3.409 \(\int x^{15} (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=161 \[ \frac{a^2 \left (a+b x^2\right )^{21/2}}{b^8}-\frac{35 a^3 \left (a+b x^2\right )^{19/2}}{19 b^8}+\frac{35 a^4 \left (a+b x^2\right )^{17/2}}{17 b^8}-\frac{7 a^5 \left (a+b x^2\right )^{15/2}}{5 b^8}+\frac{7 a^6 \left (a+b x^2\right )^{13/2}}{13 b^8}-\frac{a^7 \left (a+b x^2\right )^{11/2}}{11 b^8}+\frac{\left (a+b x^2\right )^{25/2}}{25 b^8}-\frac{7 a \left (a+b x^2\right )^{23/2}}{23 b^8} \]

[Out]

-(a^7*(a + b*x^2)^(11/2))/(11*b^8) + (7*a^6*(a + b*x^2)^(13/2))/(13*b^8) - (7*a^5*(a + b*x^2)^(15/2))/(5*b^8)
+ (35*a^4*(a + b*x^2)^(17/2))/(17*b^8) - (35*a^3*(a + b*x^2)^(19/2))/(19*b^8) + (a^2*(a + b*x^2)^(21/2))/b^8 -
(7*a*(a + b*x^2)^(23/2))/(23*b^8) + (a + b*x^2)^(25/2)/(25*b^8)

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Rubi [A]  time = 0.0999879, antiderivative size = 161, 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} \[ \frac{a^2 \left (a+b x^2\right )^{21/2}}{b^8}-\frac{35 a^3 \left (a+b x^2\right )^{19/2}}{19 b^8}+\frac{35 a^4 \left (a+b x^2\right )^{17/2}}{17 b^8}-\frac{7 a^5 \left (a+b x^2\right )^{15/2}}{5 b^8}+\frac{7 a^6 \left (a+b x^2\right )^{13/2}}{13 b^8}-\frac{a^7 \left (a+b x^2\right )^{11/2}}{11 b^8}+\frac{\left (a+b x^2\right )^{25/2}}{25 b^8}-\frac{7 a \left (a+b x^2\right )^{23/2}}{23 b^8} \]

Antiderivative was successfully verified.

[In]

Int[x^15*(a + b*x^2)^(9/2),x]

[Out]

-(a^7*(a + b*x^2)^(11/2))/(11*b^8) + (7*a^6*(a + b*x^2)^(13/2))/(13*b^8) - (7*a^5*(a + b*x^2)^(15/2))/(5*b^8)
+ (35*a^4*(a + b*x^2)^(17/2))/(17*b^8) - (35*a^3*(a + b*x^2)^(19/2))/(19*b^8) + (a^2*(a + b*x^2)^(21/2))/b^8 -
(7*a*(a + b*x^2)^(23/2))/(23*b^8) + (a + b*x^2)^(25/2)/(25*b^8)

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^{15} \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^7 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^7 (a+b x)^{9/2}}{b^7}+\frac{7 a^6 (a+b x)^{11/2}}{b^7}-\frac{21 a^5 (a+b x)^{13/2}}{b^7}+\frac{35 a^4 (a+b x)^{15/2}}{b^7}-\frac{35 a^3 (a+b x)^{17/2}}{b^7}+\frac{21 a^2 (a+b x)^{19/2}}{b^7}-\frac{7 a (a+b x)^{21/2}}{b^7}+\frac{(a+b x)^{23/2}}{b^7}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^7 \left (a+b x^2\right )^{11/2}}{11 b^8}+\frac{7 a^6 \left (a+b x^2\right )^{13/2}}{13 b^8}-\frac{7 a^5 \left (a+b x^2\right )^{15/2}}{5 b^8}+\frac{35 a^4 \left (a+b x^2\right )^{17/2}}{17 b^8}-\frac{35 a^3 \left (a+b x^2\right )^{19/2}}{19 b^8}+\frac{a^2 \left (a+b x^2\right )^{21/2}}{b^8}-\frac{7 a \left (a+b x^2\right )^{23/2}}{23 b^8}+\frac{\left (a+b x^2\right )^{25/2}}{25 b^8}\\ \end{align*}

