3.619 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{16}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 x^{15} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*x^15*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x
^13*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (10*a^2*b^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(9*x^9*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a + b*x^2)) - (
b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2))

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Rubi [A]  time = 0.0580007, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 x^{15} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^16,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*x^15*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x
^13*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (10*a^2*b^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(9*x^9*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a + b*x^2)) - (
b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(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^{16}} \, 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^{16}} \, 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^{16}}+\frac{5 a^4 b^6}{x^{14}}+\frac{10 a^3 b^7}{x^{12}}+\frac{10 a^2 b^8}{x^{10}}+\frac{5 a b^9}{x^8}+\frac{b^{10}}{x^6}\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}}{15 x^{15} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0175777, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (50050 a^2 b^3 x^6+40950 a^3 b^2 x^4+17325 a^4 b x^2+3003 a^5+32175 a b^4 x^8+9009 b^5 x^{10}\right )}{45045 x^{15} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^16,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(3003*a^5 + 17325*a^4*b*x^2 + 40950*a^3*b^2*x^4 + 50050*a^2*b^3*x^6 + 32175*a*b^4*x^8 +
9009*b^5*x^10))/(45045*x^15*(a + b*x^2))

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Maple [A]  time = 0.17, size = 80, normalized size = 0.3 \begin{align*} -{\frac{9009\,{b}^{5}{x}^{10}+32175\,a{b}^{4}{x}^{8}+50050\,{a}^{2}{b}^{3}{x}^{6}+40950\,{b}^{2}{a}^{3}{x}^{4}+17325\,{a}^{4}b{x}^{2}+3003\,{a}^{5}}{45045\,{x}^{15} \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^16,x)

[Out]

-1/45045*(9009*b^5*x^10+32175*a*b^4*x^8+50050*a^2*b^3*x^6+40950*a^3*b^2*x^4+17325*a^4*b*x^2+3003*a^5)*((b*x^2+
a)^2)^(5/2)/x^15/(b*x^2+a)^5

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Maxima [A]  time = 1.00314, size = 80, normalized size = 0.31 \begin{align*} -\frac{9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \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^16,x, algorithm="maxima")

[Out]

-1/45045*(9009*b^5*x^10 + 32175*a*b^4*x^8 + 50050*a^2*b^3*x^6 + 40950*a^3*b^2*x^4 + 17325*a^4*b*x^2 + 3003*a^5
)/x^15

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Fricas [A]  time = 1.36302, size = 159, normalized size = 0.62 \begin{align*} -\frac{9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \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^16,x, algorithm="fricas")

[Out]

-1/45045*(9009*b^5*x^10 + 32175*a*b^4*x^8 + 50050*a^2*b^3*x^6 + 40950*a^3*b^2*x^4 + 17325*a^4*b*x^2 + 3003*a^5
)/x^15

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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^{16}}\, 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**16,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**16, x)

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Giac [A]  time = 1.1232, size = 144, normalized size = 0.56 \begin{align*} -\frac{9009 \, b^{5} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + 32175 \, a b^{4} x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 50050 \, a^{2} b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 40950 \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 17325 \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 3003 \, a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{45045 \, x^{15}} \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^16,x, algorithm="giac")

[Out]

-1/45045*(9009*b^5*x^10*sgn(b*x^2 + a) + 32175*a*b^4*x^8*sgn(b*x^2 + a) + 50050*a^2*b^3*x^6*sgn(b*x^2 + a) + 4
0950*a^3*b^2*x^4*sgn(b*x^2 + a) + 17325*a^4*b*x^2*sgn(b*x^2 + a) + 3003*a^5*sgn(b*x^2 + a))/x^15