3.415 \(\int x^3 (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^2\right )^{13/2}}{13 b^2}-\frac{a \left (a+b x^2\right )^{11/2}}{11 b^2} \]

[Out]

-(a*(a + b*x^2)^(11/2))/(11*b^2) + (a + b*x^2)^(13/2)/(13*b^2)

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Rubi [A]  time = 0.0239998, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^2\right )^{13/2}}{13 b^2}-\frac{a \left (a+b x^2\right )^{11/2}}{11 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^(9/2),x]

[Out]

-(a*(a + b*x^2)^(11/2))/(11*b^2) + (a + b*x^2)^(13/2)/(13*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 )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{9/2}}{b}+\frac{(a+b x)^{11/2}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{a \left (a+b x^2\right )^{11/2}}{11 b^2}+\frac{\left (a+b x^2\right )^{13/2}}{13 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0182053, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{11/2} \left (11 b x^2-2 a\right )}{143 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(-2*a + 11*b*x^2))/(143*b^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-11\,b{x}^{2}+2\,a}{143\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^(9/2),x)

[Out]

-1/143*(b*x^2+a)^(11/2)*(-11*b*x^2+2*a)/b^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62493, size = 174, normalized size = 4.58 \begin{align*} \frac{{\left (11 \, b^{6} x^{12} + 53 \, a b^{5} x^{10} + 100 \, a^{2} b^{4} x^{8} + 90 \, a^{3} b^{3} x^{6} + 35 \, a^{4} b^{2} x^{4} + a^{5} b x^{2} - 2 \, a^{6}\right )} \sqrt{b x^{2} + a}}{143 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/143*(11*b^6*x^12 + 53*a*b^5*x^10 + 100*a^2*b^4*x^8 + 90*a^3*b^3*x^6 + 35*a^4*b^2*x^4 + a^5*b*x^2 - 2*a^6)*sq
rt(b*x^2 + a)/b^2

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Sympy [A]  time = 18.0402, size = 156, normalized size = 4.11 \begin{align*} \begin{cases} - \frac{2 a^{6} \sqrt{a + b x^{2}}}{143 b^{2}} + \frac{a^{5} x^{2} \sqrt{a + b x^{2}}}{143 b} + \frac{35 a^{4} x^{4} \sqrt{a + b x^{2}}}{143} + \frac{90 a^{3} b x^{6} \sqrt{a + b x^{2}}}{143} + \frac{100 a^{2} b^{2} x^{8} \sqrt{a + b x^{2}}}{143} + \frac{53 a b^{3} x^{10} \sqrt{a + b x^{2}}}{143} + \frac{b^{4} x^{12} \sqrt{a + b x^{2}}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-2*a**6*sqrt(a + b*x**2)/(143*b**2) + a**5*x**2*sqrt(a + b*x**2)/(143*b) + 35*a**4*x**4*sqrt(a + b*
x**2)/143 + 90*a**3*b*x**6*sqrt(a + b*x**2)/143 + 100*a**2*b**2*x**8*sqrt(a + b*x**2)/143 + 53*a*b**3*x**10*sq
rt(a + b*x**2)/143 + b**4*x**12*sqrt(a + b*x**2)/13, Ne(b, 0)), (a**(9/2)*x**4/4, True))

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Giac [B]  time = 2.4424, size = 406, normalized size = 10.68 \begin{align*} \frac{\frac{3003 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a^{4}}{b} + \frac{1716 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{3}}{b} + \frac{858 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{2}}{b} + \frac{52 \,{\left (315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}\right )} a}{b} + \frac{5 \,{\left (693 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5}\right )}}{b}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/45045*(3003*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*a^4/b + 1716*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a
)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*a^3/b + 858*(35*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2
+ a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*a^2/b + 52*(315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2
970*(b*x^2 + a)^(7/2)*a^2 - 2772*(b*x^2 + a)^(5/2)*a^3 + 1155*(b*x^2 + a)^(3/2)*a^4)*a/b + 5*(693*(b*x^2 + a)^
(13/2) - 4095*(b*x^2 + a)^(11/2)*a + 10010*(b*x^2 + a)^(9/2)*a^2 - 12870*(b*x^2 + a)^(7/2)*a^3 + 9009*(b*x^2 +
a)^(5/2)*a^4 - 3003*(b*x^2 + a)^(3/2)*a^5)/b)/b