∫ (b + 2cx + 3dx2)(a + bx + cx2 + dx3)7 dx [192]

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1602} \begin {gather*} \frac {1}{8} \left (a+b x+c x^2+d x^3\right )^8 \end {gather*}

Int[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +

1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,

q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol

\begin {align*} \int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^7 \, dx &=\frac {1}{8} \left (a+b x+c x^2+d x^3\right )^8\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(21)=42\).
time = 0.11, size = 143, normalized size = 6.81 \begin {gather*} \frac {1}{8} x (b+x (c+d x)) \left (8 a^7+28 a^6 x (b+x (c+d x))+56 a^5 x^2 (b+x (c+d x))^2+70 a^4 x^3 (b+x (c+d x))^3+56 a^3 x^4 (b+x (c+d x))^4+28 a^2 x^5 (b+x (c+d x))^5+8 a x^6 (b+x (c+d x))^6+x^7 (b+x (c+d x))^7\right ) \end {gather*}

Integrate[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x]

(x*(b + x*(c + d*x))*(8*a^7 + 28*a^6*x*(b + x*(c + d*x)) + 56*a^5*x^2*(b + x*(c + d*x))^2 + 70*a^4*x^3*(b + x*

(c + d*x))^3 + 56*a^3*x^4*(b + x*(c + d*x))^4 + 28*a^2*x^5*(b + x*(c + d*x))^5 + 8*a*x^6*(b + x*(c + d*x))^6 +

________________________________________________________________________________________


method	result	size

default	\(\frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{8}}{8}\)	\(20\)
norman	\(\text {Expression too large to display}\)	\(1579\)
gosper	\(\text {Expression too large to display}\)	\(1957\)
risch	\(\text {Expression too large to display}\)	\(1962\)

int((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x,method=_RETURNVERBOSE)

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x + a\right )}^{8} \end {gather*}

integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="maxima")

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1528 vs. \(2 (19) = 38\).
time = 0.38, size = 1528, normalized size = 72.76 \begin {gather*} \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {1}{2} \, {\left (7 \, c^{2} d^{6} + 2 \, b d^{7}\right )} x^{22} + {\left (7 \, c^{3} d^{5} + 7 \, b c d^{6} + a d^{7}\right )} x^{21} + \frac {7}{4} \, {\left (5 \, c^{4} d^{4} + 12 \, b c^{2} d^{5} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{6}\right )} x^{20} + 7 \, {\left (c^{5} d^{3} + 5 \, b c^{3} d^{4} + a b d^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{19} + \frac {7}{2} \, {\left (c^{6} d^{2} + 10 \, b c^{4} d^{3} + a^{2} d^{6} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} + 5 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4}\right )} x^{18} + {\left (c^{7} d + 21 \, b c^{5} d^{2} + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + 35 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + 35 \, {\left (2 \, b^{2} c^{3} + a c^{4}\right )} d^{3}\right )} x^{17} + \frac {1}{8} \, {\left (c^{8} + 56 \, b c^{6} d + 168 \, a^{2} b d^{5} + 70 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 560 \, {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{3} + 84 \, {\left (5 \, b^{2} c^{4} + 2 \, a c^{5}\right )} d^{2}\right )} x^{16} + {\left (b c^{7} + 7 \, a^{3} d^{5} + 35 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 35 \, {\left (b^{4} c + 6 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} + 35 \, {\left (2 \, b^{3} c^{3} + 3 \, a b c^{4}\right )} d^{2} + 7 \, {\left (3 \, b^{2} c^{5} + a c^{6}\right )} d\right )} x^{15} + \frac {1}{2} \, {\left (7 \, b^{2} c^{6} + 2 \, a c^{7} + 35 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4} + 14 \, {\left (b^{5} + 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} d^{3} + 105 \, {\left (b^{4} c^{2} + 4 \, a b^{2} c^{3} + a^{2} c^{4}\right )} d^{2} + 14 \, {\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} d\right )} x^{14} + 7 \, {\left (b^{3} c^{5} + a b c^{6} + 5 \, a^{3} b d^{4} + 5 \, {\left (a b^{4} + 6 \, a^{2} b^{2} c + 2 \, a^{3} c^{2}\right )} d^{3} + 3 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 10 \, a^{2} b c^{3}\right )} d^{2} + {\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} d\right )} x^{13} + \frac {7}{4} \, {\left (5 \, b^{4} c^{4} + 12 \, a b^{2} c^{5} + 2 \, a^{2} c^{6} + 5 \, a^{4} d^{4} + 40 \, {\left (a^{2} b^{3} + 2 \, a^{3} b c\right )} d^{3} + 2 \, {\left (b^{6} + 30 \, a b^{4} c + 90 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3}\right )} d^{2} + 4 \, {\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} d\right )} x^{12} + 7 \, {\left (b^{5} c^{3} + 5 \, a b^{3} c^{4} + 3 \, a^{2} b c^{5} + 5 \, {\left (2 \, a^{3} b^{2} + a^{4} c\right )} d^{3} + 3 \, {\left (a b^{5} + 10 \, a^{2} b^{3} c + 10 \, a^{3} b c^{2}\right )} d^{2} + {\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} d\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} c^{2} + 70 \, a b^{4} c^{3} + 105 \, a^{2} b^{2} c^{4} + 14 \, a^{3} c^{5} + 70 \, a^{4} b d^{3} + 105 \, {\left (a^{2} b^{4} + 4 \, a^{3} b^{2} c + a^{4} c^{2}\right )} d^{2} + 2 \, {\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} d\right )} x^{10} + {\left (b^{7} c + 21 \, a b^{5} c^{2} + 70 \, a^{2} b^{3} c^{3} + 35 \, a^{3} b c^{4} + 7 \, a^{5} d^{3} + 35 \, {\left (2 \, a^{3} b^{3} + 3 \, a^{4} b c\right )} d^{2} + 7 \, {\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} d\right )} x^{9} + a^{7} b x + \frac {1}{8} \, {\left (b^{8} + 56 \, a b^{6} c + 420 \, a^{2} b^{4} c^{2} + 560 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4} + 84 \, {\left (5 \, a^{4} b^{2} + 2 \, a^{5} c\right )} d^{2} + 56 \, {\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} d\right )} x^{8} + {\left (a b^{7} + 21 \, a^{2} b^{5} c + 70 \, a^{3} b^{3} c^{2} + 35 \, a^{4} b c^{3} + 21 \, a^{5} b d^{2} + 7 \, {\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} d\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{6} + 10 \, a^{3} b^{4} c + 15 \, a^{4} b^{2} c^{2} + 2 \, a^{5} c^{3} + a^{6} d^{2} + 2 \, {\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} d\right )} x^{6} + 7 \, {\left (a^{3} b^{5} + 5 \, a^{4} b^{3} c + 3 \, a^{5} b c^{2} + {\left (3 \, a^{5} b^{2} + a^{6} c\right )} d\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{4} + 12 \, a^{5} b^{2} c + 2 \, a^{6} c^{2} + 4 \, a^{6} b d\right )} x^{4} + {\left (7 \, a^{5} b^{3} + 7 \, a^{6} b c + a^{7} d\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b^{2} + 2 \, a^{7} c\right )} x^{2} \end {gather*}

integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="fricas")

1/8*d^8*x^24 + c*d^7*x^23 + 1/2*(7*c^2*d^6 + 2*b*d^7)*x^22 + (7*c^3*d^5 + 7*b*c*d^6 + a*d^7)*x^21 + 7/4*(5*c^4

*d^4 + 12*b*c^2*d^5 + 2*(b^2 + 2*a*c)*d^6)*x^20 + 7*(c^5*d^3 + 5*b*c^3*d^4 + a*b*d^6 + 3*(b^2*c + a*c^2)*d^5)*

x^19 + 7/2*(c^6*d^2 + 10*b*c^4*d^3 + a^2*d^6 + 2*(b^3 + 6*a*b*c)*d^5 + 5*(3*b^2*c^2 + 2*a*c^3)*d^4)*x^18 + (c^

7*d + 21*b*c^5*d^2 + 21*(a*b^2 + a^2*c)*d^5 + 35*(b^3*c + 3*a*b*c^2)*d^4 + 35*(2*b^2*c^3 + a*c^4)*d^3)*x^17 +

1/8*(c^8 + 56*b*c^6*d + 168*a^2*b*d^5 + 70*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 560*(b^3*c^2 + 2*a*b*c^3)*d^3

+ 84*(5*b^2*c^4 + 2*a*c^5)*d^2)*x^16 + (b*c^7 + 7*a^3*d^5 + 35*(a*b^3 + 3*a^2*b*c)*d^4 + 35*(b^4*c + 6*a*b^2*c

^2 + 2*a^2*c^3)*d^3 + 35*(2*b^3*c^3 + 3*a*b*c^4)*d^2 + 7*(3*b^2*c^5 + a*c^6)*d)*x^15 + 1/2*(7*b^2*c^6 + 2*a*c^

7 + 35*(3*a^2*b^2 + 2*a^3*c)*d^4 + 14*(b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^3 + 105*(b^4*c^2 + 4*a*b^2*c^3 + a^2

*c^4)*d^2 + 14*(5*b^3*c^4 + 6*a*b*c^5)*d)*x^14 + 7*(b^3*c^5 + a*b*c^6 + 5*a^3*b*d^4 + 5*(a*b^4 + 6*a^2*b^2*c +

2*a^3*c^2)*d^3 + 3*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^2 + (5*b^4*c^3 + 15*a*b^2*c^4 + 3*a^2*c^5)*d)*x^13

+ 7/4*(5*b^4*c^4 + 12*a*b^2*c^5 + 2*a^2*c^6 + 5*a^4*d^4 + 40*(a^2*b^3 + 2*a^3*b*c)*d^3 + 2*(b^6 + 30*a*b^4*c

+ 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^2 + 4*(3*b^5*c^2 + 20*a*b^3*c^3 + 15*a^2*b*c^4)*d)*x^12 + 7*(b^5*c^3 + 5*a*b^

