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

Optimal. Leaf size=101 \[ \frac{2 a^2 \left (a+b x^2\right )^{15/2}}{5 b^5}-\frac{4 a^3 \left (a+b x^2\right )^{13/2}}{13 b^5}+\frac{a^4 \left (a+b x^2\right )^{11/2}}{11 b^5}+\frac{\left (a+b x^2\right )^{19/2}}{19 b^5}-\frac{4 a \left (a+b x^2\right )^{17/2}}{17 b^5} \]

[Out]

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

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Rubi [A]  time = 0.0575089, antiderivative size = 101, 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} \[ \frac{2 a^2 \left (a+b x^2\right )^{15/2}}{5 b^5}-\frac{4 a^3 \left (a+b x^2\right )^{13/2}}{13 b^5}+\frac{a^4 \left (a+b x^2\right )^{11/2}}{11 b^5}+\frac{\left (a+b x^2\right )^{19/2}}{19 b^5}-\frac{4 a \left (a+b x^2\right )^{17/2}}{17 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^9*(a + b*x^2)^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0334344, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^2\right )^{11/2} \left (2288 a^2 b^2 x^4-704 a^3 b x^2+128 a^4-5720 a b^3 x^6+12155 b^4 x^8\right )}{230945 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(128*a^4 - 704*a^3*b*x^2 + 2288*a^2*b^2*x^4 - 5720*a*b^3*x^6 + 12155*b^4*x^8))/(230945*b^5
)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.6 \begin{align*}{\frac{12155\,{x}^{8}{b}^{4}-5720\,a{x}^{6}{b}^{3}+2288\,{a}^{2}{x}^{4}{b}^{2}-704\,{a}^{3}{x}^{2}b+128\,{a}^{4}}{230945\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/230945*(b*x^2+a)^(11/2)*(12155*b^4*x^8-5720*a*b^3*x^6+2288*a^2*b^2*x^4-704*a^3*b*x^2+128*a^4)/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6932, size = 277, normalized size = 2.74 \begin{align*} \frac{{\left (12155 \, b^{9} x^{18} + 55055 \, a b^{8} x^{16} + 95238 \, a^{2} b^{7} x^{14} + 75086 \, a^{3} b^{6} x^{12} + 23063 \, a^{4} b^{5} x^{10} + 35 \, a^{5} b^{4} x^{8} - 40 \, a^{6} b^{3} x^{6} + 48 \, a^{7} b^{2} x^{4} - 64 \, a^{8} b x^{2} + 128 \, a^{9}\right )} \sqrt{b x^{2} + a}}{230945 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/230945*(12155*b^9*x^18 + 55055*a*b^8*x^16 + 95238*a^2*b^7*x^14 + 75086*a^3*b^6*x^12 + 23063*a^4*b^5*x^10 + 3
5*a^5*b^4*x^8 - 40*a^6*b^3*x^6 + 48*a^7*b^2*x^4 - 64*a^8*b*x^2 + 128*a^9)*sqrt(b*x^2 + a)/b^5

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Sympy [A]  time = 46.2677, size = 230, normalized size = 2.28 \begin{align*} \begin{cases} \frac{128 a^{9} \sqrt{a + b x^{2}}}{230945 b^{5}} - \frac{64 a^{8} x^{2} \sqrt{a + b x^{2}}}{230945 b^{4}} + \frac{48 a^{7} x^{4} \sqrt{a + b x^{2}}}{230945 b^{3}} - \frac{8 a^{6} x^{6} \sqrt{a + b x^{2}}}{46189 b^{2}} + \frac{7 a^{5} x^{8} \sqrt{a + b x^{2}}}{46189 b} + \frac{23063 a^{4} x^{10} \sqrt{a + b x^{2}}}{230945} + \frac{6826 a^{3} b x^{12} \sqrt{a + b x^{2}}}{20995} + \frac{666 a^{2} b^{2} x^{14} \sqrt{a + b x^{2}}}{1615} + \frac{77 a b^{3} x^{16} \sqrt{a + b x^{2}}}{323} + \frac{b^{4} x^{18} \sqrt{a + b x^{2}}}{19} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{10}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((128*a**9*sqrt(a + b*x**2)/(230945*b**5) - 64*a**8*x**2*sqrt(a + b*x**2)/(230945*b**4) + 48*a**7*x**
4*sqrt(a + b*x**2)/(230945*b**3) - 8*a**6*x**6*sqrt(a + b*x**2)/(46189*b**2) + 7*a**5*x**8*sqrt(a + b*x**2)/(4
6189*b) + 23063*a**4*x**10*sqrt(a + b*x**2)/230945 + 6826*a**3*b*x**12*sqrt(a + b*x**2)/20995 + 666*a**2*b**2*
x**14*sqrt(a + b*x**2)/1615 + 77*a*b**3*x**16*sqrt(a + b*x**2)/323 + b**4*x**18*sqrt(a + b*x**2)/19, Ne(b, 0))
, (a**(9/2)*x**10/10, True))

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Giac [B]  time = 2.28212, size = 690, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14549535*(4199*(315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(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^4/b^4 + 6460*(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^3/b^4 + 1938*(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^2/b^4 + 532*(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/b^4 + 7*(109395*(b*x^2 + a)^(19/2) - 978120*(b*x^2 + a)^(1
7/2)*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 - 1
2932920*(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)/b^4)/b