3.88 \(\int x^7 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=72 \[ \frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4} \]

[Out]

-(a^3*(a + b*x^2)^9)/(18*b^4) + (3*a^2*(a + b*x^2)^10)/(20*b^4) - (3*a*(a + b*x^2)^11)/(22*b^4) + (a + b*x^2)^
12/(24*b^4)

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Rubi [A]  time = 0.115799, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^8,x]

[Out]

-(a^3*(a + b*x^2)^9)/(18*b^4) + (3*a^2*(a + b*x^2)^10)/(20*b^4) - (3*a*(a + b*x^2)^11)/(22*b^4) + (a + b*x^2)^
12/(24*b^4)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^7 \left (a+b x^2\right )^8 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^8}{b^3}+\frac{3 a^2 (a+b x)^9}{b^3}-\frac{3 a (a+b x)^{10}}{b^3}+\frac{(a+b x)^{11}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0028108, size = 106, normalized size = 1.47 \[ \frac{7}{5} a^2 b^6 x^{20}+\frac{28}{9} a^3 b^5 x^{18}+\frac{35}{8} a^4 b^4 x^{16}+4 a^5 b^3 x^{14}+\frac{7}{3} a^6 b^2 x^{12}+\frac{4}{5} a^7 b x^{10}+\frac{a^8 x^8}{8}+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^8,x]

[Out]

(a^8*x^8)/8 + (4*a^7*b*x^10)/5 + (7*a^6*b^2*x^12)/3 + 4*a^5*b^3*x^14 + (35*a^4*b^4*x^16)/8 + (28*a^3*b^5*x^18)
/9 + (7*a^2*b^6*x^20)/5 + (4*a*b^7*x^22)/11 + (b^8*x^24)/24

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Maple [A]  time = 0.002, size = 91, normalized size = 1.3 \begin{align*}{\frac{{b}^{8}{x}^{24}}{24}}+{\frac{4\,a{b}^{7}{x}^{22}}{11}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{20}}{5}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{18}}{9}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{16}}{8}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{7\,{a}^{6}{b}^{2}{x}^{12}}{3}}+{\frac{4\,{a}^{7}b{x}^{10}}{5}}+{\frac{{a}^{8}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^2+a)^8,x)

[Out]

1/24*b^8*x^24+4/11*a*b^7*x^22+7/5*a^2*b^6*x^20+28/9*a^3*b^5*x^18+35/8*a^4*b^4*x^16+4*a^5*b^3*x^14+7/3*a^6*b^2*
x^12+4/5*a^7*b*x^10+1/8*a^8*x^8

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Maxima [A]  time = 1.59132, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{4}{11} \, a b^{7} x^{22} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{28}{9} \, a^{3} b^{5} x^{18} + \frac{35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{4}{5} \, a^{7} b x^{10} + \frac{1}{8} \, a^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/24*b^8*x^24 + 4/11*a*b^7*x^22 + 7/5*a^2*b^6*x^20 + 28/9*a^3*b^5*x^18 + 35/8*a^4*b^4*x^16 + 4*a^5*b^3*x^14 +
7/3*a^6*b^2*x^12 + 4/5*a^7*b*x^10 + 1/8*a^8*x^8

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Fricas [A]  time = 1.37431, size = 215, normalized size = 2.99 \begin{align*} \frac{1}{24} x^{24} b^{8} + \frac{4}{11} x^{22} b^{7} a + \frac{7}{5} x^{20} b^{6} a^{2} + \frac{28}{9} x^{18} b^{5} a^{3} + \frac{35}{8} x^{16} b^{4} a^{4} + 4 x^{14} b^{3} a^{5} + \frac{7}{3} x^{12} b^{2} a^{6} + \frac{4}{5} x^{10} b a^{7} + \frac{1}{8} x^{8} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/24*x^24*b^8 + 4/11*x^22*b^7*a + 7/5*x^20*b^6*a^2 + 28/9*x^18*b^5*a^3 + 35/8*x^16*b^4*a^4 + 4*x^14*b^3*a^5 +
7/3*x^12*b^2*a^6 + 4/5*x^10*b*a^7 + 1/8*x^8*a^8

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Sympy [A]  time = 0.082515, size = 105, normalized size = 1.46 \begin{align*} \frac{a^{8} x^{8}}{8} + \frac{4 a^{7} b x^{10}}{5} + \frac{7 a^{6} b^{2} x^{12}}{3} + 4 a^{5} b^{3} x^{14} + \frac{35 a^{4} b^{4} x^{16}}{8} + \frac{28 a^{3} b^{5} x^{18}}{9} + \frac{7 a^{2} b^{6} x^{20}}{5} + \frac{4 a b^{7} x^{22}}{11} + \frac{b^{8} x^{24}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**2+a)**8,x)

[Out]

a**8*x**8/8 + 4*a**7*b*x**10/5 + 7*a**6*b**2*x**12/3 + 4*a**5*b**3*x**14 + 35*a**4*b**4*x**16/8 + 28*a**3*b**5
*x**18/9 + 7*a**2*b**6*x**20/5 + 4*a*b**7*x**22/11 + b**8*x**24/24

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Giac [A]  time = 2.10331, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{4}{11} \, a b^{7} x^{22} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{28}{9} \, a^{3} b^{5} x^{18} + \frac{35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{4}{5} \, a^{7} b x^{10} + \frac{1}{8} \, a^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/24*b^8*x^24 + 4/11*a*b^7*x^22 + 7/5*a^2*b^6*x^20 + 28/9*a^3*b^5*x^18 + 35/8*a^4*b^4*x^16 + 4*a^5*b^3*x^14 +
7/3*a^6*b^2*x^12 + 4/5*a^7*b*x^10 + 1/8*a^8*x^8