3.729 \(\int (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

[Out]

(2*a*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (2*b*(d*x)
^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^3*(a + b*x^2))

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Rubi [A]  time = 0.0834996, antiderivative size = 93, 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 \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (2*b*(d*x)
^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^3*(a + b*x^2))

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Rubi in Sympy [A]  time = 23.8377, size = 75, normalized size = 0.81 \[ \frac{8 a \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{45 d \left (a + b x^{2}\right )} + \frac{2 \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{9 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a*(d*x)**(5/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(45*d*(a + b*x**2)) + 2*(d*
x)**(5/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(9*d)

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Mathematica [A]  time = 0.0231847, size = 44, normalized size = 0.47 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (9 a+5 b x^2\right )}{45 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(9*a + 5*b*x^2))/(45*(a + b*x^2))

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Maple [A]  time = 0.004, size = 39, normalized size = 0.4 \[{\frac{2\, \left ( 5\,b{x}^{2}+9\,a \right ) x}{45\,b{x}^{2}+45\,a} \left ( dx \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*x*(5*b*x^2+9*a)*(d*x)^(3/2)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [A]  time = 0.70954, size = 30, normalized size = 0.32 \[ \frac{2}{45} \,{\left (5 \, b d^{\frac{3}{2}} x^{3} + 9 \, a d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \]

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]

2/45*(5*b*d^(3/2)*x^3 + 9*a*d^(3/2)*x)*x^(3/2)

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Fricas [A]  time = 0.25838, size = 30, normalized size = 0.32 \[ \frac{2}{45} \,{\left (5 \, b d x^{4} + 9 \, a d x^{2}\right )} \sqrt{d x} \]

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]

2/45*(5*b*d*x^4 + 9*a*d*x^2)*sqrt(d*x)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265489, size = 55, normalized size = 0.59 \[ \frac{2}{9} \, \sqrt{d x} b d x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a d x^{2}{\rm sign}\left (b x^{2} + a\right ) \]

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]

2/9*sqrt(d*x)*b*d*x^4*sign(b*x^2 + a) + 2/5*sqrt(d*x)*a*d*x^2*sign(b*x^2 + a)