*d^2+40*a*b^3*c^9*d+b^4*c^10)*x^5+1/4*(120*a^4*c^7*d^3+18
0*a^3*b*c^8*d^2+60*a^2*b^2*c^9*d+4*a*b^3*c^10)*x^4+1/3*(45*a^4*c^8*d^2+40*a^3*b*c^9*d+6*a^2*b^2*c^10)*x^3+1/2*
(10*a^4*c^9*d+4*a^3*b*c^10)*x^2+a^4*c^10*x

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maxima [B]  time = 1.42, size = 686, normalized size = 5.76 \begin {gather*} \frac {1}{15} \, b^{4} d^{10} x^{15} + a^{4} c^{10} x + \frac {1}{7} \, {\left (5 \, b^{4} c d^{9} + 2 \, a b^{3} d^{10}\right )} x^{14} + \frac {1}{13} \, {\left (45 \, b^{4} c^{2} d^{8} + 40 \, a b^{3} c d^{9} + 6 \, a^{2} b^{2} d^{10}\right )} x^{13} + \frac {1}{3} \, {\left (30 \, b^{4} c^{3} d^{7} + 45 \, a b^{3} c^{2} d^{8} + 15 \, a^{2} b^{2} c d^{9} + a^{3} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (210 \, b^{4} c^{4} d^{6} + 480 \, a b^{3} c^{3} d^{7} + 270 \, a^{2} b^{2} c^{2} d^{8} + 40 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{11} + \frac {1}{5} \, {\left (126 \, b^{4} c^{5} d^{5} + 420 \, a b^{3} c^{4} d^{6} + 360 \, a^{2} b^{2} c^{3} d^{7} + 90 \, a^{3} b c^{2} d^{8} + 5 \, a^{4} c d^{9}\right )} x^{10} + \frac {1}{3} \, {\left (70 \, b^{4} c^{6} d^{4} + 336 \, a b^{3} c^{5} d^{5} + 420 \, a^{2} b^{2} c^{4} d^{6} + 160 \, a^{3} b c^{3} d^{7} + 15 \, a^{4} c^{2} d^{8}\right )} x^{9} + 3 \, {\left (5 \, b^{4} c^{7} d^{3} + 35 \, a b^{3} c^{6} d^{4} + 63 \, a^{2} b^{2} c^{5} d^{5} + 35 \, a^{3} b c^{4} d^{6} + 5 \, a^{4} c^{3} d^{7}\right )} x^{8} + \frac {3}{7} \, {\left (15 \, b^{4} c^{8} d^{2} + 160 \, a b^{3} c^{7} d^{3} + 420 \, a^{2} b^{2} c^{6} d^{4} + 336 \, a^{3} b c^{5} d^{5} + 70 \, a^{4} c^{4} d^{6}\right )} x^{7} + \frac {1}{3} \, {\left (5 \, b^{4} c^{9} d + 90 \, a b^{3} c^{8} d^{2} + 360 \, a^{2} b^{2} c^{7} d^{3} + 420 \, a^{3} b c^{6} d^{4} + 126 \, a^{4} c^{5} d^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{10} + 40 \, a b^{3} c^{9} d + 270 \, a^{2} b^{2} c^{8} d^{2} + 480 \, a^{3} b c^{7} d^{3} + 210 \, a^{4} c^{6} d^{4}\right )} x^{5} + {\left (a b^{3} c^{10} + 15 \, a^{2} b^{2} c^{9} d + 45 \, a^{3} b c^{8} d^{2} + 30 \, a^{4} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{10} + 40 \, a^{3} b c^{9} d + 45 \, a^{4} c^{8} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{10} + 5 \, a^{4} c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/15*b^4*d^10*x^15 + a^4*c^10*x + 1/7*(5*b^4*c*d^9 + 2*a*b^3*d^10)*x^14 + 1/13*(45*b^4*c^2*d^8 + 40*a*b^3*c*d^
9 + 6*a^2*b^2*d^10)*x^13 + 1/3*(30*b^4*c^3*d^7 + 45*a*b^3*c^2*d^8 + 15*a^2*b^2*c*d^9 + a^3*b*d^10)*x^12 + 1/11
*(210*b^4*c^4*d^6 + 480*a*b^3*c^3*d^7 + 270*a^2*b^2*c^2*d^8 + 40*a^3*b*c*d^9 + a^4*d^10)*x^11 + 1/5*(126*b^4*c
^5*d^5 + 420*a*b^3*c^4*d^6 + 360*a^2*b^2*c^3*d^7 + 90*a^3*b*c^2*d^8 + 5*a^4*c*d^9)*x^10 + 1/3*(70*b^4*c^6*d^4
+ 336*a*b^3*c^5*d^5 + 420*a^2*b^2*c^4*d^6 + 160*a^3*b*c^3*d^7 + 15*a^4*c^2*d^8)*x^9 + 3*(5*b^4*c^7*d^3 + 35*a*
b^3*c^6*d^4 + 63*a^2*b^2*c^5*d^5 + 35*a^3*b*c^4*d^6 + 5*a^4*c^3*d^7)*x^8 + 3/7*(15*b^4*c^8*d^2 + 160*a*b^3*c^7
*d^3 + 420*a^2*b^2*c^6*d^4 + 336*a^3*b*c^5*d^5 + 70*a^4*c^4*d^6)*x^7 + 1/3*(5*b^4*c^9*d + 90*a*b^3*c^8*d^2 + 3
60*a^2*b^2*c^7*d^3 + 420*a^3*b*c^6*d^4 + 126*a^4*c^5*d^5)*x^6 + 1/5*(b^4*c^10 + 40*a*b^3*c^9*d + 270*a^2*b^2*c
^8*d^2 + 480*a^3*b*c^7*d^3 + 210*a^4*c^6*d^4)*x^5 + (a*b^3*c^10 + 15*a^2*b^2*c^9*d + 45*a^3*b*c^8*d^2 + 30*a^4
*c^7*d^3)*x^4 + 1/3*(6*a^2*b^2*c^10 + 40*a^3*b*c^9*d + 45*a^4*c^8*d^2)*x^3 + (2*a^3*b*c^10 + 5*a^4*c^9*d)*x^2

