3.1752 \(\int \frac{1}{\sqrt{a+b x} \sqrt [6]{c+d x}} \, dx\)

Optimal. Leaf size=780 \[ -\frac{3 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x}}{b^{2/3} \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2 b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

[Out]

(-3*(1 + Sqrt[3])*Sqrt[a + b*x]*(c + d*x)^(1/6))/(b^(2/3)*((b*c - a*d)^(1/3) - (
1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) - (3*3^(1/4)*(b*c - a*d)^(1/3)*(c + d*x)^
(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^
(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^
(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcCos[((b*c - a*d)^
(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3]
)*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-((
b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a
*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]) - (3^(3/4)*(1 - Sqrt[3])
*(b*c - a*d)^(1/3)*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))
*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(
c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*E
llipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*
c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(2*b
^(2/3)*d*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3)
)^2)])

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Rubi [A]  time = 1.23567, antiderivative size = 780, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 \left (1+\sqrt{3}\right ) \sqrt{a+b x} \sqrt [6]{c+d x}}{b^{2/3} \left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{3^{3/4} \left (1-\sqrt{3}\right ) \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2 b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{3 \sqrt [4]{3} \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*(c + d*x)^(1/6)),x]

[Out]

(-3*(1 + Sqrt[3])*Sqrt[a + b*x]*(c + d*x)^(1/6))/(b^(2/3)*((b*c - a*d)^(1/3) - (
1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))) - (3*3^(1/4)*(b*c - a*d)^(1/3)*(c + d*x)^
(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^
(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^
(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcCos[((b*c - a*d)^
(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3]
)*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-((
b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a
*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]) - (3^(3/4)*(1 - Sqrt[3])
*(b*c - a*d)^(1/3)*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))
*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(
c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*E
llipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*
c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(2*b
^(2/3)*d*Sqrt[a + b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3)
)^2)])

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Rubi in Sympy [A]  time = 50.9204, size = 685, normalized size = 0.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/6),x)

[Out]

(3 + 3*sqrt(3))*(c + d*x)**(1/6)*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(2/3)*(b**(
1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))) - 3*3**(1/4)*sqrt((b*
*(2/3)*(c + d*x)**(2/3) - b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(1/3) + (a*d -
b*c)**(2/3))/(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))**2)*
(c + d*x)**(1/6)*(a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1
/3))*elliptic_e(acos((b**(1/3)*(-sqrt(3) + 1)*(c + d*x)**(1/3) + (a*d - b*c)**(1
/3))/(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))), sqrt(3)/4
+ 1/2)/(b**(2/3)*d*sqrt(b**(1/3)*(c + d*x)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (
a*d - b*c)**(1/3))/(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3)
)**2)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 3**(3/4)*sqrt((b**(2/3)*(c + d*x)**(2/3
) - b**(1/3)*(c + d*x)**(1/3)*(a*d - b*c)**(1/3) + (a*d - b*c)**(2/3))/(b**(1/3)
*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))**2)*(-sqrt(3) + 1)*(c + d*
x)**(1/6)*(a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*el
liptic_f(acos((b**(1/3)*(-sqrt(3) + 1)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b
**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))), sqrt(3)/4 + 1/2)/
(2*b**(2/3)*d*sqrt(b**(1/3)*(c + d*x)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d -
b*c)**(1/3))/(b**(1/3)*(1 + sqrt(3))*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))**2)
*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.0650253, size = 73, normalized size = 0.09 \[ \frac{6 (c+d x)^{5/6} \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )}{5 d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(1/6)),x]

[Out]

(6*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*(c + d*x)^(5/6)*Hypergeometric2F1[1/2, 5/6
, 11/6, (b*(c + d*x))/(b*c - a*d)])/(5*d*Sqrt[a + b*x])

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt [6]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(d*x+c)^(1/6),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/6)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/6)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/6)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + a)*(d*x + c)^(1/6)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \sqrt [6]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/6),x)

[Out]

Integral(1/(sqrt(a + b*x)*(c + d*x)**(1/6)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/6)),x, algorithm="giac")

[Out]

Timed out