3.671 \(\int \frac{x^5}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=72 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.066319, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2} \sqrt{d x^2}}-\frac{a x^2}{b^2 \sqrt{d x^2}}+\frac{x^4}{3 b \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.3482, size = 70, normalized size = 0.97 \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{b^{\frac{5}{2}} \sqrt{d}} - \frac{a \sqrt{d x^{2}}}{b^{2} d} + \frac{\left (d x^{2}\right )^{\frac{3}{2}}}{3 b d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)/(d*x**2)**(1/2),x)

[Out]

a**(3/2)*atan(sqrt(b)*sqrt(d*x**2)/(sqrt(a)*sqrt(d)))/(b**(5/2)*sqrt(d)) - a*sqr
t(d*x**2)/(b**2*d) + (d*x**2)**(3/2)/(3*b*d**2)

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Mathematica [A]  time = 0.040234, size = 56, normalized size = 0.78 \[ \frac{x \left (3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 53, normalized size = 0.7 \[{\frac{x}{3\,{b}^{2}} \left ( \sqrt{ab}{x}^{3}b-3\,\sqrt{ab}xa+3\,{a}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*((a*b)^(1/2)*x^3*b-3*(a*b)^(1/2)*x*a+3*a^2*arctan(x*b/(a*b)^(1/2)))/(d*x^2
)^(1/2)/b^2/(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^5/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220001, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a d \sqrt{-\frac{a}{b d}} \log \left (\frac{2 \, b d x^{2} \sqrt{-\frac{a}{b d}} +{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{6 \, b^{2} d}, \frac{3 \, a d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}}}{d \sqrt{\frac{a}{b d}}}\right ) +{\left (b x^{2} - 3 \, a\right )} \sqrt{d x^{2}}}{3 \, b^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="fricas")

[Out]

[1/6*(3*a*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*(b*x^2 - 3*a)*sqrt(d*x^2))/(b^2*d), 1/3*(3*a*d*sqrt(a/(b*
d))*arctan(sqrt(d*x^2)/(d*sqrt(a/(b*d)))) + (b*x^2 - 3*a)*sqrt(d*x^2))/(b^2*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)/(d*x**2)**(1/2),x)

[Out]

Integral(x**5/(sqrt(d*x**2)*(a + b*x**2)), x)

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GIAC/XCAS [A]  time = 0.236454, size = 95, normalized size = 1.32 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b^{2}} + \frac{\sqrt{d x^{2}} b^{2} d^{5} x^{2} - 3 \, \sqrt{d x^{2}} a b d^{5}}{3 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="giac")

[Out]

a^2*arctan(sqrt(d*x^2)*b/sqrt(a*b*d))/(sqrt(a*b*d)*b^2) + 1/3*(sqrt(d*x^2)*b^2*d
^5*x^2 - 3*sqrt(d*x^2)*a*b*d^5)/(b^3*d^6)