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 = 0.0808946, 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 \[ \frac{2 b (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \sqrt{d x} \left (a+b x^2\right )} \]

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))

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Rubi in Sympy [A]  time = 23.1746, size = 75, normalized size = 0.82 \[ - \frac{8 a \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3 d \sqrt{d x} \left (a + b x^{2}\right )} + \frac{2 \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3 d \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**2+a)**2)**(1/2)/(d*x)**(3/2),x)

[Out]

-8*a*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(3*d*sqrt(d*x)*(a + b*x**2)) + 2*sqrt(a
**2 + 2*a*b*x**2 + b**2*x**4)/(3*d*sqrt(d*x))

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Mathematica [A]  time = 0.0277656, 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 = 0.004, size = 39, normalized size = 0.4 \[ -{\frac{2\, \left ( -b{x}^{2}+3\,a \right ) x}{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( dx \right ) ^{-{\frac{3}{2}}}} \]

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 = 0.720404, size = 32, normalized size = 0.35 \[ \frac{2 \,{\left (b \sqrt{d} x^{3} - 3 \, a \sqrt{d} x\right )}}{3 \, d^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^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 = 0.258759, size = 26, normalized size = 0.29 \[ \frac{2 \,{\left (b x^{2} - 3 \, a\right )}}{3 \, \sqrt{d x} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

2/3*(b*x^2 - 3*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.26651, size = 55, normalized size = 0.6 \[ \frac{2 \,{\left (\frac{\sqrt{d x} b x{\rm sign}\left (b x^{2} + a\right )}{d} - \frac{3 \, a{\rm sign}\left (b x^{2} + a\right )}{\sqrt{d x}}\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/(d*x)^(3/2),x, algorithm="giac")

[Out]

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