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

Optimal. Leaf size=230 \[ -\frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4}}-\frac{5}{2 a^2 \sqrt{x}}+\frac{1}{2 a \sqrt{x} \left (a+b x^2\right )} \]

[Out]

-5/(2*a^2*Sqrt[x]) + 1/(2*a*Sqrt[x]*(a + b*x^2)) + (5*b^(1/4)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)) - (5*b^(1/4)*ArcTan[1 + (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)) - (5*b^(1/4)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)) + (5*b^(1/4)*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4))

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Rubi [A]  time = 0.425311, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4}}-\frac{5}{2 a^2 \sqrt{x}}+\frac{1}{2 a \sqrt{x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-5/(2*a^2*Sqrt[x]) + 1/(2*a*Sqrt[x]*(a + b*x^2)) + (5*b^(1/4)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)) - (5*b^(1/4)*ArcTan[1 + (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)) - (5*b^(1/4)*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)) + (5*b^(1/4)*Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4))

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Rubi in Sympy [A]  time = 69.1633, size = 218, normalized size = 0.95 \[ \frac{1}{2 a \sqrt{x} \left (a + b x^{2}\right )} - \frac{5}{2 a^{2} \sqrt{x}} - \frac{5 \sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*a*sqrt(x)*(a + b*x**2)) - 5/(2*a**2*sqrt(x)) - 5*sqrt(2)*b**(1/4)*log(-sqrt
(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(9/4)) + 5*sqrt(2)*b
**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(9/4
)) + 5*sqrt(2)*b**(1/4)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4))
- 5*sqrt(2)*b**(1/4)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4))

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Mathematica [A]  time = 0.305119, size = 212, normalized size = 0.92 \[ \frac{-\frac{8 \sqrt [4]{a} b x^{3/2}}{a+b x^2}-5 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+10 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{32 \sqrt [4]{a}}{\sqrt{x}}}{16 a^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(1/4))/Sqrt[x] - (8*a^(1/4)*b*x^(3/2))/(a + b*x^2) + 10*Sqrt[2]*b^(1/4)*
ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 10*Sqrt[2]*b^(1/4)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(16*a^(9/4))

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Maple [A]  time = 0.02, size = 158, normalized size = 0.7 \[ -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{5\,\sqrt{2}}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x^2+a)^2,x)

[Out]

-2/a^2/x^(1/2)-1/2*b/a^2*x^(3/2)/(b*x^2+a)-5/16/a^2/(a/b)^(1/4)*2^(1/2)*ln((x-(a
/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2
)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-5/8/a^2/(a
/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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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)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25931, size = 259, normalized size = 1.13 \[ -\frac{20 \, b x^{2} + 20 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{x} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{125 \, a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}}}{125 \, b \sqrt{x} + \sqrt{-15625 \, a^{5} b \sqrt{-\frac{b}{a^{9}}} + 15625 \, b^{2} x}}\right ) + 5 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{x} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + 125 \, b \sqrt{x}\right ) - 5 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{x} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + 125 \, b \sqrt{x}\right ) + 16 \, a}{8 \,{\left (a^{2} b x^{2} + a^{3}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(20*b*x^2 + 20*(a^2*b*x^2 + a^3)*sqrt(x)*(-b/a^9)^(1/4)*arctan(125*a^7*(-b/
a^9)^(3/4)/(125*b*sqrt(x) + sqrt(-15625*a^5*b*sqrt(-b/a^9) + 15625*b^2*x))) + 5*
(a^2*b*x^2 + a^3)*sqrt(x)*(-b/a^9)^(1/4)*log(125*a^7*(-b/a^9)^(3/4) + 125*b*sqrt
(x)) - 5*(a^2*b*x^2 + a^3)*sqrt(x)*(-b/a^9)^(1/4)*log(-125*a^7*(-b/a^9)^(3/4) +
125*b*sqrt(x)) + 16*a)/((a^2*b*x^2 + a^3)*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224339, size = 284, normalized size = 1.23 \[ -\frac{5 \, b x^{2} + 4 \, a}{2 \,{\left (b x^{\frac{5}{2}} + a \sqrt{x}\right )} a^{2}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{2}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{2}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{2}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(5*b*x^2 + 4*a)/((b*x^(5/2) + a*sqrt(x))*a^2) - 5/8*sqrt(2)*(a*b^3)^(3/4)*a
rctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^2) - 5/8
*sqrt(2)*(a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/
b)^(1/4))/(a^3*b^2) + 5/16*sqrt(2)*(a*b^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4)
+ x + sqrt(a/b))/(a^3*b^2) - 5/16*sqrt(2)*(a*b^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(a/b
)^(1/4) + x + sqrt(a/b))/(a^3*b^2)