3.1278 \(\int (a+b x)^4 (c+d x)^7 \, dx\)

Optimal. Leaf size=119 \[ -\frac{4 b^3 (c+d x)^{11} (b c-a d)}{11 d^5}+\frac{3 b^2 (c+d x)^{10} (b c-a d)^2}{5 d^5}-\frac{4 b (c+d x)^9 (b c-a d)^3}{9 d^5}+\frac{(c+d x)^8 (b c-a d)^4}{8 d^5}+\frac{b^4 (c+d x)^{12}}{12 d^5} \]

[Out]

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

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Rubi [A]  time = 0.279351, antiderivative size = 119, 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} \[ -\frac{4 b^3 (c+d x)^{11} (b c-a d)}{11 d^5}+\frac{3 b^2 (c+d x)^{10} (b c-a d)^2}{5 d^5}-\frac{4 b (c+d x)^9 (b c-a d)^3}{9 d^5}+\frac{(c+d x)^8 (b c-a d)^4}{8 d^5}+\frac{b^4 (c+d x)^{12}}{12 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^7,x]

[Out]

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

Mathematica [B]  time = 0.049573, size = 473, normalized size = 3.97 \[ \frac{1}{10} b^2 d^5 x^{10} \left (6 a^2 d^2+28 a b c d+21 b^2 c^2\right )+\frac{1}{9} b d^4 x^9 \left (42 a^2 b c d^2+4 a^3 d^3+84 a b^2 c^2 d+35 b^3 c^3\right )+\frac{1}{8} d^3 x^8 \left (126 a^2 b^2 c^2 d^2+28 a^3 b c d^3+a^4 d^4+140 a b^3 c^3 d+35 b^4 c^4\right )+c d^2 x^7 \left (30 a^2 b^2 c^2 d^2+12 a^3 b c d^3+a^4 d^4+20 a b^3 c^3 d+3 b^4 c^4\right )+\frac{7}{6} c^2 d x^6 \left (30 a^2 b^2 c^2 d^2+20 a^3 b c d^3+3 a^4 d^4+12 a b^3 c^3 d+b^4 c^4\right )+\frac{1}{5} c^3 x^5 \left (126 a^2 b^2 c^2 d^2+140 a^3 b c d^3+35 a^4 d^4+28 a b^3 c^3 d+b^4 c^4\right )+\frac{1}{4} a c^4 x^4 \left (84 a^2 b c d^2+35 a^3 d^3+42 a b^2 c^2 d+4 b^3 c^3\right )+\frac{1}{3} a^2 c^5 x^3 \left (21 a^2 d^2+28 a b c d+6 b^2 c^2\right )+\frac{1}{2} a^3 c^6 x^2 (7 a d+4 b c)+a^4 c^7 x+\frac{1}{11} b^3 d^6 x^{11} (4 a d+7 b c)+\frac{1}{12} b^4 d^7 x^{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^7,x]

[Out]

a^4*c^7*x + (a^3*c^6*(4*b*c + 7*a*d)*x^2)/2 + (a^2*c^5*(6*b^2*c^2 + 28*a*b*c*d + 21*a^2*d^2)*x^3)/3 + (a*c^4*(
4*b^3*c^3 + 42*a*b^2*c^2*d + 84*a^2*b*c*d^2 + 35*a^3*d^3)*x^4)/4 + (c^3*(b^4*c^4 + 28*a*b^3*c^3*d + 126*a^2*b^
2*c^2*d^2 + 140*a^3*b*c*d^3 + 35*a^4*d^4)*x^5)/5 + (7*c^2*d*(b^4*c^4 + 12*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 2
0*a^3*b*c*d^3 + 3*a^4*d^4)*x^6)/6 + c*d^2*(3*b^4*c^4 + 20*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 +
a^4*d^4)*x^7 + (d^3*(35*b^4*c^4 + 140*a*b^3*c^3*d + 126*a^2*b^2*c^2*d^2 + 28*a^3*b*c*d^3 + a^4*d^4)*x^8)/8 + (
b*d^4*(35*b^3*c^3 + 84*a*b^2*c^2*d + 42*a^2*b*c*d^2 + 4*a^3*d^3)*x^9)/9 + (b^2*d^5*(21*b^2*c^2 + 28*a*b*c*d +
6*a^2*d^2)*x^10)/10 + (b^3*d^6*(7*b*c + 4*a*d)*x^11)/11 + (b^4*d^7*x^12)/12

