\(\int x^7 (b+c x+d x^2)^7 (b+2 c x+3 d x^2) \, dx\) [194]

Optimal result

Integrand size = 28, antiderivative size = 19 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} x^8 \left (b+c x+d x^2\right )^8 \]

[Out]

1/8*x^8*(d*x^2+c*x+b)^8

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1602} \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} x^8 \left (b+c x+d x^2\right )^8 \]

[In]

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

[Out]

(x^8*(b + c*x + d*x^2)^8)/8

Rule 1602

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
yQ[Qq, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^8 \left (b+c x+d x^2\right )^8 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} x^8 (b+x (c+d x))^8 \]

[In]

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

[Out]

(x^8*(b + x*(c + d*x))^8)/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(17)=34\).

Time = 0.96 (sec) , antiderivative size = 439, normalized size of antiderivative = 23.11

method	result	size
norman	\(\frac {d^{8} x^{24}}{8}+c \,d^{7} x^{23}+\left (b \,d^{7}+\frac {7}{2} c^{2} d^{6}\right ) x^{22}+\left (\frac {7}{2} b^{2} d^{6}+21 b \,c^{2} d^{5}+\frac {35}{4} c^{4} d^{4}\right ) x^{20}+\left (7 b c \,d^{6}+7 c^{3} d^{5}\right ) x^{21}+\left (35 b^{4} c \,d^{3}+70 b^{3} c^{3} d^{2}+21 b^{2} c^{5} d +b \,c^{7}\right ) x^{15}+\left (\frac {35}{4} b^{4} d^{4}+70 b^{3} c^{2} d^{3}+\frac {105}{2} b^{2} c^{4} d^{2}+7 b \,c^{6} d +\frac {1}{8} c^{8}\right ) x^{16}+\left (35 b^{3} c \,d^{4}+70 b^{2} c^{3} d^{3}+21 b \,c^{5} d^{2}+c^{7} d \right ) x^{17}+\left (7 b^{3} d^{5}+\frac {105}{2} b^{2} c^{2} d^{4}+35 b \,c^{4} d^{3}+\frac {7}{2} c^{6} d^{2}\right ) x^{18}+\left (21 b^{2} c \,d^{5}+35 b \,c^{3} d^{4}+7 c^{5} d^{3}\right ) x^{19}+\left (\frac {7}{2} b^{6} d^{2}+21 b^{5} c^{2} d +\frac {35}{4} b^{4} c^{4}\right ) x^{12}+\left (21 b^{5} c \,d^{2}+35 b^{4} c^{3} d +7 b^{3} c^{5}\right ) x^{13}+\left (7 b^{5} d^{3}+\frac {105}{2} b^{4} c^{2} d^{2}+35 b^{3} c^{4} d +\frac {7}{2} c^{6} b^{2}\right ) x^{14}+b^{7} c \,x^{9}+\left (b^{7} d +\frac {7}{2} c^{2} b^{6}\right ) x^{10}+\left (7 b^{6} c d +7 b^{5} c^{3}\right ) x^{11}+\frac {x^{8} b^{8}}{8}\)	\(439\)
gosper	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{12} b^{4} c^{4}+c \,d^{7} x^{23}+b \,c^{7} x^{15}+21 b^{2} c \,d^{5} x^{19}+35 b \,c^{3} d^{4} x^{19}+35 b^{3} c \,d^{4} x^{17}+70 b^{2} c^{3} d^{3} x^{17}+21 b \,c^{5} d^{2} x^{17}+7 c^{3} d^{5} x^{21}+7 b^{3} c^{5} x^{13}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+\frac {35}{4} x^{20} c^{4} d^{4}+x^{22} b \,d^{7}+\frac {7}{2} x^{22} c^{2} d^{6}+7 c^{5} d^{3} x^{19}+c^{7} d \,x^{17}+b^{7} c \,x^{9}+7 b^{5} c^{3} x^{11}+7 x^{14} b^{5} d^{3}+\frac {7}{2} x^{14} c^{6} b^{2}+\frac {1}{8} x^{8} b^{8}+21 x^{12} b^{5} c^{2} d +7 b^{6} c d \,x^{11}+\frac {105}{2} x^{18} b^{2} c^{2} d^{4}+21 b^{5} c \,d^{2} x^{13}+35 b^{4} c^{3} d \,x^{13}+35 b^{4} c \,d^{3} x^{15}+70 