3*c^4 + 3*a^2*b*c^5 + 5*(2*a^3*b^2 + a^4*c)*d^3 + 3*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^2 + (b^6*c + 15*a*

b^4*c^2 + 30*a^2*b^2*c^3 + 5*a^3*c^4)*d)*x^11 + 1/2*(7*b^6*c^2 + 70*a*b^4*c^3 + 105*a^2*b^2*c^4 + 14*a^3*c^5 +

70*a^4*b*d^3 + 105*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^2 + 2*(b^7 + 42*a*b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*

c^3)*d)*x^10 + (b^7*c + 21*a*b^5*c^2 + 70*a^2*b^3*c^3 + 35*a^3*b*c^4 + 7*a^5*d^3 + 35*(2*a^3*b^3 + 3*a^4*b*c)*

d^2 + 7*(a*b^6 + 15*a^2*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*d)*x^9 + a^7*b*x + 1/8*(b^8 + 56*a*b^6*c + 420*a^2

*b^4*c^2 + 560*a^3*b^2*c^3 + 70*a^4*c^4 + 84*(5*a^4*b^2 + 2*a^5*c)*d^2 + 56*(3*a^2*b^5 + 20*a^3*b^3*c + 15*a^4

*b*c^2)*d)*x^8 + (a*b^7 + 21*a^2*b^5*c + 70*a^3*b^3*c^2 + 35*a^4*b*c^3 + 21*a^5*b*d^2 + 7*(5*a^3*b^4 + 15*a^4*

b^2*c + 3*a^5*c^2)*d)*x^7 + 7/2*(a^2*b^6 + 10*a^3*b^4*c + 15*a^4*b^2*c^2 + 2*a^5*c^3 + a^6*d^2 + 2*(5*a^4*b^3

+ 6*a^5*b*c)*d)*x^6 + 7*(a^3*b^5 + 5*a^4*b^3*c + 3*a^5*b*c^2 + (3*a^5*b^2 + a^6*c)*d)*x^5 + 7/4*(5*a^4*b^4 + 1