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mupad [B]  time = 0.43, size = 664, normalized size = 5.58 \begin {gather*} x^5\,\left (42\,a^4\,c^6\,d^4+96\,a^3\,b\,c^7\,d^3+54\,a^2\,b^2\,c^8\,d^2+8\,a\,b^3\,c^9\,d+\frac {b^4\,c^{10}}{5}\right )+x^{11}\,\left (\frac {a^4\,d^{10}}{11}+\frac {40\,a^3\,b\,c\,d^9}{11}+\frac {270\,a^2\,b^2\,c^2\,d^8}{11}+\frac {480\,a\,b^3\,c^3\,d^7}{11}+\frac {210\,b^4\,c^4\,d^6}{11}\right )+x^8\,\left (15\,a^4\,c^3\,d^7+105\,a^3\,b\,c^4\,d^6+189\,a^2\,b^2\,c^5\,d^5+105\,a\,b^3\,c^6\,d^4+15\,b^4\,c^7\,d^3\right )+x^9\,\left (5\,a^4\,c^2\,d^8+\frac {160\,a^3\,b\,c^3\,d^7}{3}+140\,a^2\,b^2\,c^4\,d^6+112\,a\,b^3\,c^5\,d^5+\frac {70\,b^4\,c^6\,d^4}{3}\right )+x^7\,\left (30\,a^4\,c^4\,d^6+144\,a^3\,b\,c^5\,d^5+180\,a^2\,b^2\,c^6\,d^4+\frac {480\,a\,b^3\,c^7\,d^3}{7}+\frac {45\,b^4\,c^8\,d^2}{7}\right )+x^4\,\left (30\,a^4\,c^7\,d^3+45\,a^3\,b\,c^8\,d^2+15\,a^2\,b^2\,c^9\,d+a\,b^3\,c^{10}\right )+x^{12}\,\left (\frac {a^3\,b\,d^{10}}{3}+5\,a^2\,b^2\,c\,d^9+15\,a\,b^3\,c^2\,d^8+10\,b^4\,c^3\,d^7\right )+x^{10}\,\left (a^4\,c\,d^9+18\,a^3\,b\,c^2\,d^8+72\,a^2\,b^2\,c^3\,d^7+84\,a\,b^3\,c^4\,d^6+\frac {126\,b^4\,c^5\,d^5}{5}\right )+x^6\,\left (42\,a^4\,c^5\,d^5+140\,a^3\,b\,c^6\,d^4+120\,a^2\,b^2\,c^7\,d^3+30\,a\,b^3\,c^8\,d^2+\frac {5\,b^4\,c^9\,d}{3}\right )+a^4\,c^{10}\,x+\frac {b^4\,d^{10}\,x^{15}}{15}+a^3\,c^9\,x^2\,\left (5\,a\,d+2\,b\,c\right )+\frac {b^3\,d^9\,x^{14}\,\left (2\,a\,d+5\,b\,c\right )}{7}+\frac {a^2\,c^8\,x^3\,\left (45\,a^2\,d^2+40\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+\frac {b^2\,d^8\,x^{13}\,\left (6\,a^2\,d^2+40\,a\,b\,c\,d+45\,b^2\,c^2\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4*(c + d*x)^10,x)

