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

Optimal. Leaf size=203 \[ \frac{x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}} \]

[Out]

(x^(3/2)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/
(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])
*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(3^(1/4)*a^(
1/3)*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^
2]*Sqrt[a*x^2 + b*x^5])

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Rubi [A]  time = 0.312816, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[a*x^2 + b*x^5],x]

[Out]

(x^(3/2)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/
(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])
*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(3^(1/4)*a^(
1/3)*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^
2]*Sqrt[a*x^2 + b*x^5])

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Rubi in Sympy [A]  time = 17.2258, size = 189, normalized size = 0.93 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x^{2} + b x^{5}} F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 \sqrt [3]{a} \sqrt{x} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1
/3)*x*(1 + sqrt(3)))**2)*(a**(1/3) + b**(1/3)*x)*sqrt(a*x**2 + b*x**5)*elliptic_
f(acos((a**(1/3) + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3
)))), sqrt(3)/4 + 1/2)/(3*a**(1/3)*sqrt(x)*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*
x)/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))**2)*(a + b*x**3))

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Mathematica [C]  time = 0.200826, size = 151, normalized size = 0.74 \[ -\frac{2 i \sqrt [3]{b} x^{5/2} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-a}}{\sqrt [3]{b} x}-1\right )} \sqrt{\frac{(-a)^{2/3}}{b^{2/3} x^2}+\frac{\sqrt [3]{-a}}{\sqrt [3]{b} x}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{-a} \sqrt{x^2 \left (a+b x^3\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-2*I)*b^(1/3)*Sqrt[(-1)^(5/6)*(-1 + (-a)^(1/3)/(b^(1/3)*x))]*Sqrt[1 + (-a)^(2/
3)/(b^(2/3)*x^2) + (-a)^(1/3)/(b^(1/3)*x)]*x^(5/2)*EllipticF[ArcSin[Sqrt[-(-1)^(
5/6) - (I*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)])/(3^(1/4)*(-a)^(1/3)*Sq
rt[x^2*(a + b*x^3)])

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Maple [C]  time = 0.062, size = 437, normalized size = 2.2 \[ -4\,{\frac{{x}^{3/2} \left ( b{x}^{3}+a \right ) \left ( i\sqrt{3}{x}^{2}{b}^{2}-2\,i\sqrt [3]{-a{b}^{2}}\sqrt{3}xb+i \left ( -a{b}^{2} \right ) ^{2/3}\sqrt{3}-{b}^{2}{x}^{2}+2\,\sqrt [3]{-a{b}^{2}}xb- \left ( -a{b}^{2} \right ) ^{2/3} \right ) }{\sqrt{b{x}^{5}+a{x}^{2}}\sqrt [3]{-a{b}^{2}}b\sqrt{x \left ( b{x}^{3}+a \right ) } \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xb}{ \left ( i\sqrt{3}-1 \right ) \left ( -bx+\sqrt [3]{-a{b}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-a{b}^{2}}+2\,bx+\sqrt [3]{-a{b}^{2}}}{ \left ( i\sqrt{3}+1 \right ) \left ( -bx+\sqrt [3]{-a{b}^{2}} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-a{b}^{2}}-2\,bx-\sqrt [3]{-a{b}^{2}}}{ \left ( i\sqrt{3}-1 \right ) \left ( -bx+\sqrt [3]{-a{b}^{2}} \right ) }}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xb}{ \left ( i\sqrt{3}-1 \right ) \left ( -bx+\sqrt [3]{-a{b}^{2}} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( i\sqrt{3}-1 \right ) }{ \left ( i\sqrt{3}+1 \right ) \left ( i\sqrt{3}-3 \right ) }}} \right ){\frac{1}{\sqrt{{\frac{x \left ( -bx+\sqrt [3]{-a{b}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}+2\,bx+\sqrt [3]{-a{b}^{2}} \right ) \left ( i\sqrt{3}\sqrt [3]{-a{b}^{2}}-2\,bx-\sqrt [3]{-a{b}^{2}} \right ) }{{b}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/(b*x^5+a*x^2)^(1/2)*x^(3/2)*(b*x^3+a)/(-a*b^2)^(1/3)/b*(-(I*3^(1/2)-3)*x*b/(I
*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^
2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-
2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*
3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1
/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(I*3^(1/2)*x^2*b^2-2*I*(-a*b^2)^(1/3)
*3^(1/2)*x*b+I*(-a*b^2)^(2/3)*3^(1/2)-b^2*x^2+2*(-a*b^2)^(1/3)*x*b-(-a*b^2)^(2/3
))/(x*(b*x^3+a))^(1/2)/(I*3^(1/2)-3)/(1/b^2*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(
-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/
3)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x)/sqrt(b*x^5 + a*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x}}{\sqrt{b x^{5} + a x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x)/sqrt(b*x^5 + a*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)/sqrt(x**2*(a + b*x**3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x)/sqrt(b*x^5 + a*x^2), x)