3.12.70 \(\int (a+b x)^6 (c+d x)^7 \, dx\)

Optimal. Leaf size=173 \[ -\frac {6 b^5 (c+d x)^{13} (b c-a d)}{13 d^7}+\frac {5 b^4 (c+d x)^{12} (b c-a d)^2}{4 d^7}-\frac {20 b^3 (c+d x)^{11} (b c-a d)^3}{11 d^7}+\frac {3 b^2 (c+d x)^{10} (b c-a d)^4}{2 d^7}-\frac {2 b (c+d x)^9 (b c-a d)^5}{3 d^7}+\frac {(c+d x)^8 (b c-a d)^6}{8 d^7}+\frac {b^6 (c+d x)^{14}}{14 d^7} \]

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Rubi [A]  time = 0.43, antiderivative size = 173, 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 {6 b^5 (c+d x)^{13} (b c-a d)}{13 d^7}+\frac {5 b^4 (c+d x)^{12} (b c-a d)^2}{4 d^7}-\frac {20 b^3 (c+d x)^{11} (b c-a d)^3}{11 d^7}+\frac {3 b^2 (c+d x)^{10} (b c-a d)^4}{2 d^7}-\frac {2 b (c+d x)^9 (b c-a d)^5}{3 d^7}+\frac {(c+d x)^8 (b c-a d)^6}{8 d^7}+\frac {b^6 (c+d x)^{14}}{14 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6*(c + d*x)^7,x]

[Out]

((b*c - a*d)^6*(c + d*x)^8)/(8*d^7) - (2*b*(b*c - a*d)^5*(c + d*x)^9)/(3*d^7) + (3*b^2*(b*c - a*d)^4*(c + d*x)
^10)/(2*d^7) - (20*b^3*(b*c - a*d)^3*(c + d*x)^11)/(11*d^7) + (5*b^4*(b*c - a*d)^2*(c + d*x)^12)/(4*d^7) - (6*
b^5*(b*c - a*d)*(c + d*x)^13)/(13*d^7) + (b^6*(c + d*x)^14)/(14*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)^7 \, dx &=\int \left (\frac {(-b c+a d)^6 (c+d x)^7}{d^6}-\frac {6 b (b c-a d)^5 (c+d x)^8}{d^6}+\frac {15 b^2 (b c-a d)^4 (c+d x)^9}{d^6}-\frac {20 b^3 (b c-a d)^3 (c+d x)^{10}}{d^6}+\frac {15 b^4 (b c-a d)^2 (c+d x)^{11}}{d^6}-\frac {6 b^5 (b c-a d) (c+d x)^{12}}{d^6}+\frac {b^6 (c+d x)^{13}}{d^6}\right ) \, dx\\ &=\frac {(b c-a d)^6 (c+d x)^8}{8 d^7}-\frac {2 b (b c-a d)^5 (c+d x)^9}{3 d^7}+\frac {3 b^2 (b c-a d)^4 (c+d x)^{10}}{2 d^7}-\frac {20 b^3 (b c-a d)^3 (c+d x)^{11}}{11 d^7}+\frac {5 b^4 (b c-a d)^2 (c+d x)^{12}}{4 d^7}-\frac {6 b^5 (b c-a d) (c+d x)^{13}}{13 d^7}+\frac {b^6 (c+d x)^{14}}{14 d^7}\\ \end {align*}

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Mathematica [B]  time = 0.08, size = 684, normalized size = 3.95 \begin {gather*} a^6 c^7 x+\frac {1}{2} a^5 c^6 x^2 (7 a d+6 b c)+\frac {1}{4} b^4 d^5 x^{12} \left (5 a^2 d^2+14 a b c d+7 b^2 c^2\right )+a^4 c^5 x^3 \left (7 a^2 d^2+14 a b c d+5 b^2 c^2\right )+\frac {1}{11} b^3 d^4 x^{11} \left (20 a^3 d^3+105 a^2 b c d^2+126 a b^2 c^2 d+35 b^3 c^3\right )+\frac {1}{4} a^3 c^4 x^4 \left (35 a^3 d^3+126 a^2 b c d^2+105 a b^2 c^2 d+20 b^3 c^3\right )+\frac {1}{2} b^2 d^3 x^{10} \left (3 a^4 d^4+28 a^3 b c d^3+63 a^2 b^2 c^2 d^2+42 a b^3 c^3 d+7 b^4 c^4\right )+a^2 c^3 x^5 \left (7 a^4 d^4+42 a^3 b c d^3+63 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right )+\frac {1}{3} b d^2 x^9 \left (2 a^5 d^5+35 a^4 b c d^4+140 a^3 b^2 c^2 d^3+175 a^2 b^3 c^3 d^2+70 a b^4 c^4 d+7 b^5 c^5\right )+\frac {1}{2} a c^2 x^6 \left (7 a^5 d^5+70 a^4 b c d^4+175 a^3 b^2 c^2 d^3+140 a^2 b^3 c^3 d^2+35 a b^4 c^4 d+2 b^5 c^5\right )+\frac {1}{8} d x^8 \left (a^6 d^6+42 a^5 b c d^5+315 a^4 b^2 c^2 d^4+700 a^3 b^3 c^3 d^3+525 a^2 b^4 c^4 d^2+126 a b^5 c^5 d+7 b^6 c^6\right )+\frac {1}{7} c x^7 \left (7 a^6 d^6+126 a^5 b c d^5+525 a^4 b^2 c^2 d^4+700 a^3 b^3 c^3 d^3+315 a^2 b^4 c^4 d^2+42 a b^5 c^5 d+b^6 c^6\right )+\frac {1}{13} b^5 d^6 x^{13} (6 a d+7 b c)+\frac {1}{14} b^6 d^7 x^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^6*(c + d*x)^7,x]

