3.102 \(\int \frac{1}{x \sqrt{a x+b x^4}} \, dx\)

Optimal. Leaf size=497 \[ -\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt [4]{3} \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{a x}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt [3]{b} x \left (a+b x^3\right )}{a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{a x+b x^4}} \]

[Out]

(2*(1 + Sqrt[3])*b^(1/3)*x*(a + b*x^3))/(a*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)*S
qrt[a*x + b*x^4]) - (2*Sqrt[a*x + b*x^4])/(a*x) - (2*3^(1/4)*b^(1/3)*x*(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 + Sq
rt[3])*b^(1/3)*x)^2]*EllipticE[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])/(a^(2/3)*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 + b*x^4]) - ((
1 - Sqrt[3])*b^(1/3)*x*(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^(2/3)*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 + b*x^4])

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Rubi [A]  time = 0.935867, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt [4]{3} \sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a^{2/3} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a x+b x^4}}-\frac{2 \sqrt{a x+b x^4}}{a x}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt [3]{b} x \left (a+b x^3\right )}{a \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right ) \sqrt{a x+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a*x + b*x^4]),x]

[Out]

(2*(1 + Sqrt[3])*b^(1/3)*x*(a + b*x^3))/(a*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)*S
qrt[a*x + b*x^4]) - (2*Sqrt[a*x + b*x^4])/(a*x) - (2*3^(1/4)*b^(1/3)*x*(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 + Sq
rt[3])*b^(1/3)*x)^2]*EllipticE[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])/(a^(2/3)*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 + b*x^4]) - ((
1 - Sqrt[3])*b^(1/3)*x*(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^(2/3)*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 + b*x^4])

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Rubi in Sympy [A]  time = 55.9265, size = 445, normalized size = 0.9 \[ \frac{\sqrt [3]{b} \left (2 + 2 \sqrt{3}\right ) \sqrt{a x + b x^{4}}}{a \left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )} - \frac{2 \sqrt{a x + b x^{4}}}{a x} - \frac{2 \sqrt [4]{3} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x + b x^{4}} E\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (a + b x^{3}\right )} - \frac{3^{\frac{3}{4}} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x + b x^{4}} F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (a + b x^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**4+a*x)**(1/2),x)

[Out]

b**(1/3)*(2 + 2*sqrt(3))*sqrt(a*x + b*x**4)/(a*(a**(1/3) + b**(1/3)*x*(1 + sqrt(
3)))) - 2*sqrt(a*x + b*x**4)/(a*x) - 2*3**(1/4)*b**(1/3)*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 + b*x**4)*elliptic_e(acos((a**(1/3) + b**(1/3)*x*(-sqr
t(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sqrt(3)/4 + 1/2)/(a**(2/3)*sq
rt(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)) - 3**(3/4)*b**(1/3)*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)*(-sqrt(3) + 1)*(a**(1/3) + b**
(1/3)*x)*sqrt(a*x + b*x**4)*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**(2/3)*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 = 1.27303, size = 334, normalized size = 0.67 \[ \frac{2 \left (a^{2/3} \sqrt [3]{b} x-\sqrt [3]{a} b^{2/3} x^2+\frac{(-1)^{2/3} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2 \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt [3]{b} x}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{b} x+\sqrt [3]{a}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{b} x+\sqrt [3]{a}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{2 \left ((-1)^{2/3}-1\right )}-a\right )}{a \sqrt{x \left (a+b x^3\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x*Sqrt[a*x + b*x^4]),x]

[Out]

(2*(-a + a^(2/3)*b^(1/3)*x - a^(1/3)*b^(2/3)*x^2 + ((-1)^(2/3)*a^(1/3)*(a^(1/3)
+ b^(1/3)*x)^2*Sqrt[((1 + (-1)^(1/3))*b^(1/3)*x*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x)
)/(a^(1/3) + b^(1/3)*x)^2]*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/(a^(1/3) + b^(1
/3)*x)]*((-3 - I*Sqrt[3])*EllipticE[ArcSin[Sqrt[((3 + I*Sqrt[3])*b^(1/3)*x)/(a^(
1/3) + b^(1/3)*x)]/Sqrt[2]], (-I + Sqrt[3])/(I + Sqrt[3])] + (1 + I*Sqrt[3])*Ell
ipticF[ArcSin[Sqrt[((3 + I*Sqrt[3])*b^(1/3)*x)/(a^(1/3) + b^(1/3)*x)]/Sqrt[2]],
(-I + Sqrt[3])/(I + Sqrt[3])]))/(2*(-1 + (-1)^(2/3)))))/(a*Sqrt[x*(a + b*x^3)])

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Maple [C]  time = 0.031, size = 1083, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^4+a*x)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a x} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a*x)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{4} + a x} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x^4 + a*x)*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{x \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**4+a*x)**(1/2),x)

[Out]

Integral(1/(x*sqrt(x*(a + b*x**3))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a x} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a*x)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a*x)*x), x)