b^{3} c^{3} d^{2} x^{15}+21 b^{2} c^{5} d \,x^{15}+7 b c \,d^{6} x^{21}+\frac {1}{8} d^{8} x^{24}+35 x^{18} b \,c^{4} d^{3}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{18} c^{6} d^{2}+x^{10} b^{7} d +\frac {105}{2} x^{14} b^{4} c^{2} d^{2}+35 x^{14} b^{3} c^{4} d +70 x^{16} b^{3} c^{2} d^{3}+\frac {7}{2} x^{10} c^{2} b^{6}+\frac {105}{2} x^{16} b^{2} c^{4} d^{2}+7 x^{16} b \,c^{6} d +21 x^{20} b \,c^{2} d^{5}+\frac {1}{8} x^{16} c^{8}\)	\(497\)
risch	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{12} b^{4} c^{4}+c \,d^{7} x^{23}+b \,c^{7} x^{15}+21 b^{2} c \,d^{5} x^{19}+35 b \,c^{3} d^{4} x^{19}+35 b^{3} c \,d^{4} x^{17}+70 b^{2} c^{3} d^{3} x^{17}+21 b \,c^{5} d^{2} x^{17}+7 c^{3} d^{5} x^{21}+7 b^{3} c^{5} x^{13}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+\frac {35}{4} x^{20} c^{4} d^{4}+x^{22} b \,d^{7}+\frac {7}{2} x^{22} c^{2} d^{6}+7 c^{5} d^{3} x^{19}+c^{7} d \,x^{17}+b^{7} c \,x^{9}+7 b^{5} c^{3} x^{11}+7 x^{14} b^{5} d^{3}+\frac {7}{2} x^{14} c^{6} b^{2}+\frac {1}{8} x^{8} b^{8}+21 x^{12} b^{5} c^{2} d +7 b^{6} c d \,x^{11}+\frac {105}{2} x^{18} b^{2} c^{2} d^{4}+21 b^{5} c \,d^{2} x^{13}+35 b^{4} c^{3} d \,x^{13}+35 b^{4} c \,d^{3} x^{15}+70 b^{3} c^{3} d^{2} x^{15}+21 b^{2} c^{5} d \,x^{15}+7 b c \,d^{6} x^{21}+\frac {1}{8} d^{8} x^{24}+35 x^{18} b \,c^{4} d^{3}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{18} c^{6} d^{2}+x^{10} b^{7} d +\frac {105}{2} x^{14} b^{4} c^{2} d^{2}+35 x^{14} b^{3} c^{4} d +70 x^{16} b^{3} c^{2} d^{3}+\frac {7}{2} x^{10} c^{2} b^{6}+\frac {105}{2} x^{16} b^{2} c^{4} d^{2}+7 x^{16} b \,c^{6} d +21 x^{20} b \,c^{2} d^{5}+\frac {1}{8} x^{16} c^{8}\)	\(497\)
parallelrisch	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{12} b^{4} c^{4}+c \,d^{7} x^{23}+b \,c^{7} x^{15}+21 b^{2} c \,d^{5} x^{19}+35 b \,c^{3} d^{4} x^{19}+35 b^{3} c \,d^{4} x^{17}+70 b^{2} c^{3} d^{3} x^{17}+21 b \,c^{5} d^{2} x^{17}+7 c^{3} d^{5} x^{21}+7 b^{3} c^{5} x^{13}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+\frac {35}{4} x^{20} c^{4} d^{4}+x^{22} b \,d^{7}+\frac {7}{2} x^{22} c^{2} d^{6}+7 c^{5} d^{3} x^{19}+c^{7} d \,x^{17}+b^{7} c \,x^{9}+7 b^{5} c^{3} x^{11}+7 x^{14} b^{5} d^{3}+\frac {7}{2} x^{14} c^{6} b^{2}+\frac {1}{8} x^{8} b^{8}+21 x^{12} b^{5} c^{2} d +7 b^{6} c d \,x^{11}+\frac {105}{2} x^{18} b^{2} c^{2} d^{4}+21 b^{5} c \,d^{2} x^{13}+35 b^{4} c^{3} d \,x^{13}+35 b^{4} c \,d^{3} x^{15}+70 b^{3} c^{3} d^{2} x^{15}+21 b^{2} c^{5} d \,x^{15}+7 b c \,d^{6} x^{21}+\frac {1}{8} d^{8} x^{24}+35 x^{18} b \,c^{4} d^{3}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{18} c^{6} d^{2}+x^{10} b^{7} d +\frac {105}{2} x^{14} b^{4} c^{2} d^{2}+35 x^{14} b^{3} c^{4} d +70 x^{16} b^{3} c^{2} d^{3}+\frac {7}{2} x^{10} c^{2} b^{6}+\frac {105}{2} x^{16} b^{2} c^{4} d^{2}+7 x^{16} b \,c^{6} d +21 x^{20} b \,c^{2} d^{5}+\frac {1}{8} x^{16} c^{8}\)	\(497\)
default	\(\text {Expression too large to display}\)	\(5596\)

