3.197 \(\int x^7 (b+d x^2)^7 (b+3 d x^2) \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{8} x^8 \left (b+d x^2\right )^8 \]

[Out]

(x^8*(b + d*x^2)^8)/8

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Rubi [A]  time = 0.0280873, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {446, 74} \[ \frac{1}{8} x^8 \left (b+d x^2\right )^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^8*(b + d*x^2)^8)/8

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [B]  time = 0.0023896, size = 98, normalized size = 6.12 \[ \frac{7}{2} b^2 d^6 x^{20}+7 b^3 d^5 x^{18}+\frac{35}{4} b^4 d^4 x^{16}+7 b^5 d^3 x^{14}+\frac{7}{2} b^6 d^2 x^{12}+b^7 d x^{10}+\frac{b^8 x^8}{8}+b d^7 x^{22}+\frac{d^8 x^{24}}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^8*x^8)/8 + b^7*d*x^10 + (7*b^6*d^2*x^12)/2 + 7*b^5*d^3*x^14 + (35*b^4*d^4*x^16)/4 + 7*b^3*d^5*x^18 + (7*b^2
*d^6*x^20)/2 + b*d^7*x^22 + (d^8*x^24)/8

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Maple [B]  time = 0., size = 89, normalized size = 5.6 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+b{d}^{7}{x}^{22}+{\frac{7\,{b}^{2}{d}^{6}{x}^{20}}{2}}+7\,{b}^{3}{d}^{5}{x}^{18}+{\frac{35\,{b}^{4}{d}^{4}{x}^{16}}{4}}+7\,{b}^{5}{d}^{3}{x}^{14}+{\frac{7\,{b}^{6}{d}^{2}{x}^{12}}{2}}+d{b}^{7}{x}^{10}+{\frac{{b}^{8}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24+b*d^7*x^22+7/2*b^2*d^6*x^20+7*b^3*d^5*x^18+35/4*b^4*d^4*x^16+7*b^5*d^3*x^14+7/2*b^6*d^2*x^12+d*b^
7*x^10+1/8*b^8*x^8

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Maxima [B]  time = 0.986427, size = 119, normalized size = 7.44 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac{35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac{7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac{1}{8} \, b^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24 + b*d^7*x^22 + 7/2*b^2*d^6*x^20 + 7*b^3*d^5*x^18 + 35/4*b^4*d^4*x^16 + 7*b^5*d^3*x^14 + 7/2*b^6*d
^2*x^12 + b^7*d*x^10 + 1/8*b^8*x^8

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Fricas [B]  time = 1.20792, size = 197, normalized size = 12.31 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{22} d^{7} b + \frac{7}{2} x^{20} d^{6} b^{2} + 7 x^{18} d^{5} b^{3} + \frac{35}{4} x^{16} d^{4} b^{4} + 7 x^{14} d^{3} b^{5} + \frac{7}{2} x^{12} d^{2} b^{6} + x^{10} d b^{7} + \frac{1}{8} x^{8} b^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^24*d^8 + x^22*d^7*b + 7/2*x^20*d^6*b^2 + 7*x^18*d^5*b^3 + 35/4*x^16*d^4*b^4 + 7*x^14*d^3*b^5 + 7/2*x^12*
d^2*b^6 + x^10*d*b^7 + 1/8*x^8*b^8

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Sympy [B]  time = 0.083103, size = 97, normalized size = 6.06 \begin{align*} \frac{b^{8} x^{8}}{8} + b^{7} d x^{10} + \frac{7 b^{6} d^{2} x^{12}}{2} + 7 b^{5} d^{3} x^{14} + \frac{35 b^{4} d^{4} x^{16}}{4} + 7 b^{3} d^{5} x^{18} + \frac{7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \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+b)**7*(3*d*x**2+b),x)

[Out]

b**8*x**8/8 + b**7*d*x**10 + 7*b**6*d**2*x**12/2 + 7*b**5*d**3*x**14 + 35*b**4*d**4*x**16/4 + 7*b**3*d**5*x**1
8 + 7*b**2*d**6*x**20/2 + b*d**7*x**22 + d**8*x**24/8

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Giac [B]  time = 1.32276, size = 119, normalized size = 7.44 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac{35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac{7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac{1}{8} \, b^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24 + b*d^7*x^22 + 7/2*b^2*d^6*x^20 + 7*b^3*d^5*x^18 + 35/4*b^4*d^4*x^16 + 7*b^5*d^3*x^14 + 7/2*b^6*d
^2*x^12 + b^7*d*x^10 + 1/8*b^8*x^8