[Out]

a^6*c^7*x + (a^5*c^6*(6*b*c + 7*a*d)*x^2)/2 + a^4*c^5*(5*b^2*c^2 + 14*a*b*c*d + 7*a^2*d^2)*x^3 + (a^3*c^4*(20*
b^3*c^3 + 105*a*b^2*c^2*d + 126*a^2*b*c*d^2 + 35*a^3*d^3)*x^4)/4 + a^2*c^3*(3*b^4*c^4 + 28*a*b^3*c^3*d + 63*a^
2*b^2*c^2*d^2 + 42*a^3*b*c*d^3 + 7*a^4*d^4)*x^5 + (a*c^2*(2*b^5*c^5 + 35*a*b^4*c^4*d + 140*a^2*b^3*c^3*d^2 + 1
75*a^3*b^2*c^2*d^3 + 70*a^4*b*c*d^4 + 7*a^5*d^5)*x^6)/2 + (c*(b^6*c^6 + 42*a*b^5*c^5*d + 315*a^2*b^4*c^4*d^2 +
700*a^3*b^3*c^3*d^3 + 525*a^4*b^2*c^2*d^4 + 126*a^5*b*c*d^5 + 7*a^6*d^6)*x^7)/7 + (d*(7*b^6*c^6 + 126*a*b^5*c
^5*d + 525*a^2*b^4*c^4*d^2 + 700*a^3*b^3*c^3*d^3 + 315*a^4*b^2*c^2*d^4 + 42*a^5*b*c*d^5 + a^6*d^6)*x^8)/8 + (b
*d^2*(7*b^5*c^5 + 70*a*b^4*c^4*d + 175*a^2*b^3*c^3*d^2 + 140*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 + 2*a^5*d^5)*x^9
)/3 + (b^2*d^3*(7*b^4*c^4 + 42*a*b^3*c^3*d + 63*a^2*b^2*c^2*d^2 + 28*a^3*b*c*d^3 + 3*a^4*d^4)*x^10)/2 + (b^3*d
^4*(35*b^3*c^3 + 126*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 20*a^3*d^3)*x^11)/11 + (b^4*d^5*(7*b^2*c^2 + 14*a*b*c*d +
5*a^2*d^2)*x^12)/4 + (b^5*d^6*(7*b*c + 6*a*d)*x^13)/13 + (b^6*d^7*x^14)/14

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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)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^6*(c + d*x)^7,x]

[Out]

IntegrateAlgebraic[(a + b*x)^6*(c + d*x)^7, x]