[In]

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

[Out]

1/8*d^8*x^24+c*d^7*x^23+(b*d^7+7/2*c^2*d^6)*x^22+(7/2*b^2*d^6+21*b*c^2*d^5+35/4*c^4*d^4)*x^20+(7*b*c*d^6+7*c^3
*d^5)*x^21+(35*b^4*c*d^3+70*b^3*c^3*d^2+21*b^2*c^5*d+b*c^7)*x^15+(35/4*b^4*d^4+70*b^3*c^2*d^3+105/2*b^2*c^4*d^
2+7*b*c^6*d+1/8*c^8)*x^16+(35*b^3*c*d^4+70*b^2*c^3*d^3+21*b*c^5*d^2+c^7*d)*x^17+(7*b^3*d^5+105/2*b^2*c^2*d^4+3
5*b*c^4*d^3+7/2*c^6*d^2)*x^18+(21*b^2*c*d^5+35*b*c^3*d^4+7*c^5*d^3)*x^19+(7/2*b^6*d^2+21*b^5*c^2*d+35/4*b^4*c^
4)*x^12+(21*b^5*c*d^2+35*b^4*c^3*d+7*b^3*c^5)*x^13+(7*b^5*d^3+105/2*b^4*c^2*d^2+35*b^3*c^4*d+7/2*c^6*b^2)*x^14
+b^7*c*x^9+(b^7*d+7/2*c^2*b^6)*x^10+(7*b^6*c*d+7*b^5*c^3)*x^11+1/8*x^8*b^8

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 441, normalized size of antiderivative = 23.21 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {1}{2} \, {\left (7 \, c^{2} d^{6} + 2 \, b d^{7}\right )} x^{22} + 7 \, {\left (c^{3} d^{5} + b c d^{6}\right )} x^{21} + \frac {7}{4} \, {\left (5 \, c^{4} d^{4} + 12 \, b c^{2} d^{5} + 2 \, b^{2} d^{6}\right )} x^{20} + 7 \, {\left (c^{5} d^{3} + 5 \, b c^{3} d^{4} + 3 \, b^{2} c d^{5}\right )} x^{19} + \frac {7}{2} \, {\left (c^{6} d^{2} + 10 \, b c^{4} d^{3} + 15 \, b^{2} c^{2} d^{4} + 2 \, b^{3} d^{5}\right )} x^{18} + {\left (c^{7} d + 21 \, b c^{5} d^{2} + 70 \, b^{2} c^{3} d^{3} + 35 \, b^{3} c d^{4}\right )} x^{17} + b^{7} c x^{9} + \frac {1}{8} \, {\left (c^{8} + 56 \, b c^{6} d + 420 \, b^{2} c^{4} d^{2} + 560 \, b^{3} c^{2} d^{3} + 70 \, b^{4} d^{4}\right )} x^{16} + \frac {1}{8} \, b^{8} x^{8} + {\left (b c^{7} + 21 \, b^{2} c^{5} d + 70 \, b^{3} c^{3} d^{2} + 35 \, b^{4} c d^{3}\right )} x^{15} + \frac {7}{2} \, {\left (b^{2} c^{6} + 10 \, b^{3} c^{4} d + 15 \, b^{4} c^{2} d^{2} + 2 \, b^{5} d^{3}\right )} x^{14} + 7 \, {\left (b^{3} c^{5} + 5 \, b^{4} c^{3} d + 3 \, b^{5} c d^{2}\right )} x^{13} + \frac {7}{4} \, {\left (5 \, b^{4} c^{4} + 12 \, b^{5} c^{2} d + 2 \, b^{6} d^{2}\right )} x^{12} + 7 \, {\left (b^{5} c^{3} + b^{6} c d\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} c^{2} + 2 \, b^{7} d\right )} x^{10} \]