2*a^5*b^2*c + 2*a^6*c^2 + 4*a^6*b*d)*x^4 + (7*a^5*b^3 + 7*a^6*b*c + a^7*d)*x^3 + 1/2*(7*a^6*b^2 + 2*a^7*c)*x^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1771 vs. \(2 (17) = 34\).
time = 0.17, size = 1771, normalized size = 84.33 \begin {gather*} a^{7} b x + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} + x^{22} \left (b d^{7} + \frac {7 c^{2} d^{6}}{2}\right ) + x^{21} \left (a d^{7} + 7 b c d^{6} + 7 c^{3} d^{5}\right ) + x^{20} \cdot \left (7 a c d^{6} + \frac {7 b^{2} d^{6}}{2} + 21 b c^{2} d^{5} + \frac {35 c^{4} d^{4}}{4}\right ) + x^{19} \cdot \left (7 a b d^{6} + 21 a c^{2} d^{5} + 21 b^{2} c d^{5} + 35 b c^{3} d^{4} + 7 c^{5} d^{3}\right ) + x^{18} \cdot \left (\frac {7 a^{2} d^{6}}{2} + 42 a b c d^{5} + 35 a c^{3} d^{4} + 7 b^{3} d^{5} + \frac {105 b^{2} c^{2} d^{4}}{2} + 35 b c^{4} d^{3} + \frac {7 c^{6} d^{2}}{2}\right ) + x^{17} \cdot \left (21 a^{2} c d^{5} + 21 a b^{2} d^{5} + 105 a b c^{2} d^{4} + 35 a c^{4} d^{3} + 35 b^{3} c d^{4} + 70 b^{2} c^{3} d^{3} + 21 b c^{5} d^{2} + c^{7} d\right ) + x^{16} \cdot \left (21 a^{2} b d^{5} + \frac {105 a^{2} c^{2} d^{4}}{2} + 105 a b^{2} c d^{4} + 140 a b c^{3} d^{3} + 21 a c^{5} d^{2} + \frac {35 b^{4} d^{4}}{4} + 70 b^{3} c^{2} d^{3} + \frac {105 b^{2} c^{4} d^{2}}{2} + 7 b c^{6} d + \frac {c^{8}}{8}\right ) + x^{15} \cdot \left (7 a^{3} d^{5} + 105 a^{2} b c d^{4} + 70 a^{2} c^{3} d^{3} + 35 a b^{3} d^{4} + 210 a b^{2} c^{2} d^{3} + 105 a b c^{4} d^{2} + 7 a c^{6} d + 35 b^{4} c d^{3} + 70 b^{3} c^{3} d^{2} + 21 b^{2} c^{5} d + b c^{7}\right ) + x^{14} \cdot \left (35 a^{3} c d^{4} + \frac {105 a^{2} b^{2} d^{4}}{2} + 210 a^{2} b c^{2} d^{3} + \frac {105 a^{2} c^{4} d^{2}}{2} + 140 a b^{3} c d^{3} + 210 a b^{2} c^{3} d^{2} + 42 a b c^{5} d + a c^{7} + 7 b^{5} d^{3} + \frac {105 b^{4} c^{2} d^{2}}{2} + 35 b^{3} c^{4} d + \frac {7 b^{2} c^{6}}{2}\right ) + x^{13} \cdot \left (35 a^{3} b d^{4} + 70 a^{3} c^{2} d^{3} + 210 a^{2} b^{2} c d^{3} + 210 a^{2} b c^{3} d^{2} + 21 a^{2} c^{5} d + 35 a b^{4} d^{3} + 210 a b^{3} c^{2} d^{2} + 105 a b^{2} c^{4} d + 7 a b c^{6} + 21 b^{5} c d^{2} + 35 b^{4} c^{3} d + 7 b^{3} c^{5}\right ) + x^{12} \cdot \left (\frac {35 a^{4} d^{4}}{4} + 140 a^{3} b c d^{3} + 70 a^{3} c^{3} d^{2} + 70 a^{2} b^{3} d^{3} + 315 a^{2} b^{2} c^{2} d^{2} + 105 a^{2} b c^{4} d + \frac {7 a^{2} c^{6}}{2} + 105 a b^{4} c d^{2} + 140 a b^{3} c^{3} d + 21 a b^{2} c^{5} + \frac {7 b^{6} d^{2}}{2} + 21 b^{5} c^{2} d + \frac {35 b^{4} c^{4}}{4}\right ) + x^{11} \cdot \left (35 a^{4} c d^{3} + 70 a^{3} b^{2} d^{3} + 210 a^{3} b c^{2} d^{2} + 35 a^{3} c^{4} d + 210 a^{2} b^{3} c d^{2} + 210 a^{2} b^{2} c^{3} d + 21 a^{2} b c^{5} + 21 a b^{5} d^{2} + 105 a b^{4} c^{2} d + 35 a b^{3} c^{4} + 7 b^{6} c d + 7 b^{5} c^{3}\right ) + x^{10} \cdot \left (35 a^{4} b d^{3} + \frac {105 a^{4} c^{2} d^{2}}{2} + 210 a^{3} b^{2} c d^{2} + 140 a^{3} b c^{3} d + 7 a^{3} c^{5} + \frac {105 a^{2} b^{4} d^{2}}{2} + 210 a^{2} b^{3} c^{2} d + \frac {105 a^{2} b^{2} c^{4}}{2} + 42 a b^{5} c d + 35 a b^{4} c^{3} + b^{7} d + \frac {7 b^{6} c^{2}}{2}\right ) + x^{9} \cdot \left (7 a^{5} d^{3} + 105 a^{4} b c d^{2} + 35 a^{4} c^{3} d + 70 a^{3} b^{3} d^{2} + 210 a^{3} b^{2} c^{2} d + 35 a^{3} b c^{4} + 105 a^{2} b^{4} c d + 70 a^{2} b^{3} c^{3} + 7 a b^{6} d + 21 a b^{5} c^{2} + b^{7} c\right ) + x^{8} \cdot \left (21 a^{5} c d^{2} + \frac {105 a^{4} b^{2} d^{2}}{2} + 105 a^{4} b c^{2} d + \frac {35 a^{4} c^{4}}{4} + 140 a^{3} b^{3} c d + 70 a^{3} b^{2} c^{3} + 21 a^{2} b^{5} d + \frac {105 a^{2} b^{4} c^{2}}{2} + 7 a b^{6} c + \frac {b^{8}}{8}\right ) + x^{7} \cdot \left (21 a^{5} b d^{2} + 21 a^{5} c^{2} d + 105 a^{4} b^{2} c d + 35 a^{4} b c^{3} + 35 a^{3} b^{4} d + 70 a^{3} b^{3} c^{2} + 21 a^{2} b^{5} c + a b^{7}\right ) + x^{6} \cdot \left (\frac {7 a^{6} d^{2}}{2} + 42 a^{5} b c d + 7 a^{5} c^{3} + 35 a^{4} b^{3} d + \frac {105 a^{4} b^{2} c^{2}}{2} + 35 a^{3} b^{4} c + \frac {7 a^{2} b^{6}}{2}\right ) + x^{5} \cdot \left (7 a^{6} c d + 21 a^{5} b^{2} d + 21 a^{5} b c^{2} + 35 a^{4} b^{3} c + 7 a^{3} b^{5}\right ) + x^{4} \cdot \left (7 a^{6} b d + \frac {7 a^{6} c^{2}}{2} + 21 a^{5} b^{2} c + \frac {35 a^{4} b^{4}}{4}\right ) + x^{3} \left (a^{7} d + 7 a^{6} b c + 7 a^{5} b^{3}\right ) + x^{2} \left (a^{7} c + \frac {7 a^{6} b^{2}}{2}\right ) \end {gather*}

integrate((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x+a)**7,x)

a**7*b*x + c*d**7*x**23 + d**8*x**24/8 + x**22*(b*d**7 + 7*c**2*d**6/2) + x**21*(a*d**7 + 7*b*c*d**6 + 7*c**3*

d**5) + x**20*(7*a*c*d**6 + 7*b**2*d**6/2 + 21*b*c**2*d**5 + 35*c**4*d**4/4) + x**19*(7*a*b*d**6 + 21*a*c**2*d

