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

Optimal. Leaf size=161 \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{3 a b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0403986, antiderivative size = 161, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{3 a b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0131741, size = 59, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (9 a^2 b x^2+a^3-9 a b^2 x^4-b^3 x^6\right )}{3 x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(a + b*x^2)^2]*(a^3 + 9*a^2*b*x^2 - 9*a*b^2*x^4 - b^3*x^6))/(3*x^3*(a + b*x^2))

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Maple [A]  time = 0.166, size = 56, normalized size = 0.4 \begin{align*} -{\frac{-{b}^{3}{x}^{6}-9\,a{x}^{4}{b}^{2}+9\,{a}^{2}b{x}^{2}+{a}^{3}}{3\,{x}^{3} \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^4,x)

[Out]

-1/3*(-b^3*x^6-9*a*b^2*x^4+9*a^2*b*x^2+a^3)*((b*x^2+a)^2)^(3/2)/x^3/(b*x^2+a)^3

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Maxima [A]  time = 1.01653, size = 49, normalized size = 0.3 \begin{align*} \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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Fricas [A]  time = 1.45455, size = 72, normalized size = 0.45 \begin{align*} \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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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^{4}}\, 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**4,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**4, x)

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Giac [A]  time = 1.13596, size = 90, normalized size = 0.56 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, a b^{2} x \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{9 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/3*b^3*x^3*sgn(b*x^2 + a) + 3*a*b^2*x*sgn(b*x^2 + a) - 1/3*(9*a^2*b*x^2*sgn(b*x^2 + a) + a^3*sgn(b*x^2 + a))/
x^3