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

Optimal. Leaf size=218 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{b}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{\sqrt{x}}{2 a \left (a+b x^2\right )} \]

[Out]

Sqrt[x]/(2*a*(a + b*x^2)) - (3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(7/4)*b^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(7/4)*b^(1/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(1/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(1/4))

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Rubi [A]  time = 0.365604, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} \sqrt [4]{b}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} \sqrt [4]{b}}+\frac{\sqrt{x}}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/(2*a*(a + b*x^2)) - (3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(7/4)*b^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(7/4)*b^(1/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(1/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(1/4))

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Rubi in Sympy [A]  time = 63.3291, size = 204, normalized size = 0.94 \[ \frac{\sqrt{x}}{2 a \left (a + b x^{2}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{7}{4}} \sqrt [4]{b}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{7}{4}} \sqrt [4]{b}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/(2*a*(a + b*x**2)) - 3*sqrt(2)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) +
sqrt(a) + sqrt(b)*x)/(16*a**(7/4)*b**(1/4)) + 3*sqrt(2)*log(sqrt(2)*a**(1/4)*b**
(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(7/4)*b**(1/4)) - 3*sqrt(2)*atan(1 -
sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(7/4)*b**(1/4)) + 3*sqrt(2)*atan(1 + s
qrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(7/4)*b**(1/4))

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Mathematica [A]  time = 0.279785, size = 199, normalized size = 0.91 \[ \frac{\frac{8 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}}{16 a^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(3/4)*Sqrt[x])/(a + b*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/b^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/b^(1/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]
)/b^(1/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/b^(1/4))/(16*a^(7/4))

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Maple [A]  time = 0.012, size = 149, normalized size = 0.7 \[{\frac{1}{2\,a \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{3\,\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^2/x^(1/2),x)

[Out]

1/2*x^(1/2)/a/(b*x^2+a)+3/16/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)))+3/8/a^2*(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/a^2*(a/b)^(1/4)*2^(1/2)*a
rctan(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*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251762, size = 221, normalized size = 1.01 \[ -\frac{12 \,{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}}}{\sqrt{a^{4} \sqrt{-\frac{1}{a^{7} b}} + x} + \sqrt{x}}\right ) - 3 \,{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} \log \left (a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 3 \,{\left (a b x^{2} + a^{2}\right )} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} \log \left (-a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + \sqrt{x}\right ) - 4 \, \sqrt{x}}{8 \,{\left (a b x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(12*(a*b*x^2 + a^2)*(-1/(a^7*b))^(1/4)*arctan(a^2*(-1/(a^7*b))^(1/4)/(sqrt(
a^4*sqrt(-1/(a^7*b)) + x) + sqrt(x))) - 3*(a*b*x^2 + a^2)*(-1/(a^7*b))^(1/4)*log
(a^2*(-1/(a^7*b))^(1/4) + sqrt(x)) + 3*(a*b*x^2 + a^2)*(-1/(a^7*b))^(1/4)*log(-a
^2*(-1/(a^7*b))^(1/4) + sqrt(x)) - 4*sqrt(x))/(a*b*x^2 + a^2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219346, size = 269, normalized size = 1.23 \[ \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b} + \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b} + \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b} - \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b} + \frac{\sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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