**5 + 21*b**2*c*d**5 + 35*b*c**3*d**4 + 7*c**5*d**3) + x**18*(7*a**2*d**6/2 + 42*a*b*c*d**5 + 35*a*c**3*d**4 +

7*b**3*d**5 + 105*b**2*c**2*d**4/2 + 35*b*c**4*d**3 + 7*c**6*d**2/2) + x**17*(21*a**2*c*d**5 + 21*a*b**2*d**5

+ 105*a*b*c**2*d**4 + 35*a*c**4*d**3 + 35*b**3*c*d**4 + 70*b**2*c**3*d**3 + 21*b*c**5*d**2 + c**7*d) + x**16*

(21*a**2*b*d**5 + 105*a**2*c**2*d**4/2 + 105*a*b**2*c*d**4 + 140*a*b*c**3*d**3 + 21*a*c**5*d**2 + 35*b**4*d**4

/4 + 70*b**3*c**2*d**3 + 105*b**2*c**4*d**2/2 + 7*b*c**6*d + c**8/8) + x**15*(7*a**3*d**5 + 105*a**2*b*c*d**4

+ 70*a**2*c**3*d**3 + 35*a*b**3*d**4 + 210*a*b**2*c**2*d**3 + 105*a*b*c**4*d**2 + 7*a*c**6*d + 35*b**4*c*d**3

+ 70*b**3*c**3*d**2 + 21*b**2*c**5*d + b*c**7) + x**14*(35*a**3*c*d**4 + 105*a**2*b**2*d**4/2 + 210*a**2*b*c**

2*d**3 + 105*a**2*c**4*d**2/2 + 140*a*b**3*c*d**3 + 210*a*b**2*c**3*d**2 + 42*a*b*c**5*d + a*c**7 + 7*b**5*d**

3 + 105*b**4*c**2*d**2/2 + 35*b**3*c**4*d + 7*b**2*c**6/2) + x**13*(35*a**3*b*d**4 + 70*a**3*c**2*d**3 + 210*a

**2*b**2*c*d**3 + 210*a**2*b*c**3*d**2 + 21*a**2*c**5*d + 35*a*b**4*d**3 + 210*a*b**3*c**2*d**2 + 105*a*b**2*c

**4*d + 7*a*b*c**6 + 21*b**5*c*d**2 + 35*b**4*c**3*d + 7*b**3*c**5) + x**12*(35*a**4*d**4/4 + 140*a**3*b*c*d**

3 + 70*a**3*c**3*d**2 + 70*a**2*b**3*d**3 + 315*a**2*b**2*c**2*d**2 + 105*a**2*b*c**4*d + 7*a**2*c**6/2 + 105*

a*b**4*c*d**2 + 140*a*b**3*c**3*d + 21*a*b**2*c**5 + 7*b**6*d**2/2 + 21*b**5*c**2*d + 35*b**4*c**4/4) + x**11*

(35*a**4*c*d**3 + 70*a**3*b**2*d**3 + 210*a**3*b*c**2*d**2 + 35*a**3*c**4*d + 210*a**2*b**3*c*d**2 + 210*a**2*

b**2*c**3*d + 21*a**2*b*c**5 + 21*a*b**5*d**2 + 105*a*b**4*c**2*d + 35*a*b**3*c**4 + 7*b**6*c*d + 7*b**5*c**3)

+ x**10*(35*a**4*b*d**3 + 105*a**4*c**2*d**2/2 + 210*a**3*b**2*c*d**2 + 140*a**3*b*c**3*d + 7*a**3*c**5 + 105

*a**2*b**4*d**2/2 + 210*a**2*b**3*c**2*d + 105*a**2*b**2*c**4/2 + 42*a*b**5*c*d + 35*a*b**4*c**3 + b**7*d + 7*

b**6*c**2/2) + x**9*(7*a**5*d**3 + 105*a**4*b*c*d**2 + 35*a**4*c**3*d + 70*a**3*b**3*d**2 + 210*a**3*b**2*c**2

*d + 35*a**3*b*c**4 + 105*a**2*b**4*c*d + 70*a**2*b**3*c**3 + 7*a*b**6*d + 21*a*b**5*c**2 + b**7*c) + x**8*(21

*a**5*c*d**2 + 105*a**4*b**2*d**2/2 + 105*a**4*b*c**2*d + 35*a**4*c**4/4 + 140*a**3*b**3*c*d + 70*a**3*b**2*c*

*3 + 21*a**2*b**5*d + 105*a**2*b**4*c**2/2 + 7*a*b**6*c + b**8/8) + x**7*(21*a**5*b*d**2 + 21*a**5*c**2*d + 10

5*a**4*b**2*c*d + 35*a**4*b*c**3 + 35*a**3*b**4*d + 70*a**3*b**3*c**2 + 21*a**2*b**5*c + a*b**7) + x**6*(7*a**

