3.307 \(\int \frac{\sqrt{x}}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=239 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac{x^{3/2}}{4 a \left (a+b x^2\right )^2} \]

[Out]

x^(3/2)/(4*a*(a + b*x^2)^2) + (5*x^(3/2))/(16*a^2*(a + b*x^2)) - (5*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(3/4)) + (5*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(3/4)) + (5*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(3/4)) -
(5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9
/4)*b^(3/4))

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Rubi [A]  time = 0.43039, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} b^{3/4}}+\frac{5 x^{3/2}}{16 a^2 \left (a+b x^2\right )}+\frac{x^{3/2}}{4 a \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b*x^2)^3,x]

[Out]

x^(3/2)/(4*a*(a + b*x^2)^2) + (5*x^(3/2))/(16*a^2*(a + b*x^2)) - (5*ArcTan[1 - (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(3/4)) + (5*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(3/4)) + (5*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(3/4)) -
(5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9
/4)*b^(3/4))

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Rubi in Sympy [A]  time = 71.2657, size = 224, normalized size = 0.94 \[ \frac{x^{\frac{3}{2}}}{4 a \left (a + b x^{2}\right )^{2}} + \frac{5 x^{\frac{3}{2}}}{16 a^{2} \left (a + b x^{2}\right )} + \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{9}{4}} b^{\frac{3}{4}}} - \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{9}{4}} b^{\frac{3}{4}}} - \frac{5 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} b^{\frac{3}{4}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(3/2)/(4*a*(a + b*x**2)**2) + 5*x**(3/2)/(16*a**2*(a + b*x**2)) + 5*sqrt(2)*l
og(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(9/4)*b**(3
/4)) - 5*sqrt(2)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(1
28*a**(9/4)*b**(3/4)) - 5*sqrt(2)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(6
4*a**(9/4)*b**(3/4)) + 5*sqrt(2)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64
*a**(9/4)*b**(3/4))

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Mathematica [A]  time = 0.218929, size = 220, normalized size = 0.92 \[ \frac{\frac{32 a^{5/4} x^{3/2}}{\left (a+b x^2\right )^2}+\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{40 \sqrt [4]{a} x^{3/2}}{a+b x^2}}{128 a^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

((32*a^(5/4)*x^(3/2))/(a + b*x^2)^2 + (40*a^(1/4)*x^(3/2))/(a + b*x^2) - (10*Sqr
t[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (10*Sqrt[2]*ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(3/4))/(128*a^(9/4))

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Maple [A]  time = 0.011, size = 175, normalized size = 0.7 \[{\frac{1}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{5}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{5\,\sqrt{2}}{128\,{a}^{2}b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x^2+a)^3,x)

[Out]

1/4*x^(3/2)/a/(b*x^2+a)^2+5/16*x^(3/2)/a^2/(b*x^2+a)+5/128/a^2/b/(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)))+5/64/a^2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/
2)+1)+5/64/a^2/b/(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(sqrt(x)/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255096, size = 321, normalized size = 1.34 \[ \frac{20 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{7} b^{2} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{3}{4}}}{\sqrt{-a^{5} b \sqrt{-\frac{1}{a^{9} b^{3}}} + x} + \sqrt{x}}\right ) + 5 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{1}{4}} \log \left (a^{7} b^{2} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - 5 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{1}{4}} \log \left (-a^{7} b^{2} \left (-\frac{1}{a^{9} b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 4 \,{\left (5 \, b x^{3} + 9 \, a x\right )} \sqrt{x}}{64 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(20*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*b^3))^(1/4)*arctan(a^7*b^2*(
-1/(a^9*b^3))^(3/4)/(sqrt(-a^5*b*sqrt(-1/(a^9*b^3)) + x) + sqrt(x))) + 5*(a^2*b^
2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*b^3))^(1/4)*log(a^7*b^2*(-1/(a^9*b^3))^(3/4)
+ sqrt(x)) - 5*(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4)*(-1/(a^9*b^3))^(1/4)*log(-a^7*
b^2*(-1/(a^9*b^3))^(3/4) + sqrt(x)) + 4*(5*b*x^3 + 9*a*x)*sqrt(x))/(a^2*b^2*x^4
+ 2*a^3*b*x^2 + a^4)

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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(x**(1/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219675, size = 282, normalized size = 1.18 \[ \frac{5 \, b x^{\frac{7}{2}} + 9 \, a x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{2}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(5*b*x^(7/2) + 9*a*x^(3/2))/((b*x^2 + a)^2*a^2) + 5/64*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^3) +
5/64*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^3) - 5/128*sqrt(2)*(a*b^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(a/b)^(
1/4) + x + sqrt(a/b))/(a^3*b^3) + 5/128*sqrt(2)*(a*b^3)^(3/4)*ln(-sqrt(2)*sqrt(x
)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3)