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

Optimal. Leaf size=14 \[ \frac{1}{8} x^{16} (c+d x)^8 \]

[Out]

(x^16*(c + d*x)^8)/8

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Rubi [A]  time = 0.227122, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1584, 845} \[ \frac{1}{8} x^{16} (c+d x)^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^16*(c + d*x)^8)/8

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 845

Int[(x_)^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(c*x^(m + 2)*(f + g*x)
^(n + 1))/(g*(m + n + 3)), x] /; FreeQ[{b, c, f, g, m, n}, x] && EqQ[c*f*(m + 2) - b*g*(m + n + 3), 0] && NeQ[
m + n + 3, 0]

Rubi steps

\begin{align*} \int x^7 \left (c x+d x^2\right )^7 \left (2 c x+3 d x^2\right ) \, dx &=\int x^{14} (c+d x)^7 \left (2 c x+3 d x^2\right ) \, dx\\ &=\frac{1}{8} x^{16} (c+d x)^8\\ \end{align*}

Mathematica [B]  time = 0.0028447, size = 98, normalized size = 7. \[ \frac{7}{2} c^2 d^6 x^{22}+7 c^3 d^5 x^{21}+\frac{35}{4} c^4 d^4 x^{20}+7 c^5 d^3 x^{19}+\frac{7}{2} c^6 d^2 x^{18}+c^7 d x^{17}+\frac{c^8 x^{16}}{8}+c d^7 x^{23}+\frac{d^8 x^{24}}{8} \]

Antiderivative was successfully verified.

[In]

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

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Maple [B]  time = 0.003, size = 89, normalized size = 6.4 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+c{d}^{7}{x}^{23}+{\frac{7\,{c}^{2}{d}^{6}{x}^{22}}{2}}+7\,{c}^{3}{d}^{5}{x}^{21}+{\frac{35\,{c}^{4}{d}^{4}{x}^{20}}{4}}+7\,{c}^{5}{d}^{3}{x}^{19}+{\frac{7\,{c}^{6}{d}^{2}{x}^{18}}{2}}+{c}^{7}d{x}^{17}+{\frac{{c}^{8}{x}^{16}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B]  time = 0.973676, size = 119, normalized size = 8.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B]  time = 1.07747, size = 198, normalized size = 14.14 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [B]  time = 0.091965, size = 97, normalized size = 6.93 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B]  time = 1.27447, size = 119, normalized size = 8.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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