Mathematica [A]  time = 0.0568496, size = 94, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^{11/2} \left (369512 a^2 b^5 x^{10}-194480 a^3 b^4 x^8+91520 a^4 b^3 x^6-36608 a^5 b^2 x^4+11264 a^6 b x^2-2048 a^7-646646 a b^6 x^{12}+1062347 b^7 x^{14}\right )}{26558675 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(-2048*a^7 + 11264*a^6*b*x^2 - 36608*a^5*b^2*x^4 + 91520*a^4*b^3*x^6 - 194480*a^3*b^4*x^8
+ 369512*a^2*b^5*x^10 - 646646*a*b^6*x^12 + 1062347*b^7*x^14))/(26558675*b^8)

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Maple [A]  time = 0.006, size = 91, normalized size = 0.6 \begin{align*} -{\frac{-1062347\,{x}^{14}{b}^{7}+646646\,a{x}^{12}{b}^{6}-369512\,{a}^{2}{x}^{10}{b}^{5}+194480\,{a}^{3}{x}^{8}{b}^{4}-91520\,{a}^{4}{x}^{6}{b}^{3}+36608\,{a}^{5}{x}^{4}{b}^{2}-11264\,{a}^{6}{x}^{2}b+2048\,{a}^{7}}{26558675\,{b}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(b*x^2+a)^(9/2),x)

[Out]

-1/26558675*(b*x^2+a)^(11/2)*(-1062347*b^7*x^14+646646*a*b^6*x^12-369512*a^2*b^5*x^10+194480*a^3*b^4*x^8-91520
*a^4*b^3*x^6+36608*a^5*b^2*x^4-11264*a^6*b*x^2+2048*a^7)/b^8

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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^15*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83018, size = 386, normalized size = 2.4 \begin{align*} \frac{{\left (1062347 \, b^{12} x^{24} + 4665089 \, a b^{11} x^{22} + 7759752 \, a^{2} b^{10} x^{20} + 5810090 \, a^{3} b^{9} x^{18} + 1659515 \, a^{4} b^{8} x^{16} + 429 \, a^{5} b^{7} x^{14} - 462 \, a^{6} b^{6} x^{12} + 504 \, a^{7} b^{5} x^{10} - 560 \, a^{8} b^{4} x^{8} + 640 \, a^{9} b^{3} x^{6} - 768 \, a^{10} b^{2} x^{4} + 1024 \, a^{11} b x^{2} - 2048 \, a^{12}\right )} \sqrt{b x^{2} + a}}{26558675 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/26558675*(1062347*b^12*x^24 + 4665089*a*b^11*x^22 + 7759752*a^2*b^10*x^20 + 5810090*a^3*b^9*x^18 + 1659515*a
^4*b^8*x^16 + 429*a^5*b^7*x^14 - 462*a^6*b^6*x^12 + 504*a^7*b^5*x^10 - 560*a^8*b^4*x^8 + 640*a^9*b^3*x^6 - 768
*a^10*b^2*x^4 + 1024*a^11*b*x^2 - 2048*a^12)*sqrt(b*x^2 + a)/b^8

