3.289 \(\int \frac{x^{5/2}}{a+b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.410086, 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 \[ -\frac{a^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{a^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{7/4}}-\frac{a^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{7/4}}+\frac{2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 65.0618, size = 192, normalized size = 0.94 \[ - \frac{\sqrt{2} a^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{7}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{7}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{7}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{7}{4}}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0540531, size = 190, normalized size = 0.93 \[ \frac{-3 \sqrt{2} a^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+3 \sqrt{2} a^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+6 \sqrt{2} a^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 b^{3/4} x^{3/2}}{12 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 143, normalized size = 0.7 \[{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{a\sqrt{2}}{4\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x^2+a),x)

[Out]

2/3*x^(3/2)/b-1/4*a/b^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*a/b^2/(a/b)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*a/b^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(x^(5/2)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232156, size = 200, normalized size = 0.98 \[ -\frac{12 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{5} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{3}{4}}}{a^{2} \sqrt{x} + \sqrt{-a^{3} b^{3} \sqrt{-\frac{a^{3}}{b^{7}}} + a^{4} x}}\right ) + 3 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (b^{5} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{3}{4}} + a^{2} \sqrt{x}\right ) - 3 \, b \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-b^{5} \left (-\frac{a^{3}}{b^{7}}\right )^{\frac{3}{4}} + a^{2} \sqrt{x}\right ) - 4 \, x^{\frac{3}{2}}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(12*b*(-a^3/b^7)^(1/4)*arctan(b^5*(-a^3/b^7)^(3/4)/(a^2*sqrt(x) + sqrt(-a^3
*b^3*sqrt(-a^3/b^7) + a^4*x))) + 3*b*(-a^3/b^7)^(1/4)*log(b^5*(-a^3/b^7)^(3/4) +
a^2*sqrt(x)) - 3*b*(-a^3/b^7)^(1/4)*log(-b^5*(-a^3/b^7)^(3/4) + a^2*sqrt(x)) -
4*x^(3/2))/b

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Sympy [A]  time = 126.62, size = 180, normalized size = 0.88 \[ \begin{cases} \tilde{\infty } x^{\frac{3}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: a = 0 \\\frac{2 x^{\frac{7}{2}}}{7 a} & \text{for}\: b = 0 \\\frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{4} \left (\frac{1}{b}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 b^{4} \left (\frac{1}{b}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} a^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{b^{4} \left (\frac{1}{b}\right )^{\frac{9}{4}}} + \frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*b), Eq(a, 0)), (2*
x**(7/2)/(7*a), Eq(b, 0)), ((-1)**(3/4)*a**(3/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)
**(1/4) + sqrt(x))/(2*b**4*(1/b)**(9/4)) - (-1)**(3/4)*a**(3/4)*log((-1)**(1/4)*
a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*b**4*(1/b)**(9/4)) - (-1)**(3/4)*a**(3/4)*at
an((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(b**4*(1/b)**(9/4)) + 2*x**(3/2)
/(3*b), True))

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GIAC/XCAS [A]  time = 0.226495, size = 240, normalized size = 1.18 \[ \frac{2 \, x^{\frac{3}{2}}}{3 \, b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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