6*d**2/2 + 42*a**5*b*c*d + 7*a**5*c**3 + 35*a**4*b**3*d + 105*a**4*b**2*c**2/2 + 35*a**3*b**4*c + 7*a**2*b**6/

2) + x**5*(7*a**6*c*d + 21*a**5*b**2*d + 21*a**5*b*c**2 + 35*a**4*b**3*c + 7*a**3*b**5) + x**4*(7*a**6*b*d + 7

*a**6*c**2/2 + 21*a**5*b**2*c + 35*a**4*b**4/4) + x**3*(a**7*d + 7*a**6*b*c + 7*a**5*b**3) + x**2*(a**7*c + 7*

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (19) = 38\).
time = 3.41, size = 160, normalized size = 7.62 \begin {gather*} \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} + {\left (d x^{3} + c x^{2} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + c x^{2} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + c x^{2} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + c x^{2} + b x\right )}^{2} a^{6} + {\left (d x^{3} + c x^{2} + b x\right )} a^{7} \end {gather*}

integrate((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x+a)^7,x, algorithm="giac")

1/8*(d*x^3 + c*x^2 + b*x)^8 + (d*x^3 + c*x^2 + b*x)^7*a + 7/2*(d*x^3 + c*x^2 + b*x)^6*a^2 + 7*(d*x^3 + c*x^2 +

b*x)^5*a^3 + 35/4*(d*x^3 + c*x^2 + b*x)^4*a^4 + 7*(d*x^3 + c*x^2 + b*x)^3*a^5 + 7/2*(d*x^3 + c*x^2 + b*x)^2*a

