3.582 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a^2*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^9*
(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a + b*x^2)) - (b^3*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 = 0.0439584, antiderivative size = 167, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2)) - (a^2*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^9*
(a + b*x^2)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*x^7*(a + b*x^2)) - (b^3*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 )^{3/2}}{x^{12}} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\left (a b+b^2 x^2\right )^3}{x^{12}} \, dx}{b^2 \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^3 b^3}{x^{12}}+\frac{3 a^2 b^4}{x^{10}}+\frac{3 a b^5}{x^8}+\frac{b^6}{x^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^9 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b^3 \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 = 0.0166941, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (385 a^2 b x^2+105 a^3+495 a b^2 x^4+231 b^3 x^6\right )}{1155 x^{11} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(105*a^3 + 385*a^2*b*x^2 + 495*a*b^2*x^4 + 231*b^3*x^6))/(1155*x^11*(a + b*x^2))

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Maple [A]  time = 0.16, size = 58, normalized size = 0.4 \begin{align*} -{\frac{231\,{b}^{3}{x}^{6}+495\,a{x}^{4}{b}^{2}+385\,{a}^{2}b{x}^{2}+105\,{a}^{3}}{1155\,{x}^{11} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/x^12,x)

[Out]

-1/1155*(231*b^3*x^6+495*a*b^2*x^4+385*a^2*b*x^2+105*a^3)*((b*x^2+a)^2)^(3/2)/x^11/(b*x^2+a)^3

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Maxima [A]  time = 1.00558, size = 50, normalized size = 0.3 \begin{align*} -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \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)^(3/2)/x^12,x, algorithm="maxima")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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Fricas [A]  time = 1.42585, size = 95, normalized size = 0.57 \begin{align*} -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \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)^(3/2)/x^12,x, algorithm="fricas")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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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{3}{2}}}{x^{12}}\, 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)**(3/2)/x**12,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**12, x)

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Giac [A]  time = 1.13327, size = 93, normalized size = 0.56 \begin{align*} -\frac{231 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 495 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 385 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 105 \, a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{1155 \, x^{11}} \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)^(3/2)/x^12,x, algorithm="giac")

[Out]

-1/1155*(231*b^3*x^6*sgn(b*x^2 + a) + 495*a*b^2*x^4*sgn(b*x^2 + a) + 385*a^2*b*x^2*sgn(b*x^2 + a) + 105*a^3*sg
n(b*x^2 + a))/x^11