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fricas [B]  time = 1.20, size = 798, normalized size = 4.61 \begin {gather*} \frac {1}{14} x^{14} d^{7} b^{6} + \frac {7}{13} x^{13} d^{6} c b^{6} + \frac {6}{13} x^{13} d^{7} b^{5} a + \frac {7}{4} x^{12} d^{5} c^{2} b^{6} + \frac {7}{2} x^{12} d^{6} c b^{5} a + \frac {5}{4} x^{12} d^{7} b^{4} a^{2} + \frac {35}{11} x^{11} d^{4} c^{3} b^{6} + \frac {126}{11} x^{11} d^{5} c^{2} b^{5} a + \frac {105}{11} x^{11} d^{6} c b^{4} a^{2} + \frac {20}{11} x^{11} d^{7} b^{3} a^{3} + \frac {7}{2} x^{10} d^{3} c^{4} b^{6} + 21 x^{10} d^{4} c^{3} b^{5} a + \frac {63}{2} x^{10} d^{5} c^{2} b^{4} a^{2} + 14 x^{10} d^{6} c b^{3} a^{3} + \frac {3}{2} x^{10} d^{7} b^{2} a^{4} + \frac {7}{3} x^{9} d^{2} c^{5} b^{6} + \frac {70}{3} x^{9} d^{3} c^{4} b^{5} a + \frac {175}{3} x^{9} d^{4} c^{3} b^{4} a^{2} + \frac {140}{3} x^{9} d^{5} c^{2} b^{3} a^{3} + \frac {35}{3} x^{9} d^{6} c b^{2} a^{4} + \frac {2}{3} x^{9} d^{7} b a^{5} + \frac {7}{8} x^{8} d c^{6} b^{6} + \frac {63}{4} x^{8} d^{2} c^{5} b^{5} a + \frac {525}{8} x^{8} d^{3} c^{4} b^{4} a^{2} + \frac {175}{2} x^{8} d^{4} c^{3} b^{3} a^{3} + \frac {315}{8} x^{8} d^{5} c^{2} b^{2} a^{4} + \frac {21}{4} x^{8} d^{6} c b a^{5} + \frac {1}{8} x^{8} d^{7} a^{6} + \frac {1}{7} x^{7} c^{7} b^{6} + 6 x^{7} d c^{6} b^{5} a + 45 x^{7} d^{2} c^{5} b^{4} a^{2} + 100 x^{7} d^{3} c^{4} b^{3} a^{3} + 75 x^{7} d^{4} c^{3} b^{2} a^{4} + 18 x^{7} d^{5} c^{2} b a^{5} + x^{7} d^{6} c a^{6} + x^{6} c^{7} b^{5} a + \frac {35}{2} x^{6} d c^{6} b^{4} a^{2} + 70 x^{6} d^{2} c^{5} b^{3} a^{3} + \frac {175}{2} x^{6} d^{3} c^{4} b^{2} a^{4} + 35 x^{6} d^{4} c^{3} b a^{5} + \frac {7}{2} x^{6} d^{5} c^{2} a^{6} + 3 x^{5} c^{7} b^{4} a^{2} + 28 x^{5} d c^{6} b^{3} a^{3} + 63 x^{5} d^{2} c^{5} b^{2} a^{4} + 42 x^{5} d^{3} c^{4} b a^{5} + 7 x^{5} d^{4} c^{3} a^{6} + 5 x^{4} c^{7} b^{3} a^{3} + \frac {105}{4} x^{4} d c^{6} b^{2} a^{4} + \frac {63}{2} x^{4} d^{2} c^{5} b a^{5} + \frac {35}{4} x^{4} d^{3} c^{4} a^{6} + 5 x^{3} c^{7} b^{2} a^{4} + 14 x^{3} d c^{6} b a^{5} + 7 x^{3} d^{2} c^{5} a^{6} + 3 x^{2} c^{7} b a^{5} + \frac {7}{2} x^{2} d c^{6} a^{6} + x c^{7} a^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/14*x^14*d^7*b^6 + 7/13*x^13*d^6*c*b^6 + 6/13*x^13*d^7*b^5*a + 7/4*x^12*d^5*c^2*b^6 + 7/2*x^12*d^6*c*b^5*a +
5/4*x^12*d^7*b^4*a^2 + 35/11*x^11*d^4*c^3*b^6 + 126/11*x^11*d^5*c^2*b^5*a + 105/11*x^11*d^6*c*b^4*a^2 + 20/11*
x^11*d^7*b^3*a^3 + 7/2*x^10*d^3*c^4*b^6 + 21*x^10*d^4*c^3*b^5*a + 63/2*x^10*d^5*c^2*b^4*a^2 + 14*x^10*d^6*c*b^
3*a^3 + 3/2*x^10*d^7*b^2*a^4 + 7/3*x^9*d^2*c^5*b^6 + 70/3*x^9*d^3*c^4*b^5*a + 175/3*x^9*d^4*c^3*b^4*a^2 + 140/
3*x^9*d^5*c^2*b^3*a^3 + 35/3*x^9*d^6*c*b^2*a^4 + 2/3*x^9*d^7*b*a^5 + 7/8*x^8*d*c^6*b^6 + 63/4*x^8*d^2*c^5*b^5*
a + 525/8*x^8*d^3*c^4*b^4*a^2 + 175/2*x^8*d^4*c^3*b^3*a^3 + 315/8*x^8*d^5*c^2*b^2*a^4 + 21/4*x^8*d^6*c*b*a^5 +
1/8*x^8*d^7*a^6 + 1/7*x^7*c^7*b^6 + 6*x^7*d*c^6*b^5*a + 45*x^7*d^2*c^5*b^4*a^2 + 100*x^7*d^3*c^4*b^3*a^3 + 75
*x^7*d^4*c^3*b^2*a^4 + 18*x^7*d^5*c^2*b*a^5 + x^7*d^6*c*a^6 + x^6*c^7*b^5*a + 35/2*x^6*d*c^6*b^4*a^2 + 70*x^6*
d^2*c^5*b^3*a^3 + 175/2*x^6*d^3*c^4*b^2*a^4 + 35*x^6*d^4*c^3*b*a^5 + 7/2*x^6*d^5*c^2*a^6 + 3*x^5*c^7*b^4*a^2 +
28*x^5*d*c^6*b^3*a^3 + 63*x^5*d^2*c^5*b^2*a^4 + 42*x^5*d^3*c^4*b*a^5 + 7*x^5*d^4*c^3*a^6 + 5*x^4*c^7*b^3*a^3
+ 105/4*x^4*d*c^6*b^2*a^4 + 63/2*x^4*d^2*c^5*b*a^5 + 35/4*x^4*d^3*c^4*a^6 + 5*x^3*c^7*b^2*a^4 + 14*x^3*d*c^6*b
*a^5 + 7*x^3*d^2*c^5*a^6 + 3*x^2*c^7*b*a^5 + 7/2*x^2*d*c^6*a^6 + x*c^7*a^6

