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

Optimal. Leaf size=247 \[ \frac{b^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0581297, antiderivative size = 247, 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} \[ \frac{b^5 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{5 a b^4 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{10 a^3 b^2 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{5 a^4 b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0203504, size = 83, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (126 a^2 b^3 x^6+210 a^3 b^2 x^4+315 a^4 b x^2-63 a^5+45 a b^4 x^8+7 b^5 x^{10}\right )}{63 x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x^2)^2]*(-63*a^5 + 315*a^4*b*x^2 + 210*a^3*b^2*x^4 + 126*a^2*b^3*x^6 + 45*a*b^4*x^8 + 7*b^5*x^10)
)/(63*x*(a + b*x^2))

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Maple [A]  time = 0.171, size = 80, normalized size = 0.3 \begin{align*} -{\frac{-7\,{b}^{5}{x}^{10}-45\,a{b}^{4}{x}^{8}-126\,{a}^{2}{b}^{3}{x}^{6}-210\,{b}^{2}{a}^{3}{x}^{4}-315\,{a}^{4}b{x}^{2}+63\,{a}^{5}}{63\,x \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^2,x)

[Out]

-1/63*(-7*b^5*x^10-45*a*b^4*x^8-126*a^2*b^3*x^6-210*a^3*b^2*x^4-315*a^4*b*x^2+63*a^5)*((b*x^2+a)^2)^(5/2)/x/(b
*x^2+a)^5

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Maxima [A]  time = 1.00605, size = 80, normalized size = 0.32 \begin{align*} \frac{7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \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^2,x, algorithm="maxima")

[Out]

1/63*(7*b^5*x^10 + 45*a*b^4*x^8 + 126*a^2*b^3*x^6 + 210*a^3*b^2*x^4 + 315*a^4*b*x^2 - 63*a^5)/x

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Fricas [A]  time = 1.18887, size = 131, normalized size = 0.53 \begin{align*} \frac{7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \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^2,x, algorithm="fricas")

[Out]

1/63*(7*b^5*x^10 + 45*a*b^4*x^8 + 126*a^2*b^3*x^6 + 210*a^3*b^2*x^4 + 315*a^4*b*x^2 - 63*a^5)/x

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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^{2}}\, 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**2,x)

[Out]

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

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Giac [A]  time = 1.11001, size = 139, normalized size = 0.56 \begin{align*} \frac{1}{9} \, b^{5} x^{9} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{5}{7} \, a b^{4} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 2 \, a^{2} b^{3} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 5 \, a^{4} b x \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{x} \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^2,x, algorithm="giac")

[Out]

1/9*b^5*x^9*sgn(b*x^2 + a) + 5/7*a*b^4*x^7*sgn(b*x^2 + a) + 2*a^2*b^3*x^5*sgn(b*x^2 + a) + 10/3*a^3*b^2*x^3*sg
n(b*x^2 + a) + 5*a^4*b*x*sgn(b*x^2 + a) - a^5*sgn(b*x^2 + a)/x