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

Optimal. Leaf size=91 \[ \frac{2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0279233, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1112, 14} \[ \frac{2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*Sqrt[d*x]*(a + b*x^2)) + (2*b*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2
*x^4])/(3*d^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{(d x)^{3/2}} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{a b+b^2 x^2}{(d x)^{3/2}} \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (\frac{a b}{(d x)^{3/2}}+\frac{b^2 \sqrt{d x}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )}+\frac{2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0143756, size = 43, normalized size = 0.47 \[ \frac{2 x \left (b x^2-3 a\right ) \sqrt{\left (a+b x^2\right )^2}}{3 (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 39, normalized size = 0.4 \begin{align*} -{\frac{2\, \left ( -b{x}^{2}+3\,a \right ) x}{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^2+a)^2)^(1/2)/(d*x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.01541, size = 32, normalized size = 0.35 \begin{align*} \frac{2 \,{\left (b \sqrt{d} x^{3} - 3 \, a \sqrt{d} x\right )}}{3 \, d^{2} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

2/3*(b*sqrt(d)*x^3 - 3*a*sqrt(d)*x)/(d^2*x^(3/2))

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Fricas [A]  time = 1.2116, size = 50, normalized size = 0.55 \begin{align*} \frac{2 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{d x}}{3 \, d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

2/3*(b*x^2 - 3*a)*sqrt(d*x)/(d^2*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**2+a)**2)**(1/2)/(d*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.25953, size = 55, normalized size = 0.6 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{d x} b x \mathrm{sgn}\left (b x^{2} + a\right )}{d} - \frac{3 \, a \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{d x}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(3/2),x, algorithm="giac")

[Out]

2/3*(sqrt(d*x)*b*x*sgn(b*x^2 + a)/d - 3*a*sgn(b*x^2 + a)/sqrt(d*x))/d