3.90 \(\int x^3 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^{10}}{20 b^2}-\frac{a \left (a+b x^2\right )^9}{18 b^2} \]

[Out]

-(a*(a + b*x^2)^9)/(18*b^2) + (a + b*x^2)^10/(20*b^2)

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Rubi [A]  time = 0.0456613, antiderivative size = 34, 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} \[ \frac{\left (a+b x^2\right )^{10}}{20 b^2}-\frac{a \left (a+b x^2\right )^9}{18 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^8,x]

[Out]

-(a*(a + b*x^2)^9)/(18*b^2) + (a + b*x^2)^10/(20*b^2)

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

Mathematica [B]  time = 0.0024039, size = 106, normalized size = 3.12 \[ \frac{7}{4} a^2 b^6 x^{16}+4 a^3 b^5 x^{14}+\frac{35}{6} a^4 b^4 x^{12}+\frac{28}{5} a^5 b^3 x^{10}+\frac{7}{2} a^6 b^2 x^8+\frac{4}{3} a^7 b x^6+\frac{a^8 x^4}{4}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^8,x]

[Out]

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

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Maple [B]  time = 0.002, size = 91, normalized size = 2.7 \begin{align*}{\frac{{b}^{8}{x}^{20}}{20}}+{\frac{4\,a{b}^{7}{x}^{18}}{9}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{16}}{4}}+4\,{a}^{3}{b}^{5}{x}^{14}+{\frac{35\,{a}^{4}{b}^{4}{x}^{12}}{6}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{8}}{2}}+{\frac{4\,{a}^{7}b{x}^{6}}{3}}+{\frac{{a}^{8}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^8,x)

[Out]

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

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Maxima [B]  time = 1.82346, size = 122, normalized size = 3.59 \begin{align*} \frac{1}{20} \, b^{8} x^{20} + \frac{4}{9} \, a b^{7} x^{18} + \frac{7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{4}{3} \, a^{7} b x^{6} + \frac{1}{4} \, a^{8} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.3071, size = 211, normalized size = 6.21 \begin{align*} \frac{1}{20} x^{20} b^{8} + \frac{4}{9} x^{18} b^{7} a + \frac{7}{4} x^{16} b^{6} a^{2} + 4 x^{14} b^{5} a^{3} + \frac{35}{6} x^{12} b^{4} a^{4} + \frac{28}{5} x^{10} b^{3} a^{5} + \frac{7}{2} x^{8} b^{2} a^{6} + \frac{4}{3} x^{6} b a^{7} + \frac{1}{4} x^{4} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.082124, size = 105, normalized size = 3.09 \begin{align*} \frac{a^{8} x^{4}}{4} + \frac{4 a^{7} b x^{6}}{3} + \frac{7 a^{6} b^{2} x^{8}}{2} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{35 a^{4} b^{4} x^{12}}{6} + 4 a^{3} b^{5} x^{14} + \frac{7 a^{2} b^{6} x^{16}}{4} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{20}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**8,x)

[Out]

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

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Giac [B]  time = 2.12975, size = 122, normalized size = 3.59 \begin{align*} \frac{1}{20} \, b^{8} x^{20} + \frac{4}{9} \, a b^{7} x^{18} + \frac{7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{4}{3} \, a^{7} b x^{6} + \frac{1}{4} \, a^{8} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="giac")

[Out]

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