[Out]

x^5*((b^4*c^10)/5 + 42*a^4*c^6*d^4 + 96*a^3*b*c^7*d^3 + 54*a^2*b^2*c^8*d^2 + 8*a*b^3*c^9*d) + x^11*((a^4*d^10)
/11 + (210*b^4*c^4*d^6)/11 + (480*a*b^3*c^3*d^7)/11 + (270*a^2*b^2*c^2*d^8)/11 + (40*a^3*b*c*d^9)/11) + x^8*(1
5*a^4*c^3*d^7 + 15*b^4*c^7*d^3 + 105*a*b^3*c^6*d^4 + 105*a^3*b*c^4*d^6 + 189*a^2*b^2*c^5*d^5) + x^9*(5*a^4*c^2
*d^8 + (70*b^4*c^6*d^4)/3 + 112*a*b^3*c^5*d^5 + (160*a^3*b*c^3*d^7)/3 + 140*a^2*b^2*c^4*d^6) + x^7*(30*a^4*c^4
*d^6 + (45*b^4*c^8*d^2)/7 + (480*a*b^3*c^7*d^3)/7 + 144*a^3*b*c^5*d^5 + 180*a^2*b^2*c^6*d^4) + x^4*(a*b^3*c^10
+ 30*a^4*c^7*d^3 + 15*a^2*b^2*c^9*d + 45*a^3*b*c^8*d^2) + x^12*((a^3*b*d^10)/3 + 10*b^4*c^3*d^7 + 15*a*b^3*c^
2*d^8 + 5*a^2*b^2*c*d^9) + x^10*(a^4*c*d^9 + (126*b^4*c^5*d^5)/5 + 84*a*b^3*c^4*d^6 + 18*a^3*b*c^2*d^8 + 72*a^
2*b^2*c^3*d^7) + x^6*((5*b^4*c^9*d)/3 + 42*a^4*c^5*d^5 + 30*a*b^3*c^8*d^2 + 140*a^3*b*c^6*d^4 + 120*a^2*b^2*c^
7*d^3) + a^4*c^10*x + (b^4*d^10*x^15)/15 + a^3*c^9*x^2*(5*a*d + 2*b*c) + (b^3*d^9*x^14*(2*a*d + 5*b*c))/7 + (a
^2*c^8*x^3*(45*a^2*d^2 + 6*b^2*c^2 + 40*a*b*c*d))/3 + (b^2*d^8*x^13*(6*a^2*d^2 + 45*b^2*c^2 + 40*a*b*c*d))/13