[In]

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

[Out]

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 + b*c*d^6)*x^21 + 7/4*(5*c^4*d^4 + 12*
b*c^2*d^5 + 2*b^2*d^6)*x^20 + 7*(c^5*d^3 + 5*b*c^3*d^4 + 3*b^2*c*d^5)*x^19 + 7/2*(c^6*d^2 + 10*b*c^4*d^3 + 15*
b^2*c^2*d^4 + 2*b^3*d^5)*x^18 + (c^7*d + 21*b*c^5*d^2 + 70*b^2*c^3*d^3 + 35*b^3*c*d^4)*x^17 + b^7*c*x^9 + 1/8*
(c^8 + 56*b*c^6*d + 420*b^2*c^4*d^2 + 560*b^3*c^2*d^3 + 70*b^4*d^4)*x^16 + 1/8*b^8*x^8 + (b*c^7 + 21*b^2*c^5*d
+ 70*b^3*c^3*d^2 + 35*b^4*c*d^3)*x^15 + 7/2*(b^2*c^6 + 10*b^3*c^4*d + 15*b^4*c^2*d^2 + 2*b^5*d^3)*x^14 + 7*(b
^3*c^5 + 5*b^4*c^3*d + 3*b^5*c*d^2)*x^13 + 7/4*(5*b^4*c^4 + 12*b^5*c^2*d + 2*b^6*d^2)*x^12 + 7*(b^5*c^3 + b^6*
c*d)*x^11 + 1/2*(7*b^6*c^2 + 2*b^7*d)*x^10

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (15) = 30\).

Time = 0.08 (sec) , antiderivative size = 469, normalized size of antiderivative = 24.68 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {b^{8} x^{8}}{8} + b^{7} c x^{9} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} + x^{22} \left (b d^{7} + \frac {7 c^{2} d^{6}}{2}\right ) + x^{21} \cdot \left (7 b c d^{6} + 7 c^{3} d^{5}\right ) + x^{20} \cdot \left (\frac {7 b^{2} d^{6}}{2} + 21 b c^{2} d^{5} + \frac {35 c^{4} d^{4}}{4}\right ) + x^{19} \cdot \left (21 b^{2} c d^{5} + 35 b c^{3} d^{4} + 7 c^{5} d^{3}\right ) + x^{18} \cdot \left (7 b^{3} d^{5} + \frac {105 b^{2} c^{2} d^{4}}{2} + 35 b c^{4} d^{3} + \frac {7 c^{6} d^{2}}{2}\right ) + x^{17} \cdot \left (35 b^{3} c d^{4} + 70 b^{2} c^{3} d^{3} + 21 b c^{5} d^{2} + c^{7} d\right ) + x^{16} \cdot \left (\frac {35 b^{4} d^{4}}{4} + 70 b^{3} c^{2} d^{3} + \frac {105 b^{2} c^{4} d^{2}}{2} + 7 b c^{6} d + \frac {c^{8}}{8}\right ) + x^{15} \cdot \left (35 b^{4} c d^{3} + 70 b^{3} c^{3} d^{2} + 21 b^{2} c^{5} d + b c^{7}\right ) + x^{14} \cdot \left (7 b^{5} d^{3} + \frac {105 b^{4} c^{2} d^{2}}{2} + 35 b^{3} c^{4} d + \frac {7 b^{2} c^{6}}{2}\right ) + x^{13} \cdot \left (21 b^{5} c d^{2} + 35 b^{4} c^{3} d + 7 b^{3} c^{5}\right ) + x^{12} \cdot \left (\frac {7 b^{6} d^{2}}{2} + 21 b^{5} c^{2} d + \frac {35 b^{4} c^{4}}{4}\right ) + x^{11} \cdot \left (7 b^{6} c d + 7 b^{5} c^{3}\right ) + x^{10} \left (b^{7} d + \frac {7 b^{6} c^{2}}{2}\right ) \]

[In]

