3.614 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=249 \[ -\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*x^5*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*
(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4])/(a + b*x^2) + (5*a*b^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (b^5*x^5*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2))

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Rubi [A]  time = 0.0578224, antiderivative size = 249, 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0193039, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (150 a^2 b^3 x^6-150 a^3 b^2 x^4-25 a^4 b x^2-3 a^5+25 a b^4 x^8+3 b^5 x^{10}\right )}{15 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^6,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(-3*a^5 - 25*a^4*b*x^2 - 150*a^3*b^2*x^4 + 150*a^2*b^3*x^6 + 25*a*b^4*x^8 + 3*b^5*x^10))/
(15*x^5*(a + b*x^2))

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Maple [A]  time = 0.163, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-3\,{b}^{5}{x}^{10}-25\,a{b}^{4}{x}^{8}-150\,{a}^{2}{b}^{3}{x}^{6}+150\,{b}^{2}{a}^{3}{x}^{4}+25\,{a}^{4}b{x}^{2}+3\,{a}^{5}}{15\,{x}^{5} \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^6,x)

[Out]

-1/15*(-3*b^5*x^10-25*a*b^4*x^8-150*a^2*b^3*x^6+150*a^3*b^2*x^4+25*a^4*b*x^2+3*a^5)*((b*x^2+a)^2)^(5/2)/x^5/(b
*x^2+a)^5

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Maxima [A]  time = 1.01609, size = 80, normalized size = 0.32 \begin{align*} \frac{3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x^2 - 3*a^5)/x^5

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Fricas [A]  time = 1.34473, size = 131, normalized size = 0.53 \begin{align*} \frac{3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x^2 - 3*a^5)/x^5

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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^{6}}\, 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**6,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**6, x)

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Giac [A]  time = 1.11859, size = 143, normalized size = 0.57 \begin{align*} \frac{1}{5} \, b^{5} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{3} \, a b^{4} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 10 \, a^{2} b^{3} x \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{150 \, a^{3} b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 25 \, a^{4} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{15 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

1/5*b^5*x^5*sgn(b*x^2 + a) + 5/3*a*b^4*x^3*sgn(b*x^2 + a) + 10*a^2*b^3*x*sgn(b*x^2 + a) - 1/15*(150*a^3*b^2*x^
4*sgn(b*x^2 + a) + 25*a^4*b*x^2*sgn(b*x^2 + a) + 3*a^5*sgn(b*x^2 + a))/x^5