3.675 \(\int \frac{1}{x^3 \sqrt{d x^2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=68 \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]

[Out]

b/(a^2*Sqrt[d*x^2]) - 1/(3*a*x^2*Sqrt[d*x^2]) + (b^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sq
rt[a]])/(a^(5/2)*Sqrt[d*x^2])

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Rubi [A]  time = 0.0611417, antiderivative size = 68, 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 \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

b/(a^2*Sqrt[d*x^2]) - 1/(3*a*x^2*Sqrt[d*x^2]) + (b^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sq
rt[a]])/(a^(5/2)*Sqrt[d*x^2])

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Rubi in Sympy [A]  time = 23.9533, size = 66, normalized size = 0.97 \[ - \frac{d}{3 a \left (d x^{2}\right )^{\frac{3}{2}}} + \frac{b}{a^{2} \sqrt{d x^{2}}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{a^{\frac{5}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0358477, size = 58, normalized size = 0.85 \[ \frac{d \left (3 b^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \left (a-3 b x^2\right )\right )}{3 a^{5/2} \left (d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 58, normalized size = 0.9 \[{\frac{1}{3\,{a}^{2}{x}^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}+3\,b{x}^{2}\sqrt{ab}-a\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245265, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b d x^{4} \sqrt{-\frac{b}{a d}} \log \left (\frac{2 \, a d x^{2} \sqrt{-\frac{b}{a d}} +{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) + 2 \,{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{6 \, a^{2} d x^{4}}, \frac{3 \, b d x^{4} \sqrt{\frac{b}{a d}} \arctan \left (\frac{\sqrt{d x^{2}} b}{a d \sqrt{\frac{b}{a d}}}\right ) +{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{3 \, a^{2} d x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2)*x^3),x, algorithm="fricas")

[Out]

[1/6*(3*b*d*x^4*sqrt(-b/(a*d))*log((2*a*d*x^2*sqrt(-b/(a*d)) + (b*x^2 - a)*sqrt(
d*x^2))/(b*x^3 + a*x)) + 2*(3*b*x^2 - a)*sqrt(d*x^2))/(a^2*d*x^4), 1/3*(3*b*d*x^
4*sqrt(b/(a*d))*arctan(sqrt(d*x^2)*b/(a*d*sqrt(b/(a*d)))) + (3*b*x^2 - a)*sqrt(d
*x^2))/(a^2*d*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232168, size = 92, normalized size = 1.35 \[ \frac{1}{3} \, d^{2}{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a^{2} d^{2}} + \frac{3 \, b d x^{2} - a d}{\sqrt{d x^{2}} a^{2} d^{3} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2)*x^3),x, algorithm="giac")

[Out]

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