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Maple [B]  time = 0.002, size = 493, normalized size = 4.1 \begin{align*}{\frac{{b}^{4}{d}^{7}{x}^{12}}{12}}+{\frac{ \left ( 4\,a{b}^{3}{d}^{7}+7\,{b}^{4}c{d}^{6} \right ){x}^{11}}{11}}+{\frac{ \left ( 6\,{b}^{2}{a}^{2}{d}^{7}+28\,a{b}^{3}c{d}^{6}+21\,{b}^{4}{c}^{2}{d}^{5} \right ){x}^{10}}{10}}+{\frac{ \left ( 4\,{a}^{3}b{d}^{7}+42\,{b}^{2}{a}^{2}c{d}^{6}+84\,a{b}^{3}{c}^{2}{d}^{5}+35\,{b}^{4}{c}^{3}{d}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{4}{d}^{7}+28\,{a}^{3}bc{d}^{6}+126\,{b}^{2}{a}^{2}{c}^{2}{d}^{5}+140\,a{b}^{3}{c}^{3}{d}^{4}+35\,{b}^{4}{c}^{4}{d}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{4}c{d}^{6}+84\,{a}^{3}b{c}^{2}{d}^{5}+210\,{b}^{2}{a}^{2}{c}^{3}{d}^{4}+140\,a{b}^{3}{c}^{4}{d}^{3}+21\,{b}^{4}{c}^{5}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{4}{c}^{2}{d}^{5}+140\,{a}^{3}b{c}^{3}{d}^{4}+210\,{b}^{2}{a}^{2}{c}^{4}{d}^{3}+84\,a{b}^{3}{c}^{5}{d}^{2}+7\,{b}^{4}{c}^{6}d \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{4}{c}^{3}{d}^{4}+140\,{a}^{3}b{c}^{4}{d}^{3}+126\,{b}^{2}{a}^{2}{c}^{5}{d}^{2}+28\,a{b}^{3}{c}^{6}d+{b}^{4}{c}^{7} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{4}{c}^{4}{d}^{3}+84\,{a}^{3}b{c}^{5}{d}^{2}+42\,{b}^{2}{a}^{2}{c}^{6}d+4\,a{b}^{3}{c}^{7} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{4}{c}^{5}{d}^{2}+28\,{a}^{3}b{c}^{6}d+6\,{b}^{2}{a}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{4}{c}^{6}d+4\,{a}^{3}b{c}^{7} \right ){x}^{2}}{2}}+{a}^{4}{c}^{7}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^7,x)

[Out]

1/12*b^4*d^7*x^12+1/11*(4*a*b^3*d^7+7*b^4*c*d^6)*x^11+1/10*(6*a^2*b^2*d^7+28*a*b^3*c*d^6+21*b^4*c^2*d^5)*x^10+
1/9*(4*a^3*b*d^7+42*a^2*b^2*c*d^6+84*a*b^3*c^2*d^5+35*b^4*c^3*d^4)*x^9+1/8*(a^4*d^7+28*a^3*b*c*d^6+126*a^2*b^2
*c^2*d^5+140*a*b^3*c^3*d^4+35*b^4*c^4*d^3)*x^8+1/7*(7*a^4*c*d^6+84*a^3*b*c^2*d^5+210*a^2*b^2*c^3*d^4+140*a*b^3
*c^4*d^3+21*b^4*c^5*d^2)*x^7+1/6*(21*a^4*c^2*d^5+140*a^3*b*c^3*d^4+210*a^2*b^2*c^4*d^3+84*a*b^3*c^5*d^2+7*b^4*
c^6*d)*x^6+1/5*(35*a^4*c^3*d^4+140*a^3*b*c^4*d^3+126*a^2*b^2*c^5*d^2+28*a*b^3*c^6*d+b^4*c^7)*x^5+1/4*(35*a^4*c
^4*d^3+84*a^3*b*c^5*d^2+42*a^2*b^2*c^6*d+4*a*b^3*c^7)*x^4+1/3*(21*a^4*c^5*d^2+28*a^3*b*c^6*d+6*a^2*b^2*c^7)*x^
3+1/2*(7*a^4*c^6*d+4*a^3*b*c^7)*x^2+a^4*c^7*x