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giac [B]  time = 0.95, size = 798, normalized size = 4.61 \begin {gather*} \frac {1}{14} \, b^{6} d^{7} x^{14} + \frac {7}{13} \, b^{6} c d^{6} x^{13} + \frac {6}{13} \, a b^{5} d^{7} x^{13} + \frac {7}{4} \, b^{6} c^{2} d^{5} x^{12} + \frac {7}{2} \, a b^{5} c d^{6} x^{12} + \frac {5}{4} \, a^{2} b^{4} d^{7} x^{12} + \frac {35}{11} \, b^{6} c^{3} d^{4} x^{11} + \frac {126}{11} \, a b^{5} c^{2} d^{5} x^{11} + \frac {105}{11} \, a^{2} b^{4} c d^{6} x^{11} + \frac {20}{11} \, a^{3} b^{3} d^{7} x^{11} + \frac {7}{2} \, b^{6} c^{4} d^{3} x^{10} + 21 \, a b^{5} c^{3} d^{4} x^{10} + \frac {63}{2} \, a^{2} b^{4} c^{2} d^{5} x^{10} + 14 \, a^{3} b^{3} c d^{6} x^{10} + \frac {3}{2} \, a^{4} b^{2} d^{7} x^{10} + \frac {7}{3} \, b^{6} c^{5} d^{2} x^{9} + \frac {70}{3} \, a b^{5} c^{4} d^{3} x^{9} + \frac {175}{3} \, a^{2} b^{4} c^{3} d^{4} x^{9} + \frac {140}{3} \, a^{3} b^{3} c^{2} d^{5} x^{9} + \frac {35}{3} \, a^{4} b^{2} c d^{6} x^{9} + \frac {2}{3} \, a^{5} b d^{7} x^{9} + \frac {7}{8} \, b^{6} c^{6} d x^{8} + \frac {63}{4} \, a b^{5} c^{5} d^{2} x^{8} + \frac {525}{8} \, a^{2} b^{4} c^{4} d^{3} x^{8} + \frac {175}{2} \, a^{3} b^{3} c^{3} d^{4} x^{8} + \frac {315}{8} \, a^{4} b^{2} c^{2} d^{5} x^{8} + \frac {21}{4} \, a^{5} b c d^{6} x^{8} + \frac {1}{8} \, a^{6} d^{7} x^{8} + \frac {1}{7} \, b^{6} c^{7} x^{7} + 6 \, a b^{5} c^{6} d x^{7} + 45 \, a^{2} b^{4} c^{5} d^{2} x^{7} + 100 \, a^{3} b^{3} c^{4} d^{3} x^{7} + 75 \, a^{4} b^{2} c^{3} d^{4} x^{7} + 18 \, a^{5} b c^{2} d^{5} x^{7} + a^{6} c d^{6} x^{7} + a b^{5} c^{7} x^{6} + \frac {35}{2} \, a^{2} b^{4} c^{6} d x^{6} + 70 \, a^{3} b^{3} c^{5} d^{2} x^{6} + \frac {175}{2} \, a^{4} b^{2} c^{4} d^{3} x^{6} + 35 \, a^{5} b c^{3} d^{4} x^{6} + \frac {7}{2} \, a^{6} c^{2} d^{5} x^{6} + 3 \, a^{2} b^{4} c^{7} x^{5} + 28 \, a^{3} b^{3} c^{6} d x^{5} + 63 \, a^{4} b^{2} c^{5} d^{2} x^{5} + 42 \, a^{5} b c^{4} d^{3} x^{5} + 7 \, a^{6} c^{3} d^{4} x^{5} + 5 \, a^{3} b^{3} c^{7} x^{4} + \frac {105}{4} \, a^{4} b^{2} c^{6} d x^{4} + \frac {63}{2} \, a^{5} b c^{5} d^{2} x^{4} + \frac {35}{4} \, a^{6} c^{4} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{7} x^{3} + 14 \, a^{5} b c^{6} d x^{3} + 7 \, a^{6} c^{5} d^{2} x^{3} + 3 \, a^{5} b c^{7} x^{2} + \frac {7}{2} \, a^{6} c^{6} d x^{2} + a^{6} c^{7} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(d*x+c)^7,x, algorithm="giac")

[Out]

1/14*b^6*d^7*x^14 + 7/13*b^6*c*d^6*x^13 + 6/13*a*b^5*d^7*x^13 + 7/4*b^6*c^2*d^5*x^12 + 7/2*a*b^5*c*d^6*x^12 +
5/4*a^2*b^4*d^7*x^12 + 35/11*b^6*c^3*d^4*x^11 + 126/11*a*b^5*c^2*d^5*x^11 + 105/11*a^2*b^4*c*d^6*x^11 + 20/11*
a^3*b^3*d^7*x^11 + 7/2*b^6*c^4*d^3*x^10 + 21*a*b^5*c^3*d^4*x^10 + 63/2*a^2*b^4*c^2*d^5*x^10 + 14*a^3*b^3*c*d^6
*x^10 + 3/2*a^4*b^2*d^7*x^10 + 7/3*b^6*c^5*d^2*x^9 + 70/3*a*b^5*c^4*d^3*x^9 + 175/3*a^2*b^4*c^3*d^4*x^9 + 140/
3*a^3*b^3*c^2*d^5*x^9 + 35/3*a^4*b^2*c*d^6*x^9 + 2/3*a^5*b*d^7*x^9 + 7/8*b^6*c^6*d*x^8 + 63/4*a*b^5*c^5*d^2*x^
8 + 525/8*a^2*b^4*c^4*d^3*x^8 + 175/2*a^3*b^3*c^3*d^4*x^8 + 315/8*a^4*b^2*c^2*d^5*x^8 + 21/4*a^5*b*c*d^6*x^8 +
1/8*a^6*d^7*x^8 + 1/7*b^6*c^7*x^7 + 6*a*b^5*c^6*d*x^7 + 45*a^2*b^4*c^5*d^2*x^7 + 100*a^3*b^3*c^4*d^3*x^7 + 75
*a^4*b^2*c^3*d^4*x^7 + 18*a^5*b*c^2*d^5*x^7 + a^6*c*d^6*x^7 + a*b^5*c^7*x^6 + 35/2*a^2*b^4*c^6*d*x^6 + 70*a^3*
b^3*c^5*d^2*x^6 + 175/2*a^4*b^2*c^4*d^3*x^6 + 35*a^5*b*c^3*d^4*x^6 + 7/2*a^6*c^2*d^5*x^6 + 3*a^2*b^4*c^7*x^5 +
28*a^3*b^3*c^6*d*x^5 + 63*a^4*b^2*c^5*d^2*x^5 + 42*a^5*b*c^4*d^3*x^5 + 7*a^6*c^3*d^4*x^5 + 5*a^3*b^3*c^7*x^4
+ 105/4*a^4*b^2*c^6*d*x^4 + 63/2*a^5*b*c^5*d^2*x^4 + 35/4*a^6*c^4*d^3*x^4 + 5*a^4*b^2*c^7*x^3 + 14*a^5*b*c^6*d
*x^3 + 7*a^6*c^5*d^2*x^3 + 3*a^5*b*c^7*x^2 + 7/2*a^6*c^6*d*x^2 + a^6*c^7*x