________________________________________________________________________________________

Mupad [B]
time = 2.98, size = 1576, normalized size = 75.05 \begin {gather*} x^{12}\,\left (\frac {35\,a^4\,d^4}{4}+140\,a^3\,b\,c\,d^3+70\,a^3\,c^3\,d^2+70\,a^2\,b^3\,d^3+315\,a^2\,b^2\,c^2\,d^2+105\,a^2\,b\,c^4\,d+\frac {7\,a^2\,c^6}{2}+105\,a\,b^4\,c\,d^2+140\,a\,b^3\,c^3\,d+21\,a\,b^2\,c^5+\frac {7\,b^6\,d^2}{2}+21\,b^5\,c^2\,d+\frac {35\,b^4\,c^4}{4}\right )+x^{11}\,\left (35\,a^4\,c\,d^3+70\,a^3\,b^2\,d^3+210\,a^3\,b\,c^2\,d^2+35\,a^3\,c^4\,d+210\,a^2\,b^3\,c\,d^2+210\,a^2\,b^2\,c^3\,d+21\,a^2\,b\,c^5+21\,a\,b^5\,d^2+105\,a\,b^4\,c^2\,d+35\,a\,b^3\,c^4+7\,b^6\,c\,d+7\,b^5\,c^3\right )+x^{13}\,\left (35\,a^3\,b\,d^4+70\,a^3\,c^2\,d^3+210\,a^2\,b^2\,c\,d^3+210\,a^2\,b\,c^3\,d^2+21\,a^2\,c^5\,d+35\,a\,b^4\,d^3+210\,a\,b^3\,c^2\,d^2+105\,a\,b^2\,c^4\,d+7\,a\,b\,c^6+21\,b^5\,c\,d^2+35\,b^4\,c^3\,d+7\,b^3\,c^5\right )+x^5\,\left (7\,d\,a^6\,c+21\,d\,a^5\,b^2+21\,a^5\,b\,c^2+35\,a^4\,b^3\,c+7\,a^3\,b^5\right )+x^{19}\,\left (21\,b^2\,c\,d^5+35\,b\,c^3\,d^4+7\,a\,b\,d^6+7\,c^5\,d^3+21\,a\,c^2\,d^5\right )+x^8\,\left (21\,a^5\,c\,d^2+\frac {105\,a^4\,b^2\,d^2}{2}+105\,a^4\,b\,c^2\,d+\frac {35\,a^4\,c^4}{4}+140\,a^3\,b^3\,c\,d+70\,a^3\,b^2\,c^3+21\,a^2\,b^5\,d+\frac {105\,a^2\,b^4\,c^2}{2}+7\,a\,b^6\,c+\frac {b^8}{8}\right )+x^9\,\left (7\,a^5\,d^3+105\,a^4\,b\,c\,d^2+35\,a^4\,c^3\,d+70\,a^3\,b^3\,d^2+210\,a^3\,b^2\,c^2\,d+35\,a^3\,b\,c^4+105\,a^2\,b^4\,c\,d+70\,a^2\,b^3\,c^3+7\,a\,b^6\,d+21\,a\,b^5\,c^2+b^7\,c\right )+x^{16}\,\left (21\,a^2\,b\,d^5+\frac {105\,a^2\,c^2\,d^4}{2}+105\,a\,b^2\,c\,d^4+140\,a\,b\,c^3\,d^3+21\,a\,c^5\,d^2+\frac {35\,b^4\,d^4}{4}+70\,b^3\,c^2\,d^3+\frac {105\,b^2\,c^4\,d^2}{2}+7\,b\,c^6\,d+\frac {c^8}{8}\right )+x^{10}\,\left (35\,a^4\,b\,d^3+\frac {105\,a^4\,c^2\,d^2}{2}+210\,a^3\,b^2\,c\,d^2+140\,a^3\,b\,c^3\,d+7\,a^3\,c^5+\frac {105\,a^2\,b^4\,d^2}{2}+210\,a^2\,b^3\,c^2\,d+\frac {105\,a^2\,b^2\,c^4}{2}+42\,a\,b^5\,c\,d+35\,a\,b^4\,c^3+b^7\,d+\frac {7\,b^6\,c^2}{2}\right )+x^{15}\,\left (7\,a^3\,d^5+105\,a^2\,b\,c\,d^4+70\,a^2\,c^3\,d^3+35\,a\,b^3\,d^4+210\,a\,b^2\,c^2\,d^3+105\,a\,b\,c^4\,d^2+7\,a\,c^6\,d+35\,b^4\,c\,d^3+70\,b^3\,c^3\,d^2+21\,b^2\,c^5\,d+b\,c^7\right )+x^{14}\,\left (35\,a^3\,c\,d^4+\frac {105\,a^2\,b^2\,d^4}{2}+210\,a^2\,b\,c^2\,d^3+\frac {105\,a^2\,c^4\,d^2}{2}+140\,a\,b^3\,c\,d^3+210\,a\,b^2\,c^3\,d^2+42\,a\,b\,c^5\,d+a\,c^7+7\,b^5\,d^3+\frac {105\,b^4\,c^2\,d^2}{2}+35\,b^3\,c^4\,d+\frac {7\,b^2\,c^6}{2}\right )+x^4\,\left (7\,d\,a^6\,b+\frac {7\,a^6\,c^2}{2}+21\,a^5\,b^2\,c+\frac {35\,a^4\,b^4}{4}\right )+x^{20}\,\left (\frac {7\,b^2\,d^6}{2}+21\,b\,c^2\,d^5+\frac {35\,c^4\,d^4}{4}+7\,a\,c\,d^6\right )+x^6\,\left (\frac {7\,a^6\,d^2}{2}+42\,a^5\,b\,c\,d+7\,a^5\,c^3+35\,a^4\,b^3\,d+\frac {105\,a^4\,b^2\,c^2}{2}+35\,a^3\,b^4\,c+\frac {7\,a^2\,b^6}{2}\right )+x^7\,\left (21\,a^5\,b\,d^2+21\,a^5\,c^2\,d+105\,a^4\,b^2\,c\,d+35\,a^4\,b\,c^3+35\,a^3\,b^4\,d+70\,a^3\,b^3\,c^2+21\,a^2\,b^5\,c+a\,b^7\right )+x^{18}\,\left (\frac {7\,a^2\,d^6}{2}+42\,a\,b\,c\,d^5+35\,a\,c^3\,d^4+7\,b^3\,d^5+\frac {105\,b^2\,c^2\,d^4}{2}+35\,b\,c^4\,d^3+\frac {7\,c^6\,d^2}{2}\right )+x^{17}\,\left (21\,a^2\,c\,d^5+21\,a\,b^2\,d^5+105\,a\,b\,c^2\,d^4+35\,a\,c^4\,d^3+35\,b^3\,c\,d^4+70\,b^2\,c^3\,d^3+21\,b\,c^5\,d^2+c^7\,d\right )+x^3\,\left (d\,a^7+7\,c\,a^6\,b+7\,a^5\,b^3\right )+\frac {d^8\,x^{24}}{8}+x^2\,\left (c\,a^7+\frac {7\,a^6\,b^2}{2}\right )+c\,d^7\,x^{23}+d^5\,x^{21}\,\left (7\,c^3+7\,b\,c\,d+a\,d^2\right )+\frac {d^6\,x^{22}\,\left (7\,c^2+2\,b\,d\right )}{2}+a^7\,b\,x \end {gather*}

int((b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7,x)

x^12*((7*a^2*c^6)/2 + (35*a^4*d^4)/4 + (35*b^4*c^4)/4 + (7*b^6*d^2)/2 + 21*a*b^2*c^5 + 21*b^5*c^2*d + 70*a^2*b

^3*d^3 + 70*a^3*c^3*d^2 + 315*a^2*b^2*c^2*d^2 + 140*a*b^3*c^3*d + 105*a*b^4*c*d^2 + 105*a^2*b*c^4*d + 140*a^3*

b*c*d^3) + x^11*(7*b^5*c^3 + 35*a*b^3*c^4 + 21*a^2*b*c^5 + 21*a*b^5*d^2 + 35*a^3*c^4*d + 35*a^4*c*d^3 + 70*a^3

*b^2*d^3 + 7*b^6*c*d + 210*a^2*b^2*c^3*d + 210*a^2*b^3*c*d^2 + 210*a^3*b*c^2*d^2 + 105*a*b^4*c^2*d) + x^13*(7*

b^3*c^5 + 35*a*b^4*d^3 + 35*a^3*b*d^4 + 21*a^2*c^5*d + 35*b^4*c^3*d + 21*b^5*c*d^2 + 70*a^3*c^2*d^3 + 7*a*b*c^

