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]

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

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Rubi [A]  time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 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]]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.80, size = 23, normalized size = 0.25 \[ \frac {2 \, {\left (3 \, b x^{2} - a\right )} \sqrt {d x}}{3 \, d^{3} x^{2}} \]

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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giac [A]  time = 0.16, size = 42, normalized size = 0.46 \[ \frac {2 \, {\left (3 \, \sqrt {d x} b \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {a d \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {d x} x}\right )}}{3 \, d^{3}} \]

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

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maple [A]  time = 0.00, size = 37, normalized size = 0.41 \[ -\frac {2 \left (-3 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{3 \left (b \,x^{2}+a \right ) \left (d x \right )^{\frac {5}{2}}} \]

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 = 1.40, size = 24, normalized size = 0.26 \[ -\frac {2 \, {\left (\frac {a}{\left (d x\right )^{\frac {3}{2}}} - \frac {3 \, \sqrt {d x} b}{d^{2}}\right )}}{3 \, d} \]

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

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mupad [B]  time = 4.38, size = 53, normalized size = 0.58 \[ \frac {\left (\frac {2\,x^2}{d^2}-\frac {2\,a}{3\,b\,d^2}\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^3\,\sqrt {d\,x}+\frac {a\,x\,\sqrt {d\,x}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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