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maple [B]  time = 0.00, size = 709, normalized size = 4.10 \begin {gather*} \frac {b^{6} d^{7} x^{14}}{14}+a^{6} c^{7} x +\frac {\left (6 a \,b^{5} d^{7}+7 b^{6} c \,d^{6}\right ) x^{13}}{13}+\frac {\left (15 a^{2} b^{4} d^{7}+42 a \,b^{5} c \,d^{6}+21 b^{6} c^{2} d^{5}\right ) x^{12}}{12}+\frac {\left (20 a^{3} b^{3} d^{7}+105 a^{2} b^{4} c \,d^{6}+126 a \,b^{5} c^{2} d^{5}+35 b^{6} c^{3} d^{4}\right ) x^{11}}{11}+\frac {\left (15 a^{4} b^{2} d^{7}+140 a^{3} b^{3} c \,d^{6}+315 a^{2} b^{4} c^{2} d^{5}+210 a \,b^{5} c^{3} d^{4}+35 b^{6} c^{4} d^{3}\right ) x^{10}}{10}+\frac {\left (6 a^{5} b \,d^{7}+105 a^{4} b^{2} c \,d^{6}+420 a^{3} b^{3} c^{2} d^{5}+525 a^{2} b^{4} c^{3} d^{4}+210 a \,b^{5} c^{4} d^{3}+21 b^{6} c^{5} d^{2}\right ) x^{9}}{9}+\frac {\left (a^{6} d^{7}+42 a^{5} b c \,d^{6}+315 a^{4} b^{2} c^{2} d^{5}+700 a^{3} b^{3} c^{3} d^{4}+525 a^{2} b^{4} c^{4} d^{3}+126 a \,b^{5} c^{5} d^{2}+7 b^{6} c^{6} d \right ) x^{8}}{8}+\frac {\left (7 a^{6} c \,d^{6}+126 a^{5} b \,c^{2} d^{5}+525 a^{4} b^{2} c^{3} d^{4}+700 a^{3} b^{3} c^{4} d^{3}+315 a^{2} b^{4} c^{5} d^{2}+42 a \,b^{5} c^{6} d +b^{6} c^{7}\right ) x^{7}}{7}+\frac {\left (21 a^{6} c^{2} d^{5}+210 a^{5} b \,c^{3} d^{4}+525 a^{4} b^{2} c^{4} d^{3}+420 a^{3} b^{3} c^{5} d^{2}+105 a^{2} b^{4} c^{6} d +6 a \,b^{5} c^{7}\right ) x^{6}}{6}+\frac {\left (35 a^{6} c^{3} d^{4}+210 a^{5} b \,c^{4} d^{3}+315 a^{4} b^{2} c^{5} d^{2}+140 a^{3} b^{3} c^{6} d +15 a^{2} b^{4} c^{7}\right ) x^{5}}{5}+\frac {\left (35 a^{6} c^{4} d^{3}+126 a^{5} b \,c^{5} d^{2}+105 a^{4} b^{2} c^{6} d +20 a^{3} b^{3} c^{7}\right ) x^{4}}{4}+\frac {\left (21 a^{6} c^{5} d^{2}+42 a^{5} b \,c^{6} d +15 a^{4} b^{2} c^{7}\right ) x^{3}}{3}+\frac {\left (7 a^{6} c^{6} d +6 a^{5} b \,c^{7}\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)^7,x)

[Out]

1/14*b^6*d^7*x^14+1/13*(6*a*b^5*d^7+7*b^6*c*d^6)*x^13+1/12*(15*a^2*b^4*d^7+42*a*b^5*c*d^6+21*b^6*c^2*d^5)*x^12
+1/11*(20*a^3*b^3*d^7+105*a^2*b^4*c*d^6+126*a*b^5*c^2*d^5+35*b^6*c^3*d^4)*x^11+1/10*(15*a^4*b^2*d^7+140*a^3*b^
3*c*d^6+315*a^2*b^4*c^2*d^5+210*a*b^5*c^3*d^4+35*b^6*c^4*d^3)*x^10+1/9*(6*a^5*b*d^7+105*a^4*b^2*c*d^6+420*a^3*
b^3*c^2*d^5+525*a^2*b^4*c^3*d^4+210*a*b^5*c^4*d^3+21*b^6*c^5*d^2)*x^9+1/8*(a^6*d^7+42*a^5*b*c*d^6+315*a^4*b^2*
c^2*d^5+700*a^3*b^3*c^3*d^4+525*a^2*b^4*c^4*d^3+126*a*b^5*c^5*d^2+7*b^6*c^6*d)*x^8+1/7*(7*a^6*c*d^6+126*a^5*b*
c^2*d^5+525*a^4*b^2*c^3*d^4+700*a^3*b^3*c^4*d^3+315*a^2*b^4*c^5*d^2+42*a*b^5*c^6*d+b^6*c^7)*x^7+1/6*(21*a^6*c^
2*d^5+210*a^5*b*c^3*d^4+525*a^4*b^2*c^4*d^3+420*a^3*b^3*c^5*d^2+105*a^2*b^4*c^6*d+6*a*b^5*c^7)*x^6+1/5*(35*a^6
*c^3*d^4+210*a^5*b*c^4*d^3+315*a^4*b^2*c^5*d^2+140*a^3*b^3*c^6*d+15*a^2*b^4*c^7)*x^5+1/4*(35*a^6*c^4*d^3+126*a
^5*b*c^5*d^2+105*a^4*b^2*c^6*d+20*a^3*b^3*c^7)*x^4+1/3*(21*a^6*c^5*d^2+42*a^5*b*c^6*d+15*a^4*b^2*c^7)*x^3+1/2*
(7*a^6*c^6*d+6*a^5*b*c^7)*x^2+a^6*c^7*x

