3.86 \(\int x^{11} (a+b x^2)^8 \, dx\)

Optimal. Leaf size=110 \[ -\frac {a^5 \left (a+b x^2\right )^9}{18 b^6}+\frac {a^4 \left (a+b x^2\right )^{10}}{4 b^6}-\frac {5 a^3 \left (a+b x^2\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^2\right )^{12}}{12 b^6}+\frac {\left (a+b x^2\right )^{14}}{28 b^6}-\frac {5 a \left (a+b x^2\right )^{13}}{26 b^6} \]

[Out]

-1/18*a^5*(b*x^2+a)^9/b^6+1/4*a^4*(b*x^2+a)^10/b^6-5/11*a^3*(b*x^2+a)^11/b^6+5/12*a^2*(b*x^2+a)^12/b^6-5/26*a*
(b*x^2+a)^13/b^6+1/28*(b*x^2+a)^14/b^6

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Rubi [A]  time = 0.17, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {5 a^2 \left (a+b x^2\right )^{12}}{12 b^6}-\frac {5 a^3 \left (a+b x^2\right )^{11}}{11 b^6}+\frac {a^4 \left (a+b x^2\right )^{10}}{4 b^6}-\frac {a^5 \left (a+b x^2\right )^9}{18 b^6}+\frac {\left (a+b x^2\right )^{14}}{28 b^6}-\frac {5 a \left (a+b x^2\right )^{13}}{26 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^2)^8,x]

[Out]

-(a^5*(a + b*x^2)^9)/(18*b^6) + (a^4*(a + b*x^2)^10)/(4*b^6) - (5*a^3*(a + b*x^2)^11)/(11*b^6) + (5*a^2*(a + b
*x^2)^12)/(12*b^6) - (5*a*(a + b*x^2)^13)/(26*b^6) + (a + b*x^2)^14/(28*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 )^8 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^5 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^5 (a+b x)^8}{b^5}+\frac {5 a^4 (a+b x)^9}{b^5}-\frac {10 a^3 (a+b x)^{10}}{b^5}+\frac {10 a^2 (a+b x)^{11}}{b^5}-\frac {5 a (a+b x)^{12}}{b^5}+\frac {(a+b x)^{13}}{b^5}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 \left (a+b x^2\right )^9}{18 b^6}+\frac {a^4 \left (a+b x^2\right )^{10}}{4 b^6}-\frac {5 a^3 \left (a+b x^2\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^2\right )^{12}}{12 b^6}-\frac {5 a \left (a+b x^2\right )^{13}}{26 b^6}+\frac {\left (a+b x^2\right )^{14}}{28 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 108, normalized size = 0.98 \[ \frac {a^8 x^{12}}{12}+\frac {4}{7} a^7 b x^{14}+\frac {7}{4} a^6 b^2 x^{16}+\frac {28}{9} a^5 b^3 x^{18}+\frac {7}{2} a^4 b^4 x^{20}+\frac {28}{11} a^3 b^5 x^{22}+\frac {7}{6} a^2 b^6 x^{24}+\frac {4}{13} a b^7 x^{26}+\frac {b^8 x^{28}}{28} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^2)^8,x]

[Out]

(a^8*x^12)/12 + (4*a^7*b*x^14)/7 + (7*a^6*b^2*x^16)/4 + (28*a^5*b^3*x^18)/9 + (7*a^4*b^4*x^20)/2 + (28*a^3*b^5
*x^22)/11 + (7*a^2*b^6*x^24)/6 + (4*a*b^7*x^26)/13 + (b^8*x^28)/28

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fricas [A]  time = 0.60, size = 90, normalized size = 0.82 \[ \frac {1}{28} x^{28} b^{8} + \frac {4}{13} x^{26} b^{7} a + \frac {7}{6} x^{24} b^{6} a^{2} + \frac {28}{11} x^{22} b^{5} a^{3} + \frac {7}{2} x^{20} b^{4} a^{4} + \frac {28}{9} x^{18} b^{3} a^{5} + \frac {7}{4} x^{16} b^{2} a^{6} + \frac {4}{7} x^{14} b a^{7} + \frac {1}{12} x^{12} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/28*x^28*b^8 + 4/13*x^26*b^7*a + 7/6*x^24*b^6*a^2 + 28/11*x^22*b^5*a^3 + 7/2*x^20*b^4*a^4 + 28/9*x^18*b^3*a^5
+ 7/4*x^16*b^2*a^6 + 4/7*x^14*b*a^7 + 1/12*x^12*a^8