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sympy [B]  time = 0.18, size = 748, normalized size = 6.29 \begin {gather*} a^{4} c^{10} x + \frac {b^{4} d^{10} x^{15}}{15} + x^{14} \left (\frac {2 a b^{3} d^{10}}{7} + \frac {5 b^{4} c d^{9}}{7}\right ) + x^{13} \left (\frac {6 a^{2} b^{2} d^{10}}{13} + \frac {40 a b^{3} c d^{9}}{13} + \frac {45 b^{4} c^{2} d^{8}}{13}\right ) + x^{12} \left (\frac {a^{3} b d^{10}}{3} + 5 a^{2} b^{2} c d^{9} + 15 a b^{3} c^{2} d^{8} + 10 b^{4} c^{3} d^{7}\right ) + x^{11} \left (\frac {a^{4} d^{10}}{11} + \frac {40 a^{3} b c d^{9}}{11} + \frac {270 a^{2} b^{2} c^{2} d^{8}}{11} + \frac {480 a b^{3} c^{3} d^{7}}{11} + \frac {210 b^{4} c^{4} d^{6}}{11}\right ) + x^{10} \left (a^{4} c d^{9} + 18 a^{3} b c^{2} d^{8} + 72 a^{2} b^{2} c^{3} d^{7} + 84 a b^{3} c^{4} d^{6} + \frac {126 b^{4} c^{5} d^{5}}{5}\right ) + x^{9} \left (5 a^{4} c^{2} d^{8} + \frac {160 a^{3} b c^{3} d^{7}}{3} + 140 a^{2} b^{2} c^{4} d^{6} + 112 a b^{3} c^{5} d^{5} + \frac {70 b^{4} c^{6} d^{4}}{3}\right ) + x^{8} \left (15 a^{4} c^{3} d^{7} + 105 a^{3} b c^{4} d^{6} + 189 a^{2} b^{2} c^{5} d^{5} + 105 a b^{3} c^{6} d^{4} + 15 b^{4} c^{7} d^{3}\right ) + x^{7} \left (30 a^{4} c^{4} d^{6} + 144 a^{3} b c^{5} d^{5} + 180 a^{2} b^{2} c^{6} d^{4} + \frac {480 a b^{3} c^{7} d^{3}}{7} + \frac {45 b^{4} c^{8} d^{2}}{7}\right ) + x^{6} \left (42 a^{4} c^{5} d^{5} + 140 a^{3} b c^{6} d^{4} + 120 a^{2} b^{2} c^{7} d^{3} + 30 a b^{3} c^{8} d^{2} + \frac {5 b^{4} c^{9} d}{3}\right ) + x^{5} \left (42 a^{4} c^{6} d^{4} + 96 a^{3} b c^{7} d^{3} + 54 a^{2} b^{2} c^{8} d^{2} + 8 a b^{3} c^{9} d + \frac {b^{4} c^{10}}{5}\right ) + x^{4} \left (30 a^{4} c^{7} d^{3} + 45 a^{3} b c^{8} d^{2} + 15 a^{2} b^{2} c^{9} d + a b^{3} c^{10}\right ) + x^{3} \left (15 a^{4} c^{8} d^{2} + \frac {40 a^{3} b c^{9} d}{3} + 2 a^{2} b^{2} c^{10}\right ) + x^{2} \left (5 a^{4} c^{9} d + 2 a^{3} b c^{10}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c)**10,x)

[Out]

a**4*c**10*x + b**4*d**10*x**15/15 + x**14*(2*a*b**3*d**10/7 + 5*b**4*c*d**9/7) + x**13*(6*a**2*b**2*d**10/13
+ 40*a*b**3*c*d**9/13 + 45*b**4*c**2*d**8/13) + x**12*(a**3*b*d**10/3 + 5*a**2*b**2*c*d**9 + 15*a*b**3*c**2*d*
*8 + 10*b**4*c**3*d**7) + x**11*(a**4*d**10/11 + 40*a**3*b*c*d**9/11 + 270*a**2*b**2*c**2*d**8/11 + 480*a*b**3
*c**3*d**7/11 + 210*b**4*c**4*d**6/11) + x**10*(a**4*c*d**9 + 18*a**3*b*c**2*d**8 + 72*a**2*b**2*c**3*d**7 + 8
4*a*b**3*c**4*d**6 + 126*b**4*c**5*d**5/5) + x**9*(5*a**4*c**2*d**8 + 160*a**3*b*c**3*d**7/3 + 140*a**2*b**2*c
**4*d**6 + 112*a*b**3*c**5*d**5 + 70*b**4*c**6*d**4/3) + x**8*(15*a**4*c**3*d**7 + 105*a**3*b*c**4*d**6 + 189*
a**2*b**2*c**5*d**5 + 105*a*b**3*c**6*d**4 + 15*b**4*c**7*d**3) + x**7*(30*a**4*c**4*d**6 + 144*a**3*b*c**5*d*
*5 + 180*a**2*b**2*c**6*d**4 + 480*a*b**3*c**7*d**3/7 + 45*b**4*c**8*d**2/7) + x**6*(42*a**4*c**5*d**5 + 140*a
**3*b*c**6*d**4 + 120*a**2*b**2*c**7*d**3 + 30*a*b**3*c**8*d**2 + 5*b**4*c**9*d/3) + x**5*(42*a**4*c**6*d**4 +
96*a**3*b*c**7*d**3 + 54*a**2*b**2*c**8*d**2 + 8*a*b**3*c**9*d + b**4*c**10/5) + x**4*(30*a**4*c**7*d**3 + 45
*a**3*b*c**8*d**2 + 15*a**2*b**2*c**9*d + a*b**3*c**10) + x**3*(15*a**4*c**8*d**2 + 40*a**3*b*c**9*d/3 + 2*a**
2*b**2*c**10) + x**2*(5*a**4*c**9*d + 2*a**3*b*c**10)

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