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Sympy [A]  time = 99.6531, size = 301, normalized size = 1.87 \begin{align*} \begin{cases} - \frac{2048 a^{12} \sqrt{a + b x^{2}}}{26558675 b^{8}} + \frac{1024 a^{11} x^{2} \sqrt{a + b x^{2}}}{26558675 b^{7}} - \frac{768 a^{10} x^{4} \sqrt{a + b x^{2}}}{26558675 b^{6}} + \frac{128 a^{9} x^{6} \sqrt{a + b x^{2}}}{5311735 b^{5}} - \frac{112 a^{8} x^{8} \sqrt{a + b x^{2}}}{5311735 b^{4}} + \frac{504 a^{7} x^{10} \sqrt{a + b x^{2}}}{26558675 b^{3}} - \frac{42 a^{6} x^{12} \sqrt{a + b x^{2}}}{2414425 b^{2}} + \frac{3 a^{5} x^{14} \sqrt{a + b x^{2}}}{185725 b} + \frac{2321 a^{4} x^{16} \sqrt{a + b x^{2}}}{37145} + \frac{478 a^{3} b x^{18} \sqrt{a + b x^{2}}}{2185} + \frac{168 a^{2} b^{2} x^{20} \sqrt{a + b x^{2}}}{575} + \frac{101 a b^{3} x^{22} \sqrt{a + b x^{2}}}{575} + \frac{b^{4} x^{24} \sqrt{a + b x^{2}}}{25} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{16}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-2048*a**12*sqrt(a + b*x**2)/(26558675*b**8) + 1024*a**11*x**2*sqrt(a + b*x**2)/(26558675*b**7) - 7
68*a**10*x**4*sqrt(a + b*x**2)/(26558675*b**6) + 128*a**9*x**6*sqrt(a + b*x**2)/(5311735*b**5) - 112*a**8*x**8
*sqrt(a + b*x**2)/(5311735*b**4) + 504*a**7*x**10*sqrt(a + b*x**2)/(26558675*b**3) - 42*a**6*x**12*sqrt(a + b*
x**2)/(2414425*b**2) + 3*a**5*x**14*sqrt(a + b*x**2)/(185725*b) + 2321*a**4*x**16*sqrt(a + b*x**2)/37145 + 478
*a**3*b*x**18*sqrt(a + b*x**2)/2185 + 168*a**2*b**2*x**20*sqrt(a + b*x**2)/575 + 101*a*b**3*x**22*sqrt(a + b*x
**2)/575 + b**4*x**24*sqrt(a + b*x**2)/25, Ne(b, 0)), (a**(9/2)*x**16/16, True))

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Giac [B]  time = 1.74369, size = 973, normalized size = 6.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/1673196525*(15295*(6435*(b*x^2 + a)^(17/2) - 51051*(b*x^2 + a)^(15/2)*a + 176715*(b*x^2 + a)^(13/2)*a^2 - 34
8075*(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^4/b^7 + 3220*(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 - 1293
2920*(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^3/b^7 + 2070*(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 - 24942060*(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)*a^2/b^7 + 300*(969969*(b*x^2 + a)^(23/2) - 10623470*(b*x^2 + a)^(21/2)*a + 52837785*(b*
x^2 + a)^(19/2)*a^2 - 157477320*(b*x^2 + a)^(17/2)*a^3 + 312330018*(b*x^2 + a)^(15/2)*a^4 - 432456948*(b*x^2 +
a)^(13/2)*a^5 + 425904570*(b*x^2 + a)^(11/2)*a^6 - 297457160*(b*x^2 + a)^(9/2)*a^7 + 143416845*(b*x^2 + a)^(7
/2)*a^8 - 44618574*(b*x^2 + a)^(5/2)*a^9 + 7436429*(b*x^2 + a)^(3/2)*a^10)*a/b^7 + 33*(2028117*(b*x^2 + a)^(25
/2) - 24249225*(b*x^2 + a)^(23/2)*a + 132793375*(b*x^2 + a)^(21/2)*a^2 - 440314875*(b*x^2 + a)^(19/2)*a^3 + 98
4233250*(b*x^2 + a)^(17/2)*a^4 - 1561650090*(b*x^2 + a)^(15/2)*a^5 + 1801903950*(b*x^2 + a)^(13/2)*a^6 - 15210
87750*(b*x^2 + a)^(11/2)*a^7 + 929553625*(b*x^2 + a)^(9/2)*a^8 - 398380125*(b*x^2 + a)^(7/2)*a^9 + 111546435*(
b*x^2 + a)^(5/2)*a^10 - 16900975*(b*x^2 + a)^(3/2)*a^11)/b^7)/b