3.411 \(\int x^{11} (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=122 \[ \frac{10 a^2 \left (a+b x^2\right )^{17/2}}{17 b^6}-\frac{2 a^3 \left (a+b x^2\right )^{15/2}}{3 b^6}+\frac{5 a^4 \left (a+b x^2\right )^{13/2}}{13 b^6}-\frac{a^5 \left (a+b x^2\right )^{11/2}}{11 b^6}+\frac{\left (a+b x^2\right )^{21/2}}{21 b^6}-\frac{5 a \left (a+b x^2\right )^{19/2}}{19 b^6} \]

[Out]

-(a^5*(a + b*x^2)^(11/2))/(11*b^6) + (5*a^4*(a + b*x^2)^(13/2))/(13*b^6) - (2*a^3*(a + b*x^2)^(15/2))/(3*b^6)
+ (10*a^2*(a + b*x^2)^(17/2))/(17*b^6) - (5*a*(a + b*x^2)^(19/2))/(19*b^6) + (a + b*x^2)^(21/2)/(21*b^6)

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Rubi [A]  time = 0.0705596, antiderivative size = 122, 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} \[ \frac{10 a^2 \left (a+b x^2\right )^{17/2}}{17 b^6}-\frac{2 a^3 \left (a+b x^2\right )^{15/2}}{3 b^6}+\frac{5 a^4 \left (a+b x^2\right )^{13/2}}{13 b^6}-\frac{a^5 \left (a+b x^2\right )^{11/2}}{11 b^6}+\frac{\left (a+b x^2\right )^{21/2}}{21 b^6}-\frac{5 a \left (a+b x^2\right )^{19/2}}{19 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^2)^(9/2),x]

[Out]

-(a^5*(a + b*x^2)^(11/2))/(11*b^6) + (5*a^4*(a + b*x^2)^(13/2))/(13*b^6) - (2*a^3*(a + b*x^2)^(15/2))/(3*b^6)
+ (10*a^2*(a + b*x^2)^(17/2))/(17*b^6) - (5*a*(a + b*x^2)^(19/2))/(19*b^6) + (a + b*x^2)^(21/2)/(21*b^6)

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^{11} \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^5 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^{9/2}}{b^5}+\frac{5 a^4 (a+b x)^{11/2}}{b^5}-\frac{10 a^3 (a+b x)^{13/2}}{b^5}+\frac{10 a^2 (a+b x)^{15/2}}{b^5}-\frac{5 a (a+b x)^{17/2}}{b^5}+\frac{(a+b x)^{19/2}}{b^5}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 \left (a+b x^2\right )^{11/2}}{11 b^6}+\frac{5 a^4 \left (a+b x^2\right )^{13/2}}{13 b^6}-\frac{2 a^3 \left (a+b x^2\right )^{15/2}}{3 b^6}+\frac{10 a^2 \left (a+b x^2\right )^{17/2}}{17 b^6}-\frac{5 a \left (a+b x^2\right )^{19/2}}{19 b^6}+\frac{\left (a+b x^2\right )^{21/2}}{21 b^6}\\ \end{align*}

Mathematica [A]  time = 0.040828, size = 72, normalized size = 0.59 \[ \frac{\left (a+b x^2\right )^{11/2} \left (11440 a^2 b^3 x^6-4576 a^3 b^2 x^4+1408 a^4 b x^2-256 a^5-24310 a b^4 x^8+46189 b^5 x^{10}\right )}{969969 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(-256*a^5 + 1408*a^4*b*x^2 - 4576*a^3*b^2*x^4 + 11440*a^2*b^3*x^6 - 24310*a*b^4*x^8 + 4618
9*b^5*x^10))/(969969*b^6)

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Maple [A]  time = 0.005, size = 69, normalized size = 0.6 \begin{align*} -{\frac{-46189\,{b}^{5}{x}^{10}+24310\,a{b}^{4}{x}^{8}-11440\,{a}^{2}{b}^{3}{x}^{6}+4576\,{a}^{3}{b}^{2}{x}^{4}-1408\,{a}^{4}b{x}^{2}+256\,{a}^{5}}{969969\,{b}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^2+a)^(9/2),x)

[Out]

-1/969969*(b*x^2+a)^(11/2)*(-46189*b^5*x^10+24310*a*b^4*x^8-11440*a^2*b^3*x^6+4576*a^3*b^2*x^4-1408*a^4*b*x^2+
256*a^5)/b^6

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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^11*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72267, size = 309, normalized size = 2.53 \begin{align*} \frac{{\left (46189 \, b^{10} x^{20} + 206635 \, a b^{9} x^{18} + 351780 \, a^{2} b^{8} x^{16} + 271414 \, a^{3} b^{7} x^{14} + 80773 \, a^{4} b^{6} x^{12} + 63 \, a^{5} b^{5} x^{10} - 70 \, a^{6} b^{4} x^{8} + 80 \, a^{7} b^{3} x^{6} - 96 \, a^{8} b^{2} x^{4} + 128 \, a^{9} b x^{2} - 256 \, a^{10}\right )} \sqrt{b x^{2} + a}}{969969 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/969969*(46189*b^10*x^20 + 206635*a*b^9*x^18 + 351780*a^2*b^8*x^16 + 271414*a^3*b^7*x^14 + 80773*a^4*b^6*x^12
+ 63*a^5*b^5*x^10 - 70*a^6*b^4*x^8 + 80*a^7*b^3*x^6 - 96*a^8*b^2*x^4 + 128*a^9*b*x^2 - 256*a^10)*sqrt(b*x^2 +
a)/b^6