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Maxima [B]  time = 0.983395, size = 660, normalized size = 5.55 \begin{align*} \frac{1}{12} \, b^{4} d^{7} x^{12} + a^{4} c^{7} x + \frac{1}{11} \,{\left (7 \, b^{4} c d^{6} + 4 \, a b^{3} d^{7}\right )} x^{11} + \frac{1}{10} \,{\left (21 \, b^{4} c^{2} d^{5} + 28 \, a b^{3} c d^{6} + 6 \, a^{2} b^{2} d^{7}\right )} x^{10} + \frac{1}{9} \,{\left (35 \, b^{4} c^{3} d^{4} + 84 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} + 4 \, a^{3} b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (35 \, b^{4} c^{4} d^{3} + 140 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} + 28 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{8} +{\left (3 \, b^{4} c^{5} d^{2} + 20 \, a b^{3} c^{4} d^{3} + 30 \, a^{2} b^{2} c^{3} d^{4} + 12 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{7} + \frac{7}{6} \,{\left (b^{4} c^{6} d + 12 \, a b^{3} c^{5} d^{2} + 30 \, a^{2} b^{2} c^{4} d^{3} + 20 \, a^{3} b c^{3} d^{4} + 3 \, a^{4} c^{2} d^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{7} + 28 \, a b^{3} c^{6} d + 126 \, a^{2} b^{2} c^{5} d^{2} + 140 \, a^{3} b c^{4} d^{3} + 35 \, a^{4} c^{3} d^{4}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} c^{7} + 42 \, a^{2} b^{2} c^{6} d + 84 \, a^{3} b c^{5} d^{2} + 35 \, a^{4} c^{4} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} c^{7} + 28 \, a^{3} b c^{6} d + 21 \, a^{4} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c^{7} + 7 \, a^{4} c^{6} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/12*b^4*d^7*x^12 + a^4*c^7*x + 1/11*(7*b^4*c*d^6 + 4*a*b^3*d^7)*x^11 + 1/10*(21*b^4*c^2*d^5 + 28*a*b^3*c*d^6
+ 6*a^2*b^2*d^7)*x^10 + 1/9*(35*b^4*c^3*d^4 + 84*a*b^3*c^2*d^5 + 42*a^2*b^2*c*d^6 + 4*a^3*b*d^7)*x^9 + 1/8*(35
*b^4*c^4*d^3 + 140*a*b^3*c^3*d^4 + 126*a^2*b^2*c^2*d^5 + 28*a^3*b*c*d^6 + a^4*d^7)*x^8 + (3*b^4*c^5*d^2 + 20*a
*b^3*c^4*d^3 + 30*a^2*b^2*c^3*d^4 + 12*a^3*b*c^2*d^5 + a^4*c*d^6)*x^7 + 7/6*(b^4*c^6*d + 12*a*b^3*c^5*d^2 + 30
*a^2*b^2*c^4*d^3 + 20*a^3*b*c^3*d^4 + 3*a^4*c^2*d^5)*x^6 + 1/5*(b^4*c^7 + 28*a*b^3*c^6*d + 126*a^2*b^2*c^5*d^2
+ 140*a^3*b*c^4*d^3 + 35*a^4*c^3*d^4)*x^5 + 1/4*(4*a*b^3*c^7 + 42*a^2*b^2*c^6*d + 84*a^3*b*c^5*d^2 + 35*a^4*c
^4*d^3)*x^4 + 1/3*(6*a^2*b^2*c^7 + 28*a^3*b*c^6*d + 21*a^4*c^5*d^2)*x^3 + 1/2*(4*a^3*b*c^7 + 7*a^4*c^6*d)*x^2

