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

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

[Out]

-1/11*a^5*(b*x^2+a)^(11/2)/b^6+5/13*a^4*(b*x^2+a)^(13/2)/b^6-2/3*a^3*(b*x^2+a)^(15/2)/b^6+10/17*a^2*(b*x^2+a)^
(17/2)/b^6-5/19*a*(b*x^2+a)^(19/2)/b^6+1/21*(b*x^2+a)^(21/2)/b^6

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Rubi [A]  time = 0.07, antiderivative size = 122, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.04, size = 72, normalized size = 0.59 \[ \frac {\left (a+b x^2\right )^{11/2} \left (-256 a^5+1408 a^4 b x^2-4576 a^3 b^2 x^4+11440 a^2 b^3 x^6-24310 a b^4 x^8+46189 b^5 x^{10}\right )}{969969 b^6} \]

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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fricas [A]  time = 0.78, size = 123, normalized size = 1.01 \[ \frac {{\left (46189 \, b^{10} x^{20} + 206635 \, a b^{9} x^{18} + 351780 \, a^{2} b^{8} x^{16} + 271414 \, a^{3} b^{7} x^{14} + 80773 \, a^{4} b^{6} x^{12} + 63 \, a^{5} b^{5} x^{10} - 70 \, a^{6} b^{4} x^{8} + 80 \, a^{7} b^{3} x^{6} - 96 \, a^{8} b^{2} x^{4} + 128 \, a^{9} b x^{2} - 256 \, a^{10}\right )} \sqrt {b x^{2} + a}}{969969 \, b^{6}} \]

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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giac [A]  time = 1.01, size = 85, normalized size = 0.70 \[ \frac {46189 \, {\left (b x^{2} + a\right )}^{\frac {21}{2}} - 255255 \, {\left (b x^{2} + a\right )}^{\frac {19}{2}} a + 570570 \, {\left (b x^{2} + a\right )}^{\frac {17}{2}} a^{2} - 646646 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} a^{3} + 373065 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a^{4} - 88179 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{5}}{969969 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/969969*(46189*(b*x^2 + a)^(21/2) - 255255*(b*x^2 + a)^(19/2)*a + 570570*(b*x^2 + a)^(17/2)*a^2 - 646646*(b*x
^2 + a)^(15/2)*a^3 + 373065*(b*x^2 + a)^(13/2)*a^4 - 88179*(b*x^2 + a)^(11/2)*a^5)/b^6

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maple [A]  time = 0.01, size = 69, normalized size = 0.57 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-46189 b^{5} x^{10}+24310 a \,b^{4} x^{8}-11440 a^{2} b^{3} x^{6}+4576 a^{3} b^{2} x^{4}-1408 a^{4} b \,x^{2}+256 a^{5}\right )}{969969 b^{6}} \]

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 [A]  time = 1.48, size = 113, normalized size = 0.93 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{10}}{21 \, b} - \frac {10 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{8}}{399 \, b^{2}} + \frac {80 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x^{6}}{6783 \, b^{3}} - \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3} x^{4}}{6783 \, b^{4}} + \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{4} x^{2}}{88179 \, b^{5}} - \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{5}}{969969 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(b*x^2 + a)^(11/2)*x^10/b - 10/399*(b*x^2 + a)^(11/2)*a*x^8/b^2 + 80/6783*(b*x^2 + a)^(11/2)*a^2*x^6/b^3
- 32/6783*(b*x^2 + a)^(11/2)*a^3*x^4/b^4 + 128/88179*(b*x^2 + a)^(11/2)*a^4*x^2/b^5 - 256/969969*(b*x^2 + a)^(
11/2)*a^5/b^6

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mupad [B]  time = 4.81, size = 119, normalized size = 0.98 \[ \sqrt {b\,x^2+a}\,\left (\frac {1049\,a^4\,x^{12}}{12597}-\frac {256\,a^{10}}{969969\,b^6}+\frac {b^4\,x^{20}}{21}+\frac {1898\,a^3\,b\,x^{14}}{6783}+\frac {85\,a\,b^3\,x^{18}}{399}+\frac {3\,a^5\,x^{10}}{46189\,b}-\frac {10\,a^6\,x^8}{138567\,b^2}+\frac {80\,a^7\,x^6}{969969\,b^3}-\frac {32\,a^8\,x^4}{323323\,b^4}+\frac {128\,a^9\,x^2}{969969\,b^5}+\frac {820\,a^2\,b^2\,x^{16}}{2261}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(1/2)*((1049*a^4*x^12)/12597 - (256*a^10)/(969969*b^6) + (b^4*x^20)/21 + (1898*a^3*b*x^14)/6783 +
(85*a*b^3*x^18)/399 + (3*a^5*x^10)/(46189*b) - (10*a^6*x^8)/(138567*b^2) + (80*a^7*x^6)/(969969*b^3) - (32*a^8
*x^4)/(323323*b^4) + (128*a^9*x^2)/(969969*b^5) + (820*a^2*b^2*x^16)/2261)

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sympy [A]  time = 62.85, size = 253, normalized size = 2.07 \[ \begin {cases} - \frac {256 a^{10} \sqrt {a + b x^{2}}}{969969 b^{6}} + \frac {128 a^{9} x^{2} \sqrt {a + b x^{2}}}{969969 b^{5}} - \frac {32 a^{8} x^{4} \sqrt {a + b x^{2}}}{323323 b^{4}} + \frac {80 a^{7} x^{6} \sqrt {a + b x^{2}}}{969969 b^{3}} - \frac {10 a^{6} x^{8} \sqrt {a + b x^{2}}}{138567 b^{2}} + \frac {3 a^{5} x^{10} \sqrt {a + b x^{2}}}{46189 b} + \frac {1049 a^{4} x^{12} \sqrt {a + b x^{2}}}{12597} + \frac {1898 a^{3} b x^{14} \sqrt {a + b x^{2}}}{6783} + \frac {820 a^{2} b^{2} x^{16} \sqrt {a + b x^{2}}}{2261} + \frac {85 a b^{3} x^{18} \sqrt {a + b x^{2}}}{399} + \frac {b^{4} x^{20} \sqrt {a + b x^{2}}}{21} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{12}}{12} & \text {otherwise} \end {cases} \]

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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