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

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

[Out]

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

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Rubi [A]  time = 0.0280979, 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} \[ \frac{2 b \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 \left (a+b x^2\right )}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0163486, size = 42, normalized size = 0.46 \[ -\frac{2 x \left (a-3 b x^2\right ) \sqrt{\left (a+b x^2\right )^2}}{3 (d x)^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 37, normalized size = 0.4 \begin{align*} -{\frac{2\, \left ( -3\,b{x}^{2}+a \right ) x}{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( dx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x*(-3*b*x^2+a)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)/(d*x)^(5/2)

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Maxima [A]  time = 1.0045, size = 34, normalized size = 0.37 \begin{align*} \frac{2 \,{\left (3 \, b \sqrt{d} x^{3} - a \sqrt{d} x\right )}}{3 \, d^{3} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.27042, size = 53, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (3 \, b x^{2} - a\right )} \sqrt{d x}}{3 \, d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/3*(3*b*x^2 - a)*sqrt(d*x)/(d^3*x^2)

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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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16802, size = 57, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d x} b \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{a d \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{d x} x}\right )}}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^2+a)^2)^(1/2)/(d*x)^(5/2),x, algorithm="giac")

[Out]

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