3.672 \(\int \frac{x^3}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=52 \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0446219, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4194, size = 51, normalized size = 0.98 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{b^{\frac{3}{2}} \sqrt{d}} + \frac{\sqrt{d x^{2}}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0257385, size = 44, normalized size = 0.85 \[ \frac{x \left (\sqrt{b} x-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{b^{3/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 38, normalized size = 0.7 \[{\frac{x}{b} \left ( x\sqrt{ab}-a\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222269, size = 1, normalized size = 0.02 \[ \left [\frac{d \sqrt{-\frac{a}{b d}} \log \left (-\frac{2 \, b d x^{2} \sqrt{-\frac{a}{b d}} -{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \, \sqrt{d x^{2}}}{2 \, b d}, -\frac{d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}}}{d \sqrt{\frac{a}{b d}}}\right ) - \sqrt{d x^{2}}}{b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(d*sqrt(-a/(b*d))*log(-(2*b*d*x^2*sqrt(-a/(b*d)) - (b*x^2 - a)*sqrt(d*x^2))
/(b*x^3 + a*x)) + 2*sqrt(d*x^2))/(b*d), -(d*sqrt(a/(b*d))*arctan(sqrt(d*x^2)/(d*
sqrt(a/(b*d)))) - sqrt(d*x^2))/(b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236749, size = 62, normalized size = 1.19 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b} - \frac{\sqrt{d x^{2}}}{b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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