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giac [A]  time = 1.00, size = 90, normalized size = 0.82 \[ \frac {1}{28} \, b^{8} x^{28} + \frac {4}{13} \, a b^{7} x^{26} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {28}{11} \, a^{3} b^{5} x^{22} + \frac {7}{2} \, a^{4} b^{4} x^{20} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {7}{4} \, a^{6} b^{2} x^{16} + \frac {4}{7} \, a^{7} b x^{14} + \frac {1}{12} \, a^{8} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/28*b^8*x^28 + 4/13*a*b^7*x^26 + 7/6*a^2*b^6*x^24 + 28/11*a^3*b^5*x^22 + 7/2*a^4*b^4*x^20 + 28/9*a^5*b^3*x^18
+ 7/4*a^6*b^2*x^16 + 4/7*a^7*b*x^14 + 1/12*a^8*x^12

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maple [A]  time = 0.00, size = 91, normalized size = 0.83 \[ \frac {1}{28} b^{8} x^{28}+\frac {4}{13} a \,b^{7} x^{26}+\frac {7}{6} a^{2} b^{6} x^{24}+\frac {28}{11} a^{3} b^{5} x^{22}+\frac {7}{2} a^{4} b^{4} x^{20}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {7}{4} a^{6} b^{2} x^{16}+\frac {4}{7} a^{7} b \,x^{14}+\frac {1}{12} a^{8} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^2+a)^8,x)

[Out]

1/28*b^8*x^28+4/13*a*b^7*x^26+7/6*a^2*b^6*x^24+28/11*a^3*b^5*x^22+7/2*a^4*b^4*x^20+28/9*a^5*b^3*x^18+7/4*a^6*b
^2*x^16+4/7*a^7*b*x^14+1/12*a^8*x^12

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maxima [A]  time = 1.35, size = 90, normalized size = 0.82 \[ \frac {1}{28} \, b^{8} x^{28} + \frac {4}{13} \, a b^{7} x^{26} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {28}{11} \, a^{3} b^{5} x^{22} + \frac {7}{2} \, a^{4} b^{4} x^{20} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {7}{4} \, a^{6} b^{2} x^{16} + \frac {4}{7} \, a^{7} b x^{14} + \frac {1}{12} \, a^{8} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/28*b^8*x^28 + 4/13*a*b^7*x^26 + 7/6*a^2*b^6*x^24 + 28/11*a^3*b^5*x^22 + 7/2*a^4*b^4*x^20 + 28/9*a^5*b^3*x^18
+ 7/4*a^6*b^2*x^16 + 4/7*a^7*b*x^14 + 1/12*a^8*x^12

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mupad [B]  time = 4.57, size = 90, normalized size = 0.82 \[ \frac {a^8\,x^{12}}{12}+\frac {4\,a^7\,b\,x^{14}}{7}+\frac {7\,a^6\,b^2\,x^{16}}{4}+\frac {28\,a^5\,b^3\,x^{18}}{9}+\frac {7\,a^4\,b^4\,x^{20}}{2}+\frac {28\,a^3\,b^5\,x^{22}}{11}+\frac {7\,a^2\,b^6\,x^{24}}{6}+\frac {4\,a\,b^7\,x^{26}}{13}+\frac {b^8\,x^{28}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^2)^8,x)

[Out]

(a^8*x^12)/12 + (b^8*x^28)/28 + (4*a^7*b*x^14)/7 + (4*a*b^7*x^26)/13 + (7*a^6*b^2*x^16)/4 + (28*a^5*b^3*x^18)/
9 + (7*a^4*b^4*x^20)/2 + (28*a^3*b^5*x^22)/11 + (7*a^2*b^6*x^24)/6

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sympy [A]  time = 0.09, size = 107, normalized size = 0.97 \[ \frac {a^{8} x^{12}}{12} + \frac {4 a^{7} b x^{14}}{7} + \frac {7 a^{6} b^{2} x^{16}}{4} + \frac {28 a^{5} b^{3} x^{18}}{9} + \frac {7 a^{4} b^{4} x^{20}}{2} + \frac {28 a^{3} b^{5} x^{22}}{11} + \frac {7 a^{2} b^{6} x^{24}}{6} + \frac {4 a b^{7} x^{26}}{13} + \frac {b^{8} x^{28}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**2+a)**8,x)

[Out]

a**8*x**12/12 + 4*a**7*b*x**14/7 + 7*a**6*b**2*x**16/4 + 28*a**5*b**3*x**18/9 + 7*a**4*b**4*x**20/2 + 28*a**3*
b**5*x**22/11 + 7*a**2*b**6*x**24/6 + 4*a*b**7*x**26/13 + b**8*x**28/28

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