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

Optimal. Leaf size=91 \[ \frac{2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac{2 a \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0283024, 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} \[ \frac{2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac{2 a \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(a + b*x^2)) + (2*b*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*
x^4])/(5*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}}{\sqrt{d x}} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{a b+b^2 x^2}{\sqrt{d x}} \, 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}{\sqrt{d x}}+\frac{b^2 (d x)^{3/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{2 a \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac{2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0125833, size = 43, normalized size = 0.47 \[ \frac{2 \sqrt{\left (a+b x^2\right )^2} \left (5 a x+b x^3\right )}{5 \sqrt{d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x^2)^2]*(5*a*x + b*x^3))/(5*Sqrt[d*x]*(a + b*x^2))

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Maple [A]  time = 0.042, size = 38, normalized size = 0.4 \begin{align*}{\frac{2\, \left ( b{x}^{2}+5\,a \right ) x}{5\,b{x}^{2}+5\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x*(b*x^2+5*a)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/(d*x)^(1/2)

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Maxima [A]  time = 1.04161, size = 32, normalized size = 0.35 \begin{align*} \frac{2 \,{\left (b \sqrt{d} x^{3} + 5 \, a \sqrt{d} x\right )}}{5 \, d \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*sqrt(d)*x^3 + 5*a*sqrt(d)*x)/(d*sqrt(x))

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Fricas [A]  time = 1.30979, size = 42, normalized size = 0.46 \begin{align*} \frac{2 \,{\left (b x^{2} + 5 \, a\right )} \sqrt{d x}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x^2 + 5*a)*sqrt(d*x)/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\left (a + b x^{2}\right )^{2}}}{\sqrt{d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((a + b*x**2)**2)/sqrt(d*x), x)

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Giac [A]  time = 1.21156, size = 54, normalized size = 0.59 \begin{align*} \frac{2 \,{\left (\sqrt{d x} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 5 \, \sqrt{d x} a \mathrm{sgn}\left (b x^{2} + a\right )\right )}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(sqrt(d*x)*b*x^2*sgn(b*x^2 + a) + 5*sqrt(d*x)*a*sgn(b*x^2 + a))/d