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Sympy [A]  time = 62.4883, size = 253, normalized size = 2.07 \begin{align*} \begin{cases} - \frac{256 a^{10} \sqrt{a + b x^{2}}}{969969 b^{6}} + \frac{128 a^{9} x^{2} \sqrt{a + b x^{2}}}{969969 b^{5}} - \frac{32 a^{8} x^{4} \sqrt{a + b x^{2}}}{323323 b^{4}} + \frac{80 a^{7} x^{6} \sqrt{a + b x^{2}}}{969969 b^{3}} - \frac{10 a^{6} x^{8} \sqrt{a + b x^{2}}}{138567 b^{2}} + \frac{3 a^{5} x^{10} \sqrt{a + b x^{2}}}{46189 b} + \frac{1049 a^{4} x^{12} \sqrt{a + b x^{2}}}{12597} + \frac{1898 a^{3} b x^{14} \sqrt{a + b x^{2}}}{6783} + \frac{820 a^{2} b^{2} x^{16} \sqrt{a + b x^{2}}}{2261} + \frac{85 a b^{3} x^{18} \sqrt{a + b x^{2}}}{399} + \frac{b^{4} x^{20} \sqrt{a + b x^{2}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-256*a**10*sqrt(a + b*x**2)/(969969*b**6) + 128*a**9*x**2*sqrt(a + b*x**2)/(969969*b**5) - 32*a**8*
x**4*sqrt(a + b*x**2)/(323323*b**4) + 80*a**7*x**6*sqrt(a + b*x**2)/(969969*b**3) - 10*a**6*x**8*sqrt(a + b*x*
*2)/(138567*b**2) + 3*a**5*x**10*sqrt(a + b*x**2)/(46189*b) + 1049*a**4*x**12*sqrt(a + b*x**2)/12597 + 1898*a*
*3*b*x**14*sqrt(a + b*x**2)/6783 + 820*a**2*b**2*x**16*sqrt(a + b*x**2)/2261 + 85*a*b**3*x**18*sqrt(a + b*x**2
)/399 + b**4*x**20*sqrt(a + b*x**2)/21, Ne(b, 0)), (a**(9/2)*x**12/12, True))

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Giac [B]  time = 2.42774, size = 784, normalized size = 6.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/14549535*(1615*(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)*a^4/b^5 + 1292*(3003*(b*x^2 + a)
^(15/2) - 20790*(b*x^2 + a)^(13/2)*a + 61425*(b*x^2 + a)^(11/2)*a^2 - 100100*(b*x^2 + a)^(9/2)*a^3 + 96525*(b*
x^2 + a)^(7/2)*a^4 - 54054*(b*x^2 + a)^(5/2)*a^5 + 15015*(b*x^2 + a)^(3/2)*a^6)*a^3/b^5 + 798*(6435*(b*x^2 + a
)^(17/2) - 51051*(b*x^2 + a)^(15/2)*a + 176715*(b*x^2 + a)^(13/2)*a^2 - 348075*(b*x^2 + a)^(11/2)*a^3 + 425425
*(b*x^2 + a)^(9/2)*a^4 - 328185*(b*x^2 + a)^(7/2)*a^5 + 153153*(b*x^2 + a)^(5/2)*a^6 - 36465*(b*x^2 + a)^(3/2)
*a^7)*a^2/b^5 + 28*(109395*(b*x^2 + a)^(19/2) - 978120*(b*x^2 + a)^(17/2)*a + 3879876*(b*x^2 + a)^(15/2)*a^2 -
8953560*(b*x^2 + a)^(13/2)*a^3 + 13226850*(b*x^2 + a)^(11/2)*a^4 - 12932920*(b*x^2 + a)^(9/2)*a^5 + 8314020*(
b*x^2 + a)^(7/2)*a^6 - 3325608*(b*x^2 + a)^(5/2)*a^7 + 692835*(b*x^2 + a)^(3/2)*a^8)*a/b^5 + 3*(230945*(b*x^2
+ a)^(21/2) - 2297295*(b*x^2 + a)^(19/2)*a + 10270260*(b*x^2 + a)^(17/2)*a^2 - 27159132*(b*x^2 + a)^(15/2)*a^3
+ 47006190*(b*x^2 + a)^(13/2)*a^4 - 55552770*(b*x^2 + a)^(11/2)*a^5 + 45265220*(b*x^2 + a)^(9/2)*a^6 - 249420
60*(b*x^2 + a)^(7/2)*a^7 + 8729721*(b*x^2 + a)^(5/2)*a^8 - 1616615*(b*x^2 + a)^(3/2)*a^9)/b^5)/b