3.410 \(\int x^{13} (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=140 \[ \frac {a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}-\frac {6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac {a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac {20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}+\frac {\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac {2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

[Out]

1/11*a^6*(b*x^2+a)^(11/2)/b^7-6/13*a^5*(b*x^2+a)^(13/2)/b^7+a^4*(b*x^2+a)^(15/2)/b^7-20/17*a^3*(b*x^2+a)^(17/2
)/b^7+15/19*a^2*(b*x^2+a)^(19/2)/b^7-2/7*a*(b*x^2+a)^(21/2)/b^7+1/23*(b*x^2+a)^(23/2)/b^7

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Rubi [A]  time = 0.08, antiderivative size = 140, 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} \[ \frac {15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}-\frac {20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac {a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac {6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac {a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}+\frac {\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac {2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^13*(a + b*x^2)^(9/2),x]

[Out]

(a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(a + b*x^2)^(15/2))/b^7 - (20*a
^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a
+ b*x^2)^(23/2)/(23*b^7)

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^{13} \left (a+b x^2\right )^{9/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^6 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^6 (a+b x)^{9/2}}{b^6}-\frac {6 a^5 (a+b x)^{11/2}}{b^6}+\frac {15 a^4 (a+b x)^{13/2}}{b^6}-\frac {20 a^3 (a+b x)^{15/2}}{b^6}+\frac {15 a^2 (a+b x)^{17/2}}{b^6}-\frac {6 a (a+b x)^{19/2}}{b^6}+\frac {(a+b x)^{21/2}}{b^6}\right ) \, dx,x,x^2\right )\\ &=\frac {a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}-\frac {6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac {a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac {20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac {15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}-\frac {2 a \left (a+b x^2\right )^{21/2}}{7 b^7}+\frac {\left (a+b x^2\right )^{23/2}}{23 b^7}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 83, normalized size = 0.59 \[ \frac {\left (a+b x^2\right )^{11/2} \left (1024 a^6-5632 a^5 b x^2+18304 a^4 b^2 x^4-45760 a^3 b^3 x^6+97240 a^2 b^4 x^8-184756 a b^5 x^{10}+323323 b^6 x^{12}\right )}{7436429 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(1024*a^6 - 5632*a^5*b*x^2 + 18304*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 97240*a^2*b^4*x^8 - 1
84756*a*b^5*x^10 + 323323*b^6*x^12))/(7436429*b^7)

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fricas [A]  time = 0.90, size = 134, normalized size = 0.96 \[ \frac {{\left (323323 \, b^{11} x^{22} + 1431859 \, a b^{10} x^{20} + 2406690 \, a^{2} b^{9} x^{18} + 1826110 \, a^{3} b^{8} x^{16} + 530959 \, a^{4} b^{7} x^{14} + 231 \, a^{5} b^{6} x^{12} - 252 \, a^{6} b^{5} x^{10} + 280 \, a^{7} b^{4} x^{8} - 320 \, a^{8} b^{3} x^{6} + 384 \, a^{9} b^{2} x^{4} - 512 \, a^{10} b x^{2} + 1024 \, a^{11}\right )} \sqrt {b x^{2} + a}}{7436429 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/7436429*(323323*b^11*x^22 + 1431859*a*b^10*x^20 + 2406690*a^2*b^9*x^18 + 1826110*a^3*b^8*x^16 + 530959*a^4*b
^7*x^14 + 231*a^5*b^6*x^12 - 252*a^6*b^5*x^10 + 280*a^7*b^4*x^8 - 320*a^8*b^3*x^6 + 384*a^9*b^2*x^4 - 512*a^10
*b*x^2 + 1024*a^11)*sqrt(b*x^2 + a)/b^7

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giac [A]  time = 1.10, size = 99, normalized size = 0.71 \[ \frac {323323 \, {\left (b x^{2} + a\right )}^{\frac {23}{2}} - 2124694 \, {\left (b x^{2} + a\right )}^{\frac {21}{2}} a + 5870865 \, {\left (b x^{2} + a\right )}^{\frac {19}{2}} a^{2} - 8748740 \, {\left (b x^{2} + a\right )}^{\frac {17}{2}} a^{3} + 7436429 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} a^{4} - 3432198 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a^{5} + 676039 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{6}}{7436429 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/7436429*(323323*(b*x^2 + a)^(23/2) - 2124694*(b*x^2 + a)^(21/2)*a + 5870865*(b*x^2 + a)^(19/2)*a^2 - 8748740
*(b*x^2 + a)^(17/2)*a^3 + 7436429*(b*x^2 + a)^(15/2)*a^4 - 3432198*(b*x^2 + a)^(13/2)*a^5 + 676039*(b*x^2 + a)
^(11/2)*a^6)/b^7

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maple [A]  time = 0.01, size = 80, normalized size = 0.57 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (323323 x^{12} b^{6}-184756 a \,x^{10} b^{5}+97240 a^{2} x^{8} b^{4}-45760 a^{3} x^{6} b^{3}+18304 a^{4} x^{4} b^{2}-5632 a^{5} x^{2} b +1024 a^{6}\right )}{7436429 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13*(b*x^2+a)^(9/2),x)

[Out]

