3.607 \(\int x^8 (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac {b^5 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {a^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {10 a^3 b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )} \]

[Out]

1/9*a^5*x^9*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+5/11*a^4*b*x^11*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+10/13*a^3*b^2*x^13*((b
*x^2+a)^2)^(1/2)/(b*x^2+a)+2/3*a^2*b^3*x^15*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+5/17*a*b^4*x^17*((b*x^2+a)^2)^(1/2)/
(b*x^2+a)+1/19*b^5*x^19*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Rubi [A]  time = 0.06, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1112, 270} \[ \frac {b^5 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {10 a^3 b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {a^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*x^9*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*(a + b*x^2)) + (5*a^4*b*x^11*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11
*(a + b*x^2)) + (10*a^3*b^2*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a + b*x^2)) + (2*a^2*b^3*x^15*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (5*a*b^4*x^17*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*(a + b*x^2)) + (b
^5*x^19*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*(a + b*x^2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^8 \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 x^8+5 a^4 b^6 x^{10}+10 a^3 b^7 x^{12}+10 a^2 b^8 x^{14}+5 a b^9 x^{16}+b^{10} x^{18}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^5 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {10 a^3 b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {2 a^2 b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac {5 a b^4 x^{17} \sqrt {a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac {b^5 x^{19} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 83, normalized size = 0.33 \[ \frac {x^9 \sqrt {\left (a+b x^2\right )^2} \left (46189 a^5+188955 a^4 b x^2+319770 a^3 b^2 x^4+277134 a^2 b^3 x^6+122265 a b^4 x^8+21879 b^5 x^{10}\right )}{415701 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^9*Sqrt[(a + b*x^2)^2]*(46189*a^5 + 188955*a^4*b*x^2 + 319770*a^3*b^2*x^4 + 277134*a^2*b^3*x^6 + 122265*a*b^
4*x^8 + 21879*b^5*x^10))/(415701*(a + b*x^2))

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fricas [A]  time = 1.14, size = 57, normalized size = 0.22 \[ \frac {1}{19} \, b^{5} x^{19} + \frac {5}{17} \, a b^{4} x^{17} + \frac {2}{3} \, a^{2} b^{3} x^{15} + \frac {10}{13} \, a^{3} b^{2} x^{13} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{9} \, a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/19*b^5*x^19 + 5/17*a*b^4*x^17 + 2/3*a^2*b^3*x^15 + 10/13*a^3*b^2*x^13 + 5/11*a^4*b*x^11 + 1/9*a^5*x^9

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giac [A]  time = 0.16, size = 105, normalized size = 0.41 \[ \frac {1}{19} \, b^{5} x^{19} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{17} \, a b^{4} x^{17} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, a^{2} b^{3} x^{15} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {10}{13} \, a^{3} b^{2} x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{11} \, a^{4} b x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{9} \, a^{5} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/19*b^5*x^19*sgn(b*x^2 + a) + 5/17*a*b^4*x^17*sgn(b*x^2 + a) + 2/3*a^2*b^3*x^15*sgn(b*x^2 + a) + 10/13*a^3*b^
2*x^13*sgn(b*x^2 + a) + 5/11*a^4*b*x^11*sgn(b*x^2 + a) + 1/9*a^5*x^9*sgn(b*x^2 + a)

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maple [A]  time = 0.01, size = 80, normalized size = 0.31 \[ \frac {\left (21879 b^{5} x^{10}+122265 a \,b^{4} x^{8}+277134 a^{2} b^{3} x^{6}+319770 a^{3} b^{2} x^{4}+188955 a^{4} b \,x^{2}+46189 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{9}}{415701 \left (b \,x^{2}+a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/415701*x^9*(21879*b^5*x^10+122265*a*b^4*x^8+277134*a^2*b^3*x^6+319770*a^3*b^2*x^4+188955*a^4*b*x^2+46189*a^5
)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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maxima [A]  time = 1.33, size = 57, normalized size = 0.22 \[ \frac {1}{19} \, b^{5} x^{19} + \frac {5}{17} \, a b^{4} x^{17} + \frac {2}{3} \, a^{2} b^{3} x^{15} + \frac {10}{13} \, a^{3} b^{2} x^{13} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{9} \, a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/19*b^5*x^19 + 5/17*a*b^4*x^17 + 2/3*a^2*b^3*x^15 + 10/13*a^3*b^2*x^13 + 5/11*a^4*b*x^11 + 1/9*a^5*x^9

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^8\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int(x^8*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{8} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**8*((a + b*x**2)**2)**(5/2), x)

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