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

Optimal. Leaf size=204 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.373613, antiderivative size = 204, 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 \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 62.9354, size = 192, normalized size = 0.94 \[ - \frac{2}{3 a x^{\frac{3}{2}}} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{7}{4}}} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{7}{4}}} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*a*x**(3/2)) + sqrt(2)*b**(3/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sq
rt(a) + sqrt(b)*x)/(4*a**(7/4)) - sqrt(2)*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(7/4)) + sqrt(2)*b**(3/4)*atan(1 - sqrt(2)
*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(7/4)) - sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*b**
(1/4)*sqrt(x)/a**(1/4))/(2*a**(7/4))

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Mathematica [A]  time = 0.131253, size = 190, normalized size = 0.93 \[ \frac{-\frac{8 a^{3/4}}{x^{3/2}}+3 \sqrt{2} b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-3 \sqrt{2} b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+6 \sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{12 a^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-8*a^(3/4))/x^(3/2) + 6*Sqrt[2]*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)] - 6*Sqrt[2]*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 3*Sq
rt[2]*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 3*Sqr
t[2]*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(12*a^(
7/4))

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Maple [A]  time = 0.012, size = 143, normalized size = 0.7 \[ -{\frac{b\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{b\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{2}{3\,a}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^2+a),x)

[Out]

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

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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^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230928, size = 197, normalized size = 0.97 \[ \frac{12 \, a x^{\frac{3}{2}} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{a^{4} \sqrt{-\frac{b^{3}}{a^{7}}} + b^{2} x}}\right ) - 3 \, a x^{\frac{3}{2}} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) + 3 \, a x^{\frac{3}{2}} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (-a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) - 4}{6 \, a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(12*a*x^(3/2)*(-b^3/a^7)^(1/4)*arctan(a^2*(-b^3/a^7)^(1/4)/(b*sqrt(x) + sqrt
(a^4*sqrt(-b^3/a^7) + b^2*x))) - 3*a*x^(3/2)*(-b^3/a^7)^(1/4)*log(a^2*(-b^3/a^7)
^(1/4) + b*sqrt(x)) + 3*a*x^(3/2)*(-b^3/a^7)^(1/4)*log(-a^2*(-b^3/a^7)^(1/4) + b
*sqrt(x)) - 4)/(a*x^(3/2))

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Sympy [A]  time = 160.845, size = 184, normalized size = 0.9 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} + \frac{\sqrt [4]{-1} b^{7} \left (\frac{1}{b}\right )^{\frac{25}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{4}}} - \frac{\sqrt [4]{-1} b^{7} \left (\frac{1}{b}\right )^{\frac{25}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{4}}} + \frac{\sqrt [4]{-1} b^{7} \left (\frac{1}{b}\right )^{\frac{25}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{a^{\frac{7}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b*x**(7/2)), Eq(a, 0)), (-
2/(3*a*x**(3/2)), Eq(b, 0)), (-2/(3*a*x**(3/2)) + (-1)**(1/4)*b**7*(1/b)**(25/4)
*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(7/4)) - (-1)**(1/4)*b*
*7*(1/b)**(25/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(7/4)) +
(-1)**(1/4)*b**7*(1/b)**(25/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4))
)/a**(7/4), True))

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GIAC/XCAS [A]  time = 0.224249, size = 240, normalized size = 1.18 \[ -\frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2}} - \frac{2}{3 \, a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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