3.533 \(\int \left (\frac{x}{\left (a+b x^2\right )^{5/2}}+\frac{x}{\left (a+b x^2\right )^{3/2}}+\frac{x}{\left (1+x^2\right ) \sqrt{a+b x^2}}\right ) \, dx\)

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

[Out]

-1/(3*b*(a + b*x^2)^(3/2)) - 1/(b*Sqrt[a + b*x^2]) - ArcTanh[Sqrt[a + b*x^2]/Sqr
t[a - b]]/Sqrt[a - b]

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Rubi [A]  time = 0.102488, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.085 \[ -\frac{1}{b \sqrt{a+b x^2}}-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^2)^(5/2) + x/(a + b*x^2)^(3/2) + x/((1 + x^2)*Sqrt[a + b*x^2]),x]

[Out]

-1/(3*b*(a + b*x^2)^(3/2)) - 1/(b*Sqrt[a + b*x^2]) - ArcTanh[Sqrt[a + b*x^2]/Sqr
t[a - b]]/Sqrt[a - b]

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Rubi in Sympy [A]  time = 6.05442, size = 54, normalized size = 0.79 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a - b}} \right )}}{\sqrt{a - b}} - \frac{1}{b \sqrt{a + b x^{2}}} - \frac{1}{3 b \left (a + b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**2+a)**(5/2)+x/(b*x**2+a)**(3/2)+x/(x**2+1)/(b*x**2+a)**(1/2),x)

[Out]

-atanh(sqrt(a + b*x**2)/sqrt(a - b))/sqrt(a - b) - 1/(b*sqrt(a + b*x**2)) - 1/(3
*b*(a + b*x**2)**(3/2))

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Mathematica [A]  time = 0.180312, size = 63, normalized size = 0.93 \[ -\frac{3 a+3 b x^2+1}{3 b \left (a+b x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^2)^(5/2) + x/(a + b*x^2)^(3/2) + x/((1 + x^2)*Sqrt[a + b*x^2]),x]

[Out]

-(1 + 3*a + 3*b*x^2)/(3*b*(a + b*x^2)^(3/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a -
b]]/Sqrt[a - b]

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Maple [A]  time = 0.016, size = 56, normalized size = 0.8 \[ -{\frac{1}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{1\arctan \left ({1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a)^(5/2)+x/(b*x^2+a)^(3/2)+x/(x^2+1)/(b*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291352, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b x^{2} + 3 \, a + 1\right )} \sqrt{b x^{2} + a} \sqrt{a - b} - 3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \log \left (-\frac{4 \,{\left ({\left (a b - b^{2}\right )} x^{2} + 2 \, a^{2} - 3 \, a b + b^{2}\right )} \sqrt{b x^{2} + a} -{\left (b^{2} x^{4} + 2 \,{\left (4 \, a b - 3 \, b^{2}\right )} x^{2} + 8 \, a^{2} - 8 \, a b + b^{2}\right )} \sqrt{a - b}}{x^{4} + 2 \, x^{2} + 1}\right )}{12 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt{a - b}}, -\frac{2 \,{\left (3 \, b x^{2} + 3 \, a + 1\right )} \sqrt{b x^{2} + a} \sqrt{-a + b} - 3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \arctan \left (-\frac{{\left (b x^{2} + 2 \, a - b\right )} \sqrt{-a + b}}{2 \, \sqrt{b x^{2} + a}{\left (a - b\right )}}\right )}{6 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt{-a + b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^2 + a)^(3/2) + x/(sqrt(b*x^2 + a)*(x^2 + 1)) + x/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(4*(3*b*x^2 + 3*a + 1)*sqrt(b*x^2 + a)*sqrt(a - b) - 3*(b^3*x^4 + 2*a*b^2
*x^2 + a^2*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)))/((b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*sqrt(a - b)), -1/6*(2*(3*b*x^2 + 3*a +
1)*sqrt(b*x^2 + a)*sqrt(-a + b) - 3*(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)*arctan(-1/2
*(b*x^2 + 2*a - b)*sqrt(-a + b)/(sqrt(b*x^2 + a)*(a - b))))/((b^3*x^4 + 2*a*b^2*
x^2 + a^2*b)*sqrt(-a + b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (a^{2} + 2 a b x^{2} + a x^{2} + a + b^{2} x^{4} + b x^{4} + b x^{2} + x^{2} + 1\right )}{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (x^{2} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**2+a)**(5/2)+x/(b*x**2+a)**(3/2)+x/(x**2+1)/(b*x**2+a)**(1/2),x)

[Out]

Integral(x*(a**2 + 2*a*b*x**2 + a*x**2 + a + b**2*x**4 + b*x**4 + b*x**2 + x**2
+ 1)/((a + b*x**2)**(5/2)*(x**2 + 1)), x)

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GIAC/XCAS [A]  time = 0.263541, size = 74, normalized size = 1.09 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a + b}}\right )}{\sqrt{-a + b}} - \frac{1}{\sqrt{b x^{2} + a} b} - \frac{1}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^2 + a)^(3/2) + x/(sqrt(b*x^2 + a)*(x^2 + 1)) + x/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

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