3.747 \(\int x^2 \left (\frac{c}{a+b x^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=75 \[ \frac{c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{c x \sqrt{\frac{c}{a+b x^2}}}{b} \]

[Out]

-((c*x*Sqrt[c/(a + b*x^2)])/b) + (c*Sqrt[c/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcTanh[
(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)

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Rubi [A]  time = 0.237621, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c \sqrt{a+b x^2} \sqrt{\frac{c}{a+b x^2}} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{c x \sqrt{\frac{c}{a+b x^2}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

-((c*x*Sqrt[c/(a + b*x^2)])/b) + (c*Sqrt[c/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcTanh[
(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)

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Rubi in Sympy [A]  time = 7.50974, size = 63, normalized size = 0.84 \[ - \frac{c x \sqrt{\frac{c}{a + b x^{2}}}}{b} + \frac{c \sqrt{\frac{c}{a + b x^{2}}} \sqrt{a + b x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*x*sqrt(c/(a + b*x**2))/b + c*sqrt(c/(a + b*x**2))*sqrt(a + b*x**2)*atanh(sqrt
(b)*x/sqrt(a + b*x**2))/b**(3/2)

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Mathematica [A]  time = 0.0501516, size = 66, normalized size = 0.88 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}} \left (\sqrt{b} x-\sqrt{a+b x^2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((c*Sqrt[c/(a + b*x^2)]*(Sqrt[b]*x - Sqrt[a + b*x^2]*Log[b*x + Sqrt[b]*Sqrt[a +
b*x^2]]))/b^(3/2))

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Maple [A]  time = 0.017, size = 60, normalized size = 0.8 \[ -{(b{x}^{2}+a) \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}} \left ( x{b}^{{\frac{3}{2}}}-\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) b\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c/(b*x^2+a))^(3/2),x)

[Out]

-(c/(b*x^2+a))^(3/2)*(b*x^2+a)*(x*b^(3/2)-ln(x*b^(1/2)+(b*x^2+a)^(1/2))*b*(b*x^2
+a)^(1/2))/b^(5/2)

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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^2*(c/(b*x^2 + a))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285441, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, c x \sqrt{\frac{c}{b x^{2} + a}} - c \sqrt{\frac{c}{b}} \log \left (-2 \, b c x^{2} - a c - 2 \,{\left (b^{2} x^{3} + a b x\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{\frac{c}{b}}\right )}{2 \, b}, -\frac{c x \sqrt{\frac{c}{b x^{2} + a}} - c \sqrt{-\frac{c}{b}} \arctan \left (\frac{c x}{{\left (b x^{2} + a\right )} \sqrt{\frac{c}{b x^{2} + a}} \sqrt{-\frac{c}{b}}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28659, size = 96, normalized size = 1.28 \[ -{\left (\frac{c x{\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} b} + \frac{c{\rm ln}\left ({\left | -\sqrt{b c} x + \sqrt{b c x^{2} + a c} \right |}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c} b}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c*x*sign(b*x^2 + a)/(sqrt(b*c*x^2 + a*c)*b) + c*ln(abs(-sqrt(b*c)*x + sqrt(b*c
*x^2 + a*c)))*sign(b*x^2 + a)/(sqrt(b*c)*b))*c