3.621 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{20}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0582526, antiderivative size = 255, normalized size of antiderivative = 1., 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left (a+b x^2\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^20,x]

[Out]

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

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]

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]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{20}} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\left (a b+b^2 x^2\right )^5}{x^{20}} \, 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 (\frac{a^5 b^5}{x^{20}}+\frac{5 a^4 b^6}{x^{18}}+\frac{10 a^3 b^7}{x^{16}}+\frac{10 a^2 b^8}{x^{14}}+\frac{5 a b^9}{x^{12}}+\frac{b^{10}}{x^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left (a+b x^2\right )}-\frac{2 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0202025, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (319770 a^2 b^3 x^6+277134 a^3 b^2 x^4+122265 a^4 b x^2+21879 a^5+188955 a b^4 x^8+46189 b^5 x^{10}\right )}{415701 x^{19} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^20,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(21879*a^5 + 122265*a^4*b*x^2 + 277134*a^3*b^2*x^4 + 319770*a^2*b^3*x^6 + 188955*a*b^4*x
^8 + 46189*b^5*x^10))/(415701*x^19*(a + b*x^2))

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Maple [A]  time = 0.171, size = 80, normalized size = 0.3 \begin{align*} -{\frac{46189\,{b}^{5}{x}^{10}+188955\,a{b}^{4}{x}^{8}+319770\,{a}^{2}{b}^{3}{x}^{6}+277134\,{b}^{2}{a}^{3}{x}^{4}+122265\,{a}^{4}b{x}^{2}+21879\,{a}^{5}}{415701\,{x}^{19} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^20,x)

[Out]

-1/415701*(46189*b^5*x^10+188955*a*b^4*x^8+319770*a^2*b^3*x^6+277134*a^3*b^2*x^4+122265*a^4*b*x^2+21879*a^5)*(
(b*x^2+a)^2)^(5/2)/x^19/(b*x^2+a)^5

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Maxima [A]  time = 0.99646, size = 80, normalized size = 0.31 \begin{align*} -\frac{46189 \, b^{5} x^{10} + 188955 \, a b^{4} x^{8} + 319770 \, a^{2} b^{3} x^{6} + 277134 \, a^{3} b^{2} x^{4} + 122265 \, a^{4} b x^{2} + 21879 \, a^{5}}{415701 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^20,x, algorithm="maxima")

[Out]

-1/415701*(46189*b^5*x^10 + 188955*a*b^4*x^8 + 319770*a^2*b^3*x^6 + 277134*a^3*b^2*x^4 + 122265*a^4*b*x^2 + 21
879*a^5)/x^19

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Fricas [A]  time = 1.25834, size = 169, normalized size = 0.66 \begin{align*} -\frac{46189 \, b^{5} x^{10} + 188955 \, a b^{4} x^{8} + 319770 \, a^{2} b^{3} x^{6} + 277134 \, a^{3} b^{2} x^{4} + 122265 \, a^{4} b x^{2} + 21879 \, a^{5}}{415701 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^20,x, algorithm="fricas")

[Out]

-1/415701*(46189*b^5*x^10 + 188955*a*b^4*x^8 + 319770*a^2*b^3*x^6 + 277134*a^3*b^2*x^4 + 122265*a^4*b*x^2 + 21
879*a^5)/x^19

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{20}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**20,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**20, x)

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Giac [A]  time = 1.1368, size = 144, normalized size = 0.56 \begin{align*} -\frac{46189 \, b^{5} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + 188955 \, a b^{4} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 319770 \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 277134 \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 122265 \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 21879 \, a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{415701 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^20,x, algorithm="giac")

[Out]

-1/415701*(46189*b^5*x^10*sgn(b*x^2 + a) + 188955*a*b^4*x^8*sgn(b*x^2 + a) + 319770*a^2*b^3*x^6*sgn(b*x^2 + a)
+ 277134*a^3*b^2*x^4*sgn(b*x^2 + a) + 122265*a^4*b*x^2*sgn(b*x^2 + a) + 21879*a^5*sgn(b*x^2 + a))/x^19