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Fricas [B]  time = 1.87938, size = 1184, normalized size = 9.95 \begin{align*} \frac{1}{12} x^{12} d^{7} b^{4} + \frac{7}{11} x^{11} d^{6} c b^{4} + \frac{4}{11} x^{11} d^{7} b^{3} a + \frac{21}{10} x^{10} d^{5} c^{2} b^{4} + \frac{14}{5} x^{10} d^{6} c b^{3} a + \frac{3}{5} x^{10} d^{7} b^{2} a^{2} + \frac{35}{9} x^{9} d^{4} c^{3} b^{4} + \frac{28}{3} x^{9} d^{5} c^{2} b^{3} a + \frac{14}{3} x^{9} d^{6} c b^{2} a^{2} + \frac{4}{9} x^{9} d^{7} b a^{3} + \frac{35}{8} x^{8} d^{3} c^{4} b^{4} + \frac{35}{2} x^{8} d^{4} c^{3} b^{3} a + \frac{63}{4} x^{8} d^{5} c^{2} b^{2} a^{2} + \frac{7}{2} x^{8} d^{6} c b a^{3} + \frac{1}{8} x^{8} d^{7} a^{4} + 3 x^{7} d^{2} c^{5} b^{4} + 20 x^{7} d^{3} c^{4} b^{3} a + 30 x^{7} d^{4} c^{3} b^{2} a^{2} + 12 x^{7} d^{5} c^{2} b a^{3} + x^{7} d^{6} c a^{4} + \frac{7}{6} x^{6} d c^{6} b^{4} + 14 x^{6} d^{2} c^{5} b^{3} a + 35 x^{6} d^{3} c^{4} b^{2} a^{2} + \frac{70}{3} x^{6} d^{4} c^{3} b a^{3} + \frac{7}{2} x^{6} d^{5} c^{2} a^{4} + \frac{1}{5} x^{5} c^{7} b^{4} + \frac{28}{5} x^{5} d c^{6} b^{3} a + \frac{126}{5} x^{5} d^{2} c^{5} b^{2} a^{2} + 28 x^{5} d^{3} c^{4} b a^{3} + 7 x^{5} d^{4} c^{3} a^{4} + x^{4} c^{7} b^{3} a + \frac{21}{2} x^{4} d c^{6} b^{2} a^{2} + 21 x^{4} d^{2} c^{5} b a^{3} + \frac{35}{4} x^{4} d^{3} c^{4} a^{4} + 2 x^{3} c^{7} b^{2} a^{2} + \frac{28}{3} x^{3} d c^{6} b a^{3} + 7 x^{3} d^{2} c^{5} a^{4} + 2 x^{2} c^{7} b a^{3} + \frac{7}{2} x^{2} d c^{6} a^{4} + x c^{7} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/12*x^12*d^7*b^4 + 7/11*x^11*d^6*c*b^4 + 4/11*x^11*d^7*b^3*a + 21/10*x^10*d^5*c^2*b^4 + 14/5*x^10*d^6*c*b^3*a
+ 3/5*x^10*d^7*b^2*a^2 + 35/9*x^9*d^4*c^3*b^4 + 28/3*x^9*d^5*c^2*b^3*a + 14/3*x^9*d^6*c*b^2*a^2 + 4/9*x^9*d^7
*b*a^3 + 35/8*x^8*d^3*c^4*b^4 + 35/2*x^8*d^4*c^3*b^3*a + 63/4*x^8*d^5*c^2*b^2*a^2 + 7/2*x^8*d^6*c*b*a^3 + 1/8*
x^8*d^7*a^4 + 3*x^7*d^2*c^5*b^4 + 20*x^7*d^3*c^4*b^3*a + 30*x^7*d^4*c^3*b^2*a^2 + 12*x^7*d^5*c^2*b*a^3 + x^7*d
^6*c*a^4 + 7/6*x^6*d*c^6*b^4 + 14*x^6*d^2*c^5*b^3*a + 35*x^6*d^3*c^4*b^2*a^2 + 70/3*x^6*d^4*c^3*b*a^3 + 7/2*x^
6*d^5*c^2*a^4 + 1/5*x^5*c^7*b^4 + 28/5*x^5*d*c^6*b^3*a + 126/5*x^5*d^2*c^5*b^2*a^2 + 28*x^5*d^3*c^4*b*a^3 + 7*
x^5*d^4*c^3*a^4 + x^4*c^7*b^3*a + 21/2*x^4*d*c^6*b^2*a^2 + 21*x^4*d^2*c^5*b*a^3 + 35/4*x^4*d^3*c^4*a^4 + 2*x^3
*c^7*b^2*a^2 + 28/3*x^3*d*c^6*b*a^3 + 7*x^3*d^2*c^5*a^4 + 2*x^2*c^7*b*a^3 + 7/2*x^2*d*c^6*a^4 + x*c^7*a^4