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maxima [B]  time = 1.32, size = 706, normalized size = 4.08 \begin {gather*} \frac {1}{14} \, b^{6} d^{7} x^{14} + a^{6} c^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} c d^{6} + 6 \, a b^{5} d^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} c^{2} d^{5} + 14 \, a b^{5} c d^{6} + 5 \, a^{2} b^{4} d^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} c^{3} d^{4} + 126 \, a b^{5} c^{2} d^{5} + 105 \, a^{2} b^{4} c d^{6} + 20 \, a^{3} b^{3} d^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} c^{4} d^{3} + 42 \, a b^{5} c^{3} d^{4} + 63 \, a^{2} b^{4} c^{2} d^{5} + 28 \, a^{3} b^{3} c d^{6} + 3 \, a^{4} b^{2} d^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} c^{5} d^{2} + 70 \, a b^{5} c^{4} d^{3} + 175 \, a^{2} b^{4} c^{3} d^{4} + 140 \, a^{3} b^{3} c^{2} d^{5} + 35 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} c^{6} d + 126 \, a b^{5} c^{5} d^{2} + 525 \, a^{2} b^{4} c^{4} d^{3} + 700 \, a^{3} b^{3} c^{3} d^{4} + 315 \, a^{4} b^{2} c^{2} d^{5} + 42 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{7} + 42 \, a b^{5} c^{6} d + 315 \, a^{2} b^{4} c^{5} d^{2} + 700 \, a^{3} b^{3} c^{4} d^{3} + 525 \, a^{4} b^{2} c^{3} d^{4} + 126 \, a^{5} b c^{2} d^{5} + 7 \, a^{6} c d^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} c^{7} + 35 \, a^{2} b^{4} c^{6} d + 140 \, a^{3} b^{3} c^{5} d^{2} + 175 \, a^{4} b^{2} c^{4} d^{3} + 70 \, a^{5} b c^{3} d^{4} + 7 \, a^{6} c^{2} d^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} c^{7} + 28 \, a^{3} b^{3} c^{6} d + 63 \, a^{4} b^{2} c^{5} d^{2} + 42 \, a^{5} b c^{4} d^{3} + 7 \, a^{6} c^{3} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} c^{7} + 105 \, a^{4} b^{2} c^{6} d + 126 \, a^{5} b c^{5} d^{2} + 35 \, a^{6} c^{4} d^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{7} + 14 \, a^{5} b c^{6} d + 7 \, a^{6} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b c^{7} + 7 \, a^{6} c^{6} 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)^7,x, algorithm="maxima")

[Out]

1/14*b^6*d^7*x^14 + a^6*c^7*x + 1/13*(7*b^6*c*d^6 + 6*a*b^5*d^7)*x^13 + 1/4*(7*b^6*c^2*d^5 + 14*a*b^5*c*d^6 +
5*a^2*b^4*d^7)*x^12 + 1/11*(35*b^6*c^3*d^4 + 126*a*b^5*c^2*d^5 + 105*a^2*b^4*c*d^6 + 20*a^3*b^3*d^7)*x^11 + 1/
2*(7*b^6*c^4*d^3 + 42*a*b^5*c^3*d^4 + 63*a^2*b^4*c^2*d^5 + 28*a^3*b^3*c*d^6 + 3*a^4*b^2*d^7)*x^10 + 1/3*(7*b^6
*c^5*d^2 + 70*a*b^5*c^4*d^3 + 175*a^2*b^4*c^3*d^4 + 140*a^3*b^3*c^2*d^5 + 35*a^4*b^2*c*d^6 + 2*a^5*b*d^7)*x^9
+ 1/8*(7*b^6*c^6*d + 126*a*b^5*c^5*d^2 + 525*a^2*b^4*c^4*d^3 + 700*a^3*b^3*c^3*d^4 + 315*a^4*b^2*c^2*d^5 + 42*
a^5*b*c*d^6 + a^6*d^7)*x^8 + 1/7*(b^6*c^7 + 42*a*b^5*c^6*d + 315*a^2*b^4*c^5*d^2 + 700*a^3*b^3*c^4*d^3 + 525*a
^4*b^2*c^3*d^4 + 126*a^5*b*c^2*d^5 + 7*a^6*c*d^6)*x^7 + 1/2*(2*a*b^5*c^7 + 35*a^2*b^4*c^6*d + 140*a^3*b^3*c^5*
d^2 + 175*a^4*b^2*c^4*d^3 + 70*a^5*b*c^3*d^4 + 7*a^6*c^2*d^5)*x^6 + (3*a^2*b^4*c^7 + 28*a^3*b^3*c^6*d + 63*a^4
*b^2*c^5*d^2 + 42*a^5*b*c^4*d^3 + 7*a^6*c^3*d^4)*x^5 + 1/4*(20*a^3*b^3*c^7 + 105*a^4*b^2*c^6*d + 126*a^5*b*c^5
*d^2 + 35*a^6*c^4*d^3)*x^4 + (5*a^4*b^2*c^7 + 14*a^5*b*c^6*d + 7*a^6*c^5*d^2)*x^3 + 1/2*(6*a^5*b*c^7 + 7*a^6*c
^6*d)*x^2

