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

Optimal. Leaf size=202 \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4}}-\frac{2}{a \sqrt{x}} \]

[Out]

-2/(a*Sqrt[x]) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2
]*a^(5/4)) - (b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(5/4)) - (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2
*Sqrt[2]*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(2*Sqrt[2]*a^(5/4))

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Rubi [A]  time = 0.380668, antiderivative size = 202, 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 \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4}}-\frac{2}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(a + b*x^2)),x]

[Out]

-2/(a*Sqrt[x]) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2
]*a^(5/4)) - (b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(5/4)) - (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2
*Sqrt[2]*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(2*Sqrt[2]*a^(5/4))

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Rubi in Sympy [A]  time = 63.8511, size = 190, normalized size = 0.94 \[ - \frac{2}{a \sqrt{x}} - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*sqrt(x)) - sqrt(2)*b**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(
a) + sqrt(b)*x)/(4*a**(5/4)) + sqrt(2)*b**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sq
rt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(5/4)) + sqrt(2)*b**(1/4)*atan(1 - sqrt(2)*b*
*(1/4)*sqrt(x)/a**(1/4))/(2*a**(5/4)) - sqrt(2)*b**(1/4)*atan(1 + sqrt(2)*b**(1/
4)*sqrt(x)/a**(1/4))/(2*a**(5/4))

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Mathematica [A]  time = 0.110088, size = 189, normalized size = 0.94 \[ \frac{-\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{8 \sqrt [4]{a}}{\sqrt{x}}}{4 a^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 140, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{4\,a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-2\,{\frac{1}{a\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x^2+a),x)

[Out]

-1/4/a/(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)))-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1
/2)-1)-2/a/x^(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(1/((b*x^2 + a)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229676, size = 180, normalized size = 0.89 \[ -\frac{4 \, a \sqrt{x} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}}}{b \sqrt{x} + \sqrt{-a^{3} b \sqrt{-\frac{b}{a^{5}}} + b^{2} x}}\right ) + a \sqrt{x} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b \sqrt{x}\right ) - a \sqrt{x} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b \sqrt{x}\right ) + 4}{2 \, a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*a*sqrt(x)*(-b/a^5)^(1/4)*arctan(a^4*(-b/a^5)^(3/4)/(b*sqrt(x) + sqrt(-a^
3*b*sqrt(-b/a^5) + b^2*x))) + a*sqrt(x)*(-b/a^5)^(1/4)*log(a^4*(-b/a^5)^(3/4) +
b*sqrt(x)) - a*sqrt(x)*(-b/a^5)^(1/4)*log(-a^4*(-b/a^5)^(3/4) + b*sqrt(x)) + 4)/
(a*sqrt(x))

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Sympy [A]  time = 66.4366, size = 180, normalized size = 0.89 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\- \frac{2}{a \sqrt{x}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{5}{4}} b^{11} \left (\frac{1}{b}\right )^{\frac{45}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{5}{4}} b^{11} \left (\frac{1}{b}\right )^{\frac{45}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{a^{\frac{5}{4}} b^{11} \left (\frac{1}{b}\right )^{\frac{45}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b*x**(5/2)), Eq(a, 0)), (-
2/(a*sqrt(x)), Eq(b, 0)), (-2/(a*sqrt(x)) + (-1)**(3/4)*log(-(-1)**(1/4)*a**(1/4
)*(1/b)**(1/4) + sqrt(x))/(2*a**(5/4)*b**11*(1/b)**(45/4)) - (-1)**(3/4)*log((-1
)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(5/4)*b**11*(1/b)**(45/4)) - (-1
)**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(a**(5/4)*b**11*(1/b)
**(45/4)), True))

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GIAC/XCAS [A]  time = 0.214661, size = 257, normalized size = 1.27 \[ -\frac{2}{a \sqrt{x}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} b^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} b^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2} b^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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