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Sympy [B]  time = 0.129943, size = 549, normalized size = 4.61 \begin{align*} a^{4} c^{7} x + \frac{b^{4} d^{7} x^{12}}{12} + x^{11} \left (\frac{4 a b^{3} d^{7}}{11} + \frac{7 b^{4} c d^{6}}{11}\right ) + x^{10} \left (\frac{3 a^{2} b^{2} d^{7}}{5} + \frac{14 a b^{3} c d^{6}}{5} + \frac{21 b^{4} c^{2} d^{5}}{10}\right ) + x^{9} \left (\frac{4 a^{3} b d^{7}}{9} + \frac{14 a^{2} b^{2} c d^{6}}{3} + \frac{28 a b^{3} c^{2} d^{5}}{3} + \frac{35 b^{4} c^{3} d^{4}}{9}\right ) + x^{8} \left (\frac{a^{4} d^{7}}{8} + \frac{7 a^{3} b c d^{6}}{2} + \frac{63 a^{2} b^{2} c^{2} d^{5}}{4} + \frac{35 a b^{3} c^{3} d^{4}}{2} + \frac{35 b^{4} c^{4} d^{3}}{8}\right ) + x^{7} \left (a^{4} c d^{6} + 12 a^{3} b c^{2} d^{5} + 30 a^{2} b^{2} c^{3} d^{4} + 20 a b^{3} c^{4} d^{3} + 3 b^{4} c^{5} d^{2}\right ) + x^{6} \left (\frac{7 a^{4} c^{2} d^{5}}{2} + \frac{70 a^{3} b c^{3} d^{4}}{3} + 35 a^{2} b^{2} c^{4} d^{3} + 14 a b^{3} c^{5} d^{2} + \frac{7 b^{4} c^{6} d}{6}\right ) + x^{5} \left (7 a^{4} c^{3} d^{4} + 28 a^{3} b c^{4} d^{3} + \frac{126 a^{2} b^{2} c^{5} d^{2}}{5} + \frac{28 a b^{3} c^{6} d}{5} + \frac{b^{4} c^{7}}{5}\right ) + x^{4} \left (\frac{35 a^{4} c^{4} d^{3}}{4} + 21 a^{3} b c^{5} d^{2} + \frac{21 a^{2} b^{2} c^{6} d}{2} + a b^{3} c^{7}\right ) + x^{3} \left (7 a^{4} c^{5} d^{2} + \frac{28 a^{3} b c^{6} d}{3} + 2 a^{2} b^{2} c^{7}\right ) + x^{2} \left (\frac{7 a^{4} c^{6} d}{2} + 2 a^{3} b c^{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c)**7,x)

[Out]

a**4*c**7*x + b**4*d**7*x**12/12 + x**11*(4*a*b**3*d**7/11 + 7*b**4*c*d**6/11) + x**10*(3*a**2*b**2*d**7/5 + 1
4*a*b**3*c*d**6/5 + 21*b**4*c**2*d**5/10) + x**9*(4*a**3*b*d**7/9 + 14*a**2*b**2*c*d**6/3 + 28*a*b**3*c**2*d**
5/3 + 35*b**4*c**3*d**4/9) + x**8*(a**4*d**7/8 + 7*a**3*b*c*d**6/2 + 63*a**2*b**2*c**2*d**5/4 + 35*a*b**3*c**3
*d**4/2 + 35*b**4*c**4*d**3/8) + x**7*(a**4*c*d**6 + 12*a**3*b*c**2*d**5 + 30*a**2*b**2*c**3*d**4 + 20*a*b**3*
c**4*d**3 + 3*b**4*c**5*d**2) + x**6*(7*a**4*c**2*d**5/2 + 70*a**3*b*c**3*d**4/3 + 35*a**2*b**2*c**4*d**3 + 14
*a*b**3*c**5*d**2 + 7*b**4*c**6*d/6) + x**5*(7*a**4*c**3*d**4 + 28*a**3*b*c**4*d**3 + 126*a**2*b**2*c**5*d**2/
5 + 28*a*b**3*c**6*d/5 + b**4*c**7/5) + x**4*(35*a**4*c**4*d**3/4 + 21*a**3*b*c**5*d**2 + 21*a**2*b**2*c**6*d/
2 + a*b**3*c**7) + x**3*(7*a**4*c**5*d**2 + 28*a**3*b*c**6*d/3 + 2*a**2*b**2*c**7) + x**2*(7*a**4*c**6*d/2 + 2
*a**3*b*c**7)