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

[Out]

b**8*x**8/8 + b**7*c*x**9 + c*d**7*x**23 + d**8*x**24/8 + x**22*(b*d**7 + 7*c**2*d**6/2) + x**21*(7*b*c*d**6 +
7*c**3*d**5) + x**20*(7*b**2*d**6/2 + 21*b*c**2*d**5 + 35*c**4*d**4/4) + x**19*(21*b**2*c*d**5 + 35*b*c**3*d*
*4 + 7*c**5*d**3) + x**18*(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*(35*b*
*3*c*d**4 + 70*b**2*c**3*d**3 + 21*b*c**5*d**2 + c**7*d) + x**16*(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*(35*b**4*c*d**3 + 70*b**3*c**3*d**2 + 21*b**2*c**5*d + b*c**7) +
x**14*(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*(21*b**5*c*d**2 + 35*b**4*
c**3*d + 7*b**3*c**5) + x**12*(7*b**6*d**2/2 + 21*b**5*c**2*d + 35*b**4*c**4/4) + x**11*(7*b**6*c*d + 7*b**5*c
**3) + x**10*(b**7*d + 7*b**6*c**2/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (17) = 34\).

Time = 0.19 (sec) , antiderivative size = 441, normalized size of antiderivative = 23.21 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {1}{2} \, {\left (7 \, c^{2} d^{6} + 2 \, b d^{7}\right )} x^{22} + 7 \, {\left (c^{3} d^{5} + b c d^{6}\right )} x^{21} + \frac {7}{4} \, {\left (5 \, c^{4} d^{4} + 12 \, b c^{2} d^{5} + 2 \, b^{2} d^{6}\right )} x^{20} + 7 \, {\left (c^{5} d^{3} + 5 \, b c^{3} d^{4} + 3 \, b^{2} c d^{5}\right )} x^{19} + \frac {7}{2} \, {\left (c^{6} d^{2} + 10 \, b c^{4} d^{3} + 15 \, b^{2} c^{2} d^{4} + 2 \, b^{3} d^{5}\right )} x^{18} + {\left (c^{7} d + 21 \, b c^{5} d^{2} + 70 \, b^{2} c^{3} d^{3} + 35 \, b^{3} c d^{4}\right )} x^{17} + b^{7} c x^{9} + \frac {1}{8} \, {\left (c^{8} + 56 \, b c^{6} d + 420 \, b^{2} c^{4} d^{2} + 560 \, b^{3} c^{2} d^{3} + 70 \, b^{4} d^{4}\right )} x^{16} + \frac {1}{8} \, b^{8} x^{8} + {\left (b c^{7} + 21 \, b^{2} c^{5} d + 70 \, b^{3} c^{3} d^{2} + 35 \, b^{4} c d^{3}\right )} x^{15} + \frac {7}{2} \, {\left (b^{2} c^{6} + 10 \, b^{3} c^{4} d + 15 \, b^{4} c^{2} d^{2} + 2 \, b^{5} d^{3}\right )} x^{14} + 7 \, {\left (b^{3} c^{5} + 5 \, b^{4} c^{3} d + 3 \, b^{5} c d^{2}\right )} x^{13} + \frac {7}{4} \, {\left (5 \, b^{4} c^{4} + 12 \, b^{5} c^{2} d + 2 \, b^{6} d^{2}\right )} x^{12} + 7 \, {\left (b^{5} c^{3} + b^{6} c d\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} c^{2} + 2 \, b^{7} d\right )} x^{10} \]

[In]

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

[Out]

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 + b*c*d^6)*x^21 + 7/4*(5*c^4*d^4 + 12*
b*c^2*d^5 + 2*b^2*d^6)*x^20 + 7*(c^5*d^3 + 5*b*c^3*d^4 + 3*b^2*c*d^5)*x^19 + 7/2*(c^6*d^2 + 10*b*c^4*d^3 + 15*
b^2*c^2*d^4 + 2*b^3*d^5)*x^18 + (c^7*d + 21*b*c^5*d^2 + 70*b^2*c^3*d^3 + 35*b^3*c*d^4)*x^17 + b^7*c*x^9 + 1/8*
(c^8 + 56*b*c^6*d + 420*b^2*c^4*d^2 + 560*b^3*c^2*d^3 + 70*b^4*d^4)*x^16 + 1/8*b^8*x^8 + (b*c^7 + 21*b^2*c^5*d
+ 70*b^3*c^3*d^2 + 35*b^4*c*d^3)*x^15 + 7/2*(b^2*c^6 + 10*b^3*c^4*d + 15*b^4*c^2*d^2 + 2*b^5*d^3)*x^14 + 7*(b
^3*c^5 + 5*b^4*c^3*d + 3*b^5*c*d^2)*x^13 + 7/4*(5*b^4*c^4 + 12*b^5*c^2*d + 2*b^6*d^2)*x^12 + 7*(b^5*c^3 + b^6*
c*d)*x^11 + 1/2*(7*b^6*c^2 + 2*b^7*d)*x^10