1/7436429*(b*x^2+a)^(11/2)*(323323*b^6*x^12-184756*a*b^5*x^10+97240*a^2*b^4*x^8-45760*a^3*b^3*x^6+18304*a^4*b^
2*x^4-5632*a^5*b*x^2+1024*a^6)/b^7

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maxima [A]  time = 1.45, size = 133, normalized size = 0.95 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{12}}{23 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{10}}{161 \, b^{2}} + \frac {40 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x^{8}}{3059 \, b^{3}} - \frac {320 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3} x^{6}}{52003 \, b^{4}} + \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{4} x^{4}}{52003 \, b^{5}} - \frac {512 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{5} x^{2}}{676039 \, b^{6}} + \frac {1024 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{6}}{7436429 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

1/23*(b*x^2 + a)^(11/2)*x^12/b - 4/161*(b*x^2 + a)^(11/2)*a*x^10/b^2 + 40/3059*(b*x^2 + a)^(11/2)*a^2*x^8/b^3
- 320/52003*(b*x^2 + a)^(11/2)*a^3*x^6/b^4 + 128/52003*(b*x^2 + a)^(11/2)*a^4*x^4/b^5 - 512/676039*(b*x^2 + a)
^(11/2)*a^5*x^2/b^6 + 1024/7436429*(b*x^2 + a)^(11/2)*a^6/b^7

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mupad [B]  time = 4.84, size = 130, normalized size = 0.93 \[ \sqrt {b\,x^2+a}\,\left (\frac {1024\,a^{11}}{7436429\,b^7}+\frac {3713\,a^4\,x^{14}}{52003}+\frac {b^4\,x^{22}}{23}+\frac {12770\,a^3\,b\,x^{16}}{52003}+\frac {31\,a\,b^3\,x^{20}}{161}+\frac {3\,a^5\,x^{12}}{96577\,b}-\frac {36\,a^6\,x^{10}}{1062347\,b^2}+\frac {40\,a^7\,x^8}{1062347\,b^3}-\frac {320\,a^8\,x^6}{7436429\,b^4}+\frac {384\,a^9\,x^4}{7436429\,b^5}-\frac {512\,a^{10}\,x^2}{7436429\,b^6}+\frac {990\,a^2\,b^2\,x^{18}}{3059}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(1/2)*((1024*a^11)/(7436429*b^7) + (3713*a^4*x^14)/52003 + (b^4*x^22)/23 + (12770*a^3*b*x^16)/5200
3 + (31*a*b^3*x^20)/161 + (3*a^5*x^12)/(96577*b) - (36*a^6*x^10)/(1062347*b^2) + (40*a^7*x^8)/(1062347*b^3) -
(320*a^8*x^6)/(7436429*b^4) + (384*a^9*x^4)/(7436429*b^5) - (512*a^10*x^2)/(7436429*b^6) + (990*a^2*b^2*x^18)/
3059)

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sympy [A]  time = 79.37, size = 277, normalized size = 1.98 \[ \begin {cases} \frac {1024 a^{11} \sqrt {a + b x^{2}}}{7436429 b^{7}} - \frac {512 a^{10} x^{2} \sqrt {a + b x^{2}}}{7436429 b^{6}} + \frac {384 a^{9} x^{4} \sqrt {a + b x^{2}}}{7436429 b^{5}} - \frac {320 a^{8} x^{6} \sqrt {a + b x^{2}}}{7436429 b^{4}} + \frac {40 a^{7} x^{8} \sqrt {a + b x^{2}}}{1062347 b^{3}} - \frac {36 a^{6} x^{10} \sqrt {a + b x^{2}}}{1062347 b^{2}} + \frac {3 a^{5} x^{12} \sqrt {a + b x^{2}}}{96577 b} + \frac {3713 a^{4} x^{14} \sqrt {a + b x^{2}}}{52003} + \frac {12770 a^{3} b x^{16} \sqrt {a + b x^{2}}}{52003} + \frac {990 a^{2} b^{2} x^{18} \sqrt {a + b x^{2}}}{3059} + \frac {31 a b^{3} x^{20} \sqrt {a + b x^{2}}}{161} + \frac {b^{4} x^{22} \sqrt {a + b x^{2}}}{23} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{14}}{14} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((1024*a**11*sqrt(a + b*x**2)/(7436429*b**7) - 512*a**10*x**2*sqrt(a + b*x**2)/(7436429*b**6) + 384*a
**9*x**4*sqrt(a + b*x**2)/(7436429*b**5) - 320*a**8*x**6*sqrt(a + b*x**2)/(7436429*b**4) + 40*a**7*x**8*sqrt(a
+ b*x**2)/(1062347*b**3) - 36*a**6*x**10*sqrt(a + b*x**2)/(1062347*b**2) + 3*a**5*x**12*sqrt(a + b*x**2)/(965
77*b) + 3713*a**4*x**14*sqrt(a + b*x**2)/52003 + 12770*a**3*b*x**16*sqrt(a + b*x**2)/52003 + 990*a**2*b**2*x**
18*sqrt(a + b*x**2)/3059 + 31*a*b**3*x**20*sqrt(a + b*x**2)/161 + b**4*x**22*sqrt(a + b*x**2)/23, Ne(b, 0)), (
a**(9/2)*x**14/14, True))

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