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 = 0.0808466, 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 \[ \frac{2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac{2 a \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]

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))

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Rubi in Sympy [A]  time = 40.7226, size = 75, normalized size = 0.82 \[ \frac{8 a \sqrt{d x} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{5 d \left (a + b x^{2}\right )} + \frac{2 \sqrt{d x} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**2+a)**2)**(1/2)/(d*x)**(1/2),x)

[Out]

8*a*sqrt(d*x)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(5*d*(a + b*x**2)) + 2*sqrt(d*
x)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(5*d)

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Mathematica [A]  time = 0.0197705, 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 = 0.004, size = 38, normalized size = 0.4 \[{\frac{2\, \left ( b{x}^{2}+5\,a \right ) x}{5\,b{x}^{2}+5\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{dx}}}} \]

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 = 0.709548, size = 32, normalized size = 0.35 \[ \frac{2 \,{\left (b \sqrt{d} x^{3} + 5 \, a \sqrt{d} x\right )}}{5 \, d \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/sqrt(d*x),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 = 0.256726, size = 26, normalized size = 0.29 \[ \frac{2 \,{\left (b x^{2} + 5 \, a\right )} \sqrt{d x}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/sqrt(d*x),x, algorithm="fricas")

[Out]

2/5*(b*x^2 + 5*a)*sqrt(d*x)/d

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

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]

Timed out

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GIAC/XCAS [A]  time = 0.262916, size = 54, normalized size = 0.59 \[ \frac{2 \,{\left (\sqrt{d x} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 5 \, \sqrt{d x} a{\rm sign}\left (b x^{2} + a\right )\right )}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/sqrt(d*x),x, algorithm="giac")

[Out]

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