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} \]

[In]

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

[Out]

1/8*(d*x^3 + c*x^2 + b*x)^8

Mupad [B] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 418, normalized size of antiderivative = 22.00 \[ \int x^7 \left (b+c x+d x^2\right )^7 \left (b+2 c x+3 d x^2\right ) \, dx=x^{14}\,\left (7\,b^5\,d^3+\frac {105\,b^4\,c^2\,d^2}{2}+35\,b^3\,c^4\,d+\frac {7\,b^2\,c^6}{2}\right )+x^{18}\,\left (7\,b^3\,d^5+\frac {105\,b^2\,c^2\,d^4}{2}+35\,b\,c^4\,d^3+\frac {7\,c^6\,d^2}{2}\right )+x^{12}\,\left (\frac {7\,b^6\,d^2}{2}+21\,b^5\,c^2\,d+\frac {35\,b^4\,c^4}{4}\right )+x^{20}\,\left (\frac {7\,b^2\,d^6}{2}+21\,b\,c^2\,d^5+\frac {35\,c^4\,d^4}{4}\right )+x^{16}\,\left (\frac {35\,b^4\,d^4}{4}+70\,b^3\,c^2\,d^3+\frac {105\,b^2\,c^4\,d^2}{2}+7\,b\,c^6\,d+\frac {c^8}{8}\right )+\frac {b^8\,x^8}{8}+\frac {d^8\,x^{24}}{8}+x^{10}\,\left (d\,b^7+\frac {7\,b^6\,c^2}{2}\right )+b^7\,c\,x^9+c\,d^7\,x^{23}+\frac {d^6\,x^{22}\,\left (7\,c^2+2\,b\,d\right )}{2}+7\,b^3\,c\,x^{13}\,\left (3\,b^2\,d^2+5\,b\,c^2\,d+c^4\right )+7\,c\,d^3\,x^{19}\,\left (3\,b^2\,d^2+5\,b\,c^2\,d+c^4\right )+b\,c\,x^{15}\,\left (35\,b^3\,d^3+70\,b^2\,c^2\,d^2+21\,b\,c^4\,d+c^6\right )+c\,d\,x^{17}\,\left (35\,b^3\,d^3+70\,b^2\,c^2\,d^2+21\,b\,c^4\,d+c^6\right )+7\,b^5\,c\,x^{11}\,\left (c^2+b\,d\right )+7\,c\,d^5\,x^{21}\,\left (c^2+b\,d\right ) \]

[In]

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

[Out]

x^14*((7*b^2*c^6)/2 + 7*b^5*d^3 + 35*b^3*c^4*d + (105*b^4*c^2*d^2)/2) + x^18*(7*b^3*d^5 + (7*c^6*d^2)/2 + 35*b
*c^4*d^3 + (105*b^2*c^2*d^4)/2) + x^12*((35*b^4*c^4)/4 + (7*b^6*d^2)/2 + 21*b^5*c^2*d) + x^20*((7*b^2*d^6)/2 +
(35*c^4*d^4)/4 + 21*b*c^2*d^5) + x^16*(c^8/8 + (35*b^4*d^4)/4 + (105*b^2*c^4*d^2)/2 + 70*b^3*c^2*d^3 + 7*b*c^
6*d) + (b^8*x^8)/8 + (d^8*x^24)/8 + x^10*(b^7*d + (7*b^6*c^2)/2) + b^7*c*x^9 + c*d^7*x^23 + (d^6*x^22*(2*b*d +
7*c^2))/2 + 7*b^3*c*x^13*(c^4 + 3*b^2*d^2 + 5*b*c^2*d) + 7*c*d^3*x^19*(c^4 + 3*b^2*d^2 + 5*b*c^2*d) + b*c*x^1
5*(c^6 + 35*b^3*d^3 + 70*b^2*c^2*d^2 + 21*b*c^4*d) + c*d*x^17*(c^6 + 35*b^3*d^3 + 70*b^2*c^2*d^2 + 21*b*c^4*d)
+ 7*b^5*c*x^11*(b*d + c^2) + 7*c*d^5*x^21*(b*d + c^2)