6 + 210*a*b^3*c^2*d^2 + 210*a^2*b*c^3*d^2 + 210*a^2*b^2*c*d^3 + 105*a*b^2*c^4*d) + x^5*(7*a^3*b^5 + 35*a^4*b^3

*c + 21*a^5*b*c^2 + 21*a^5*b^2*d + 7*a^6*c*d) + x^19*(7*c^5*d^3 + 21*a*c^2*d^5 + 35*b*c^3*d^4 + 21*b^2*c*d^5 +

7*a*b*d^6) + x^8*(b^8/8 + (35*a^4*c^4)/4 + 21*a^2*b^5*d + 21*a^5*c*d^2 + (105*a^2*b^4*c^2)/2 + 70*a^3*b^2*c^3

+ (105*a^4*b^2*d^2)/2 + 7*a*b^6*c + 140*a^3*b^3*c*d + 105*a^4*b*c^2*d) + x^9*(b^7*c + 7*a^5*d^3 + 21*a*b^5*c^

2 + 35*a^3*b*c^4 + 35*a^4*c^3*d + 70*a^2*b^3*c^3 + 70*a^3*b^3*d^2 + 7*a*b^6*d + 210*a^3*b^2*c^2*d + 105*a^2*b^

4*c*d + 105*a^4*b*c*d^2) + x^16*(c^8/8 + (35*b^4*d^4)/4 + 21*a^2*b*d^5 + 21*a*c^5*d^2 + (105*a^2*c^2*d^4)/2 +

(105*b^2*c^4*d^2)/2 + 70*b^3*c^2*d^3 + 7*b*c^6*d + 140*a*b*c^3*d^3 + 105*a*b^2*c*d^4) + x^10*(b^7*d + 7*a^3*c^

5 + (7*b^6*c^2)/2 + 35*a*b^4*c^3 + 35*a^4*b*d^3 + (105*a^2*b^2*c^4)/2 + (105*a^2*b^4*d^2)/2 + (105*a^4*c^2*d^2

)/2 + 210*a^2*b^3*c^2*d + 210*a^3*b^2*c*d^2 + 42*a*b^5*c*d + 140*a^3*b*c^3*d) + x^15*(b*c^7 + 7*a^3*d^5 + 35*a

*b^3*d^4 + 21*b^2*c^5*d + 35*b^4*c*d^3 + 70*a^2*c^3*d^3 + 70*b^3*c^3*d^2 + 7*a*c^6*d + 210*a*b^2*c^2*d^3 + 105

*a*b*c^4*d^2 + 105*a^2*b*c*d^4) + x^14*(a*c^7 + (7*b^2*c^6)/2 + 7*b^5*d^3 + 35*a^3*c*d^4 + 35*b^3*c^4*d + (105

*a^2*b^2*d^4)/2 + (105*a^2*c^4*d^2)/2 + (105*b^4*c^2*d^2)/2 + 210*a*b^2*c^3*d^2 + 210*a^2*b*c^2*d^3 + 42*a*b*c

^5*d + 140*a*b^3*c*d^3) + x^4*((35*a^4*b^4)/4 + (7*a^6*c^2)/2 + 21*a^5*b^2*c + 7*a^6*b*d) + x^20*((7*b^2*d^6)/

2 + (35*c^4*d^4)/4 + 21*b*c^2*d^5 + 7*a*c*d^6) + x^6*((7*a^2*b^6)/2 + 7*a^5*c^3 + (7*a^6*d^2)/2 + 35*a^3*b^4*c

+ 35*a^4*b^3*d + (105*a^4*b^2*c^2)/2 + 42*a^5*b*c*d) + x^7*(a*b^7 + 21*a^2*b^5*c + 35*a^4*b*c^3 + 35*a^3*b^4*

d + 21*a^5*b*d^2 + 21*a^5*c^2*d + 70*a^3*b^3*c^2 + 105*a^4*b^2*c*d) + x^18*((7*a^2*d^6)/2 + 7*b^3*d^5 + (7*c^6

*d^2)/2 + 35*a*c^3*d^4 + 35*b*c^4*d^3 + (105*b^2*c^2*d^4)/2 + 42*a*b*c*d^5) + x^17*(c^7*d + 21*a*b^2*d^5 + 35*

a*c^4*d^3 + 21*a^2*c*d^5 + 21*b*c^5*d^2 + 35*b^3*c*d^4 + 70*b^2*c^3*d^3 + 105*a*b*c^2*d^4) + x^3*(a^7*d + 7*a^

5*b^3 + 7*a^6*b*c) + (d^8*x^24)/8 + x^2*(a^7*c + (7*a^6*b^2)/2) + c*d^7*x^23 + d^5*x^21*(a*d^2 + 7*c^3 + 7*b*c

________________________________________________________________________________________

3.2.92 \(\int (b+2 c x+3 d x^2) (a+b x+c x^2+d x^3)^7 \, dx\) [192]