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mupad [B]  time = 0.26, size = 683, normalized size = 3.95 \begin {gather*} x^5\,\left (7\,a^6\,c^3\,d^4+42\,a^5\,b\,c^4\,d^3+63\,a^4\,b^2\,c^5\,d^2+28\,a^3\,b^3\,c^6\,d+3\,a^2\,b^4\,c^7\right )+x^{10}\,\left (\frac {3\,a^4\,b^2\,d^7}{2}+14\,a^3\,b^3\,c\,d^6+\frac {63\,a^2\,b^4\,c^2\,d^5}{2}+21\,a\,b^5\,c^3\,d^4+\frac {7\,b^6\,c^4\,d^3}{2}\right )+x^6\,\left (\frac {7\,a^6\,c^2\,d^5}{2}+35\,a^5\,b\,c^3\,d^4+\frac {175\,a^4\,b^2\,c^4\,d^3}{2}+70\,a^3\,b^3\,c^5\,d^2+\frac {35\,a^2\,b^4\,c^6\,d}{2}+a\,b^5\,c^7\right )+x^9\,\left (\frac {2\,a^5\,b\,d^7}{3}+\frac {35\,a^4\,b^2\,c\,d^6}{3}+\frac {140\,a^3\,b^3\,c^2\,d^5}{3}+\frac {175\,a^2\,b^4\,c^3\,d^4}{3}+\frac {70\,a\,b^5\,c^4\,d^3}{3}+\frac {7\,b^6\,c^5\,d^2}{3}\right )+x^7\,\left (a^6\,c\,d^6+18\,a^5\,b\,c^2\,d^5+75\,a^4\,b^2\,c^3\,d^4+100\,a^3\,b^3\,c^4\,d^3+45\,a^2\,b^4\,c^5\,d^2+6\,a\,b^5\,c^6\,d+\frac {b^6\,c^7}{7}\right )+x^8\,\left (\frac {a^6\,d^7}{8}+\frac {21\,a^5\,b\,c\,d^6}{4}+\frac {315\,a^4\,b^2\,c^2\,d^5}{8}+\frac {175\,a^3\,b^3\,c^3\,d^4}{2}+\frac {525\,a^2\,b^4\,c^4\,d^3}{8}+\frac {63\,a\,b^5\,c^5\,d^2}{4}+\frac {7\,b^6\,c^6\,d}{8}\right )+x^4\,\left (\frac {35\,a^6\,c^4\,d^3}{4}+\frac {63\,a^5\,b\,c^5\,d^2}{2}+\frac {105\,a^4\,b^2\,c^6\,d}{4}+5\,a^3\,b^3\,c^7\right )+x^{11}\,\left (\frac {20\,a^3\,b^3\,d^7}{11}+\frac {105\,a^2\,b^4\,c\,d^6}{11}+\frac {126\,a\,b^5\,c^2\,d^5}{11}+\frac {35\,b^6\,c^3\,d^4}{11}\right )+a^6\,c^7\,x+\frac {b^6\,d^7\,x^{14}}{14}+\frac {a^5\,c^6\,x^2\,\left (7\,a\,d+6\,b\,c\right )}{2}+\frac {b^5\,d^6\,x^{13}\,\left (6\,a\,d+7\,b\,c\right )}{13}+a^4\,c^5\,x^3\,\left (7\,a^2\,d^2+14\,a\,b\,c\,d+5\,b^2\,c^2\right )+\frac {b^4\,d^5\,x^{12}\,\left (5\,a^2\,d^2+14\,a\,b\,c\,d+7\,b^2\,c^2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6*(c + d*x)^7,x)

[Out]

x^5*(3*a^2*b^4*c^7 + 7*a^6*c^3*d^4 + 28*a^3*b^3*c^6*d + 42*a^5*b*c^4*d^3 + 63*a^4*b^2*c^5*d^2) + x^10*((3*a^4*
b^2*d^7)/2 + (7*b^6*c^4*d^3)/2 + 21*a*b^5*c^3*d^4 + 14*a^3*b^3*c*d^6 + (63*a^2*b^4*c^2*d^5)/2) + x^6*(a*b^5*c^
7 + (7*a^6*c^2*d^5)/2 + (35*a^2*b^4*c^6*d)/2 + 35*a^5*b*c^3*d^4 + 70*a^3*b^3*c^5*d^2 + (175*a^4*b^2*c^4*d^3)/2
) + x^9*((2*a^5*b*d^7)/3 + (7*b^6*c^5*d^2)/3 + (70*a*b^5*c^4*d^3)/3 + (35*a^4*b^2*c*d^6)/3 + (175*a^2*b^4*c^3*
d^4)/3 + (140*a^3*b^3*c^2*d^5)/3) + x^7*((b^6*c^7)/7 + a^6*c*d^6 + 18*a^5*b*c^2*d^5 + 45*a^2*b^4*c^5*d^2 + 100
*a^3*b^3*c^4*d^3 + 75*a^4*b^2*c^3*d^4 + 6*a*b^5*c^6*d) + x^8*((a^6*d^7)/8 + (7*b^6*c^6*d)/8 + (63*a*b^5*c^5*d^
2)/4 + (525*a^2*b^4*c^4*d^3)/8 + (175*a^3*b^3*c^3*d^4)/2 + (315*a^4*b^2*c^2*d^5)/8 + (21*a^5*b*c*d^6)/4) + x^4
*(5*a^3*b^3*c^7 + (35*a^6*c^4*d^3)/4 + (105*a^4*b^2*c^6*d)/4 + (63*a^5*b*c^5*d^2)/2) + x^11*((20*a^3*b^3*d^7)/
11 + (35*b^6*c^3*d^4)/11 + (126*a*b^5*c^2*d^5)/11 + (105*a^2*b^4*c*d^6)/11) + a^6*c^7*x + (b^6*d^7*x^14)/14 +
(a^5*c^6*x^2*(7*a*d + 6*b*c))/2 + (b^5*d^6*x^13*(6*a*d + 7*b*c))/13 + a^4*c^5*x^3*(7*a^2*d^2 + 5*b^2*c^2 + 14*
a*b*c*d) + (b^4*d^5*x^12*(5*a^2*d^2 + 7*b^2*c^2 + 14*a*b*c*d))/4

