3.532 \(\int \frac{x \left (1+a+x^2+b x^2\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

[Out]

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

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Rubi [A]  time = 0.191742, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.9886, size = 39, normalized size = 0.78 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a - b}} \right )}}{\sqrt{a - b}} - \frac{1}{b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a + b*x**2)/sqrt(a - b))/sqrt(a - b) - 1/(b*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.0336971, size = 49, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{b-a}}\right )}{\sqrt{b-a}}-\frac{1}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(b*Sqrt[a + b*x^2])) + ArcTan[Sqrt[a + b*x^2]/Sqrt[-a + b]]/Sqrt[-a + b]

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Maple [B]  time = 0.021, size = 133, normalized size = 2.7 \[ -{\frac{1}{\sqrt{b{x}^{2}+a}}}-{\frac{1}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{a}{a-b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{a}{a-b}\arctan \left ({1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}-{\frac{b}{a-b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{b}{a-b}\arctan \left ({1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285327, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} x^{2} + a b\right )} \log \left (-\frac{4 \,{\left ({\left (a b - b^{2}\right )} x^{2} + 2 \, a^{2} - 3 \, a b + b^{2}\right )} \sqrt{b x^{2} + a} -{\left (b^{2} x^{4} + 2 \,{\left (4 \, a b - 3 \, b^{2}\right )} x^{2} + 8 \, a^{2} - 8 \, a b + b^{2}\right )} \sqrt{a - b}}{x^{4} + 2 \, x^{2} + 1}\right ) - 4 \, \sqrt{b x^{2} + a} \sqrt{a - b}}{4 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{a - b}}, \frac{{\left (b^{2} x^{2} + a b\right )} \arctan \left (-\frac{{\left (b x^{2} + 2 \, a - b\right )} \sqrt{-a + b}}{2 \, \sqrt{b x^{2} + a}{\left (a - b\right )}}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{-a + b}}{2 \,{\left (b^{2} x^{2} + a b\right )} \sqrt{-a + b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((b^2*x^2 + a*b)*log(-(4*((a*b - b^2)*x^2 + 2*a^2 - 3*a*b + b^2)*sqrt(b*x^2
+ a) - (b^2*x^4 + 2*(4*a*b - 3*b^2)*x^2 + 8*a^2 - 8*a*b + b^2)*sqrt(a - b))/(x^
4 + 2*x^2 + 1)) - 4*sqrt(b*x^2 + a)*sqrt(a - b))/((b^2*x^2 + a*b)*sqrt(a - b)),
1/2*((b^2*x^2 + a*b)*arctan(-1/2*(b*x^2 + 2*a - b)*sqrt(-a + b)/(sqrt(b*x^2 + a)
*(a - b))) - 2*sqrt(b*x^2 + a)*sqrt(-a + b))/((b^2*x^2 + a*b)*sqrt(-a + b))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.265921, size = 55, normalized size = 1.1 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a + b}}\right )}{\sqrt{-a + b}} - \frac{1}{\sqrt{b x^{2} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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