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Giac [B]  time = 1.07816, size = 737, normalized size = 6.19 \begin{align*} \frac{1}{12} \, b^{4} d^{7} x^{12} + \frac{7}{11} \, b^{4} c d^{6} x^{11} + \frac{4}{11} \, a b^{3} d^{7} x^{11} + \frac{21}{10} \, b^{4} c^{2} d^{5} x^{10} + \frac{14}{5} \, a b^{3} c d^{6} x^{10} + \frac{3}{5} \, a^{2} b^{2} d^{7} x^{10} + \frac{35}{9} \, b^{4} c^{3} d^{4} x^{9} + \frac{28}{3} \, a b^{3} c^{2} d^{5} x^{9} + \frac{14}{3} \, a^{2} b^{2} c d^{6} x^{9} + \frac{4}{9} \, a^{3} b d^{7} x^{9} + \frac{35}{8} \, b^{4} c^{4} d^{3} x^{8} + \frac{35}{2} \, a b^{3} c^{3} d^{4} x^{8} + \frac{63}{4} \, a^{2} b^{2} c^{2} d^{5} x^{8} + \frac{7}{2} \, a^{3} b c d^{6} x^{8} + \frac{1}{8} \, a^{4} d^{7} x^{8} + 3 \, b^{4} c^{5} d^{2} x^{7} + 20 \, a b^{3} c^{4} d^{3} x^{7} + 30 \, a^{2} b^{2} c^{3} d^{4} x^{7} + 12 \, a^{3} b c^{2} d^{5} x^{7} + a^{4} c d^{6} x^{7} + \frac{7}{6} \, b^{4} c^{6} d x^{6} + 14 \, a b^{3} c^{5} d^{2} x^{6} + 35 \, a^{2} b^{2} c^{4} d^{3} x^{6} + \frac{70}{3} \, a^{3} b c^{3} d^{4} x^{6} + \frac{7}{2} \, a^{4} c^{2} d^{5} x^{6} + \frac{1}{5} \, b^{4} c^{7} x^{5} + \frac{28}{5} \, a b^{3} c^{6} d x^{5} + \frac{126}{5} \, a^{2} b^{2} c^{5} d^{2} x^{5} + 28 \, a^{3} b c^{4} d^{3} x^{5} + 7 \, a^{4} c^{3} d^{4} x^{5} + a b^{3} c^{7} x^{4} + \frac{21}{2} \, a^{2} b^{2} c^{6} d x^{4} + 21 \, a^{3} b c^{5} d^{2} x^{4} + \frac{35}{4} \, a^{4} c^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{7} x^{3} + \frac{28}{3} \, a^{3} b c^{6} d x^{3} + 7 \, a^{4} c^{5} d^{2} x^{3} + 2 \, a^{3} b c^{7} x^{2} + \frac{7}{2} \, a^{4} c^{6} d x^{2} + a^{4} c^{7} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^7,x, algorithm="giac")

[Out]

1/12*b^4*d^7*x^12 + 7/11*b^4*c*d^6*x^11 + 4/11*a*b^3*d^7*x^11 + 21/10*b^4*c^2*d^5*x^10 + 14/5*a*b^3*c*d^6*x^10
+ 3/5*a^2*b^2*d^7*x^10 + 35/9*b^4*c^3*d^4*x^9 + 28/3*a*b^3*c^2*d^5*x^9 + 14/3*a^2*b^2*c*d^6*x^9 + 4/9*a^3*b*d
^7*x^9 + 35/8*b^4*c^4*d^3*x^8 + 35/2*a*b^3*c^3*d^4*x^8 + 63/4*a^2*b^2*c^2*d^5*x^8 + 7/2*a^3*b*c*d^6*x^8 + 1/8*
a^4*d^7*x^8 + 3*b^4*c^5*d^2*x^7 + 20*a*b^3*c^4*d^3*x^7 + 30*a^2*b^2*c^3*d^4*x^7 + 12*a^3*b*c^2*d^5*x^7 + a^4*c
*d^6*x^7 + 7/6*b^4*c^6*d*x^6 + 14*a*b^3*c^5*d^2*x^6 + 35*a^2*b^2*c^4*d^3*x^6 + 70/3*a^3*b*c^3*d^4*x^6 + 7/2*a^
4*c^2*d^5*x^6 + 1/5*b^4*c^7*x^5 + 28/5*a*b^3*c^6*d*x^5 + 126/5*a^2*b^2*c^5*d^2*x^5 + 28*a^3*b*c^4*d^3*x^5 + 7*
a^4*c^3*d^4*x^5 + a*b^3*c^7*x^4 + 21/2*a^2*b^2*c^6*d*x^4 + 21*a^3*b*c^5*d^2*x^4 + 35/4*a^4*c^4*d^3*x^4 + 2*a^2
*b^2*c^7*x^3 + 28/3*a^3*b*c^6*d*x^3 + 7*a^4*c^5*d^2*x^3 + 2*a^3*b*c^7*x^2 + 7/2*a^4*c^6*d*x^2 + a^4*c^7*x