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sympy [B]  time = 0.18, size = 796, normalized size = 4.60 \begin {gather*} a^{6} c^{7} x + \frac {b^{6} d^{7} x^{14}}{14} + x^{13} \left (\frac {6 a b^{5} d^{7}}{13} + \frac {7 b^{6} c d^{6}}{13}\right ) + x^{12} \left (\frac {5 a^{2} b^{4} d^{7}}{4} + \frac {7 a b^{5} c d^{6}}{2} + \frac {7 b^{6} c^{2} d^{5}}{4}\right ) + x^{11} \left (\frac {20 a^{3} b^{3} d^{7}}{11} + \frac {105 a^{2} b^{4} c d^{6}}{11} + \frac {126 a b^{5} c^{2} d^{5}}{11} + \frac {35 b^{6} c^{3} d^{4}}{11}\right ) + x^{10} \left (\frac {3 a^{4} b^{2} d^{7}}{2} + 14 a^{3} b^{3} c d^{6} + \frac {63 a^{2} b^{4} c^{2} d^{5}}{2} + 21 a b^{5} c^{3} d^{4} + \frac {7 b^{6} c^{4} d^{3}}{2}\right ) + x^{9} \left (\frac {2 a^{5} b d^{7}}{3} + \frac {35 a^{4} b^{2} c d^{6}}{3} + \frac {140 a^{3} b^{3} c^{2} d^{5}}{3} + \frac {175 a^{2} b^{4} c^{3} d^{4}}{3} + \frac {70 a b^{5} c^{4} d^{3}}{3} + \frac {7 b^{6} c^{5} d^{2}}{3}\right ) + x^{8} \left (\frac {a^{6} d^{7}}{8} + \frac {21 a^{5} b c d^{6}}{4} + \frac {315 a^{4} b^{2} c^{2} d^{5}}{8} + \frac {175 a^{3} b^{3} c^{3} d^{4}}{2} + \frac {525 a^{2} b^{4} c^{4} d^{3}}{8} + \frac {63 a b^{5} c^{5} d^{2}}{4} + \frac {7 b^{6} c^{6} d}{8}\right ) + x^{7} \left (a^{6} c d^{6} + 18 a^{5} b c^{2} d^{5} + 75 a^{4} b^{2} c^{3} d^{4} + 100 a^{3} b^{3} c^{4} d^{3} + 45 a^{2} b^{4} c^{5} d^{2} + 6 a b^{5} c^{6} d + \frac {b^{6} c^{7}}{7}\right ) + x^{6} \left (\frac {7 a^{6} c^{2} d^{5}}{2} + 35 a^{5} b c^{3} d^{4} + \frac {175 a^{4} b^{2} c^{4} d^{3}}{2} + 70 a^{3} b^{3} c^{5} d^{2} + \frac {35 a^{2} b^{4} c^{6} d}{2} + a b^{5} c^{7}\right ) + x^{5} \left (7 a^{6} c^{3} d^{4} + 42 a^{5} b c^{4} d^{3} + 63 a^{4} b^{2} c^{5} d^{2} + 28 a^{3} b^{3} c^{6} d + 3 a^{2} b^{4} c^{7}\right ) + x^{4} \left (\frac {35 a^{6} c^{4} d^{3}}{4} + \frac {63 a^{5} b c^{5} d^{2}}{2} + \frac {105 a^{4} b^{2} c^{6} d}{4} + 5 a^{3} b^{3} c^{7}\right ) + x^{3} \left (7 a^{6} c^{5} d^{2} + 14 a^{5} b c^{6} d + 5 a^{4} b^{2} c^{7}\right ) + x^{2} \left (\frac {7 a^{6} c^{6} d}{2} + 3 a^{5} b c^{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(d*x+c)**7,x)

[Out]

a**6*c**7*x + b**6*d**7*x**14/14 + x**13*(6*a*b**5*d**7/13 + 7*b**6*c*d**6/13) + x**12*(5*a**2*b**4*d**7/4 + 7
*a*b**5*c*d**6/2 + 7*b**6*c**2*d**5/4) + x**11*(20*a**3*b**3*d**7/11 + 105*a**2*b**4*c*d**6/11 + 126*a*b**5*c*
*2*d**5/11 + 35*b**6*c**3*d**4/11) + x**10*(3*a**4*b**2*d**7/2 + 14*a**3*b**3*c*d**6 + 63*a**2*b**4*c**2*d**5/
2 + 21*a*b**5*c**3*d**4 + 7*b**6*c**4*d**3/2) + x**9*(2*a**5*b*d**7/3 + 35*a**4*b**2*c*d**6/3 + 140*a**3*b**3*
c**2*d**5/3 + 175*a**2*b**4*c**3*d**4/3 + 70*a*b**5*c**4*d**3/3 + 7*b**6*c**5*d**2/3) + x**8*(a**6*d**7/8 + 21
*a**5*b*c*d**6/4 + 315*a**4*b**2*c**2*d**5/8 + 175*a**3*b**3*c**3*d**4/2 + 525*a**2*b**4*c**4*d**3/8 + 63*a*b*
*5*c**5*d**2/4 + 7*b**6*c**6*d/8) + x**7*(a**6*c*d**6 + 18*a**5*b*c**2*d**5 + 75*a**4*b**2*c**3*d**4 + 100*a**
3*b**3*c**4*d**3 + 45*a**2*b**4*c**5*d**2 + 6*a*b**5*c**6*d + b**6*c**7/7) + x**6*(7*a**6*c**2*d**5/2 + 35*a**
5*b*c**3*d**4 + 175*a**4*b**2*c**4*d**3/2 + 70*a**3*b**3*c**5*d**2 + 35*a**2*b**4*c**6*d/2 + a*b**5*c**7) + x*
*5*(7*a**6*c**3*d**4 + 42*a**5*b*c**4*d**3 + 63*a**4*b**2*c**5*d**2 + 28*a**3*b**3*c**6*d + 3*a**2*b**4*c**7)
+ x**4*(35*a**6*c**4*d**3/4 + 63*a**5*b*c**5*d**2/2 + 105*a**4*b**2*c**6*d/4 + 5*a**3*b**3*c**7) + x**3*(7*a**
6*c**5*d**2 + 14*a**5*b*c**6*d + 5*a**4*b**2*c**7) + x**2*(7*a**6*c**6*d/2 + 3*a**5*b*c**7)

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