3.204 \(\int x^8 (2 c+3 d x) (c x+d x^2)^7 \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{8} x^8 \left (c x+d x^2\right )^8 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {763} \[ \frac {1}{8} x^8 \left (c x+d x^2\right )^8 \]

Antiderivative was successfully verified.

[In]

Int[x^8*(2*c + 3*d*x)*(c*x + d*x^2)^7,x]

[Out]

(x^8*(c*x + d*x^2)^8)/8

Rule 763

Int[((e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(e*x)^m*(b*
x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] /; FreeQ[{b, c, e, f, g, m, p}, x] && EqQ[b*g*(m + p + 1) - c*f*(m +
2*p + 2), 0] && NeQ[m + 2*p + 2, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.00, size = 98, normalized size = 5.44 \[ \frac {c^8 x^{16}}{8}+c^7 d x^{17}+\frac {7}{2} c^6 d^2 x^{18}+7 c^5 d^3 x^{19}+\frac {35}{4} c^4 d^4 x^{20}+7 c^3 d^5 x^{21}+\frac {7}{2} c^2 d^6 x^{22}+c d^7 x^{23}+\frac {d^8 x^{24}}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(2*c + 3*d*x)*(c*x + d*x^2)^7,x]

[Out]

(c^8*x^16)/8 + c^7*d*x^17 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20)/4 + 7*c^3*d^5*x^21 + (7*c^
2*d^6*x^22)/2 + c*d^7*x^23 + (d^8*x^24)/8

________________________________________________________________________________________

fricas [B]  time = 0.56, size = 88, normalized size = 4.89 \[ \frac {1}{8} x^{24} d^{8} + x^{23} d^{7} c + \frac {7}{2} x^{22} d^{6} c^{2} + 7 x^{21} d^{5} c^{3} + \frac {35}{4} x^{20} d^{4} c^{4} + 7 x^{19} d^{3} c^{5} + \frac {7}{2} x^{18} d^{2} c^{6} + x^{17} d c^{7} + \frac {1}{8} x^{16} c^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(3*d*x+2*c)*(d*x^2+c*x)^7,x, algorithm="fricas")

[Out]

1/8*x^24*d^8 + x^23*d^7*c + 7/2*x^22*d^6*c^2 + 7*x^21*d^5*c^3 + 35/4*x^20*d^4*c^4 + 7*x^19*d^3*c^5 + 7/2*x^18*
d^2*c^6 + x^17*d*c^7 + 1/8*x^16*c^8

________________________________________________________________________________________

giac [B]  time = 0.29, size = 88, normalized size = 4.89 \[ \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(3*d*x+2*c)*(d*x^2+c*x)^7,x, algorithm="giac")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

________________________________________________________________________________________

maple [B]  time = 0.00, size = 89, normalized size = 4.94 \[ \frac {1}{8} d^{8} x^{24}+c \,d^{7} x^{23}+\frac {7}{2} c^{2} d^{6} x^{22}+7 c^{3} d^{5} x^{21}+\frac {35}{4} c^{4} d^{4} x^{20}+7 c^{5} d^{3} x^{19}+\frac {7}{2} c^{6} d^{2} x^{18}+c^{7} d \,x^{17}+\frac {1}{8} c^{8} x^{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(3*d*x+2*c)*(d*x^2+c*x)^7,x)

[Out]

1/8*d^8*x^24+c*d^7*x^23+7/2*c^2*d^6*x^22+7*c^3*d^5*x^21+35/4*c^4*d^4*x^20+7*c^5*d^3*x^19+7/2*c^6*d^2*x^18+c^7*
d*x^17+1/8*c^8*x^16

________________________________________________________________________________________

maxima [B]  time = 0.57, size = 88, normalized size = 4.89 \[ \frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(3*d*x+2*c)*(d*x^2+c*x)^7,x, algorithm="maxima")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 88, normalized size = 4.89 \[ \frac {c^8\,x^{16}}{8}+c^7\,d\,x^{17}+\frac {7\,c^6\,d^2\,x^{18}}{2}+7\,c^5\,d^3\,x^{19}+\frac {35\,c^4\,d^4\,x^{20}}{4}+7\,c^3\,d^5\,x^{21}+\frac {7\,c^2\,d^6\,x^{22}}{2}+c\,d^7\,x^{23}+\frac {d^8\,x^{24}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(c*x + d*x^2)^7*(2*c + 3*d*x),x)

[Out]

(c^8*x^16)/8 + (d^8*x^24)/8 + c^7*d*x^17 + c*d^7*x^23 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20
)/4 + 7*c^3*d^5*x^21 + (7*c^2*d^6*x^22)/2

________________________________________________________________________________________

sympy [B]  time = 0.10, size = 97, normalized size = 5.39 \[ \frac {c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac {7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac {35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac {7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(3*d*x+2*c)*(d*x**2+c*x)**7,x)

[Out]

c**8*x**16/8 + c**7*d*x**17 + 7*c**6*d**2*x**18/2 + 7*c**5*d**3*x**19 + 35*c**4*d**4*x**20/4 + 7*c**3*d**5*x**
21 + 7*c**2*d**6*x**22/2 + c*d**7*x**23 + d**8*x**24/8

________________________________________________________________________________________