3.1568 \(\int \frac{1}{\sqrt{a+b x} \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=762 \[ -\frac{6 \sqrt{a+b x}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{2 \sqrt{2} 3^{3/4} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

[Out]

(-6*Sqrt[a + b*x])/(b^(2/3)*((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)
^(1/3))) + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^(1/3)*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3
)*(c + d*x)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3
)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))],
-7 + 4*Sqrt[3]])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d
)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(
c + d*x)^(1/3))^2)]) - (2*Sqrt[2]*3^(3/4)*(b*c - a*d)^(1/3)*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))
], -7 + 4*Sqrt[3]])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a
*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)
*(c + d*x)^(1/3))^2)])

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Rubi [A]  time = 1.2798, antiderivative size = 762, 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 \[ -\frac{6 \sqrt{a+b x}}{b^{2/3} \left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac{2 \sqrt{2} 3^{3/4} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt{3}\right )}{b^{2/3} d \sqrt{a+b x} \sqrt{-\frac{\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*(c + d*x)^(1/3)),x]

[Out]

(-6*Sqrt[a + b*x])/(b^(2/3)*((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)
^(1/3))) + (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^(1/3)*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3
)*(c + d*x)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3
)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))],
-7 + 4*Sqrt[3]])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d
)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(
c + d*x)^(1/3))^2)]) - (2*Sqrt[2]*3^(3/4)*(b*c - a*d)^(1/3)*((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))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*(b*c - a*d)^(1/3) - b^(1
/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))
], -7 + 4*Sqrt[3]])/(b^(2/3)*d*Sqrt[a + b*x]*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a
*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)
*(c + d*x)^(1/3))^2)])

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Rubi in Sympy [A]  time = 64.1664, size = 654, normalized size = 0.86 \[ \frac{6 \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{b^{\frac{2}{3}} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )} - \frac{3 \sqrt [4]{3} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} d \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}}{\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{b^{\frac{2}{3}} d \sqrt{\frac{\sqrt [3]{a d - b c} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \sqrt [3]{c + d x} + \left (1 + \sqrt{3}\right ) \sqrt [3]{a d - b c}\right )^{2}}} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(2/3)*(b**(1/3)*(c + d*x)**(1/3) + (1 + sq
rt(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)*(c + d*x
)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(-sqrt(3) + 2)*(a*d - b*c)*
*(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))*elliptic_e(asin((b**(1/3
)*(c + d*x)**(1/3) - (-1 + sqrt(3))*(a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/
3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))), -7 - 4*sqrt(3))/(b**(2/3)*d*sqrt((a*d -
b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x
)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3))**2)*sqrt(a - b*c/d + b*(c + d*x)/d)
) + 2*sqrt(2)*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)*(c + d*x)**(1/3) + (1 + sq
rt(3))*(a*d - b*c)**(1/3))**2)*(a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/3) + (
a*d - b*c)**(1/3))*elliptic_f(asin((b**(1/3)*(c + d*x)**(1/3) - (-1 + sqrt(3))*(
a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(3))*(a*d - b*c)**(1/3)
)), -7 - 4*sqrt(3))/(b**(2/3)*d*sqrt((a*d - b*c)**(1/3)*(b**(1/3)*(c + d*x)**(1/
3) + (a*d - b*c)**(1/3))/(b**(1/3)*(c + d*x)**(1/3) + (1 + sqrt(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 = 0.066143, size = 73, normalized size = 0.1 \[ \frac{3 (c+d x)^{2/3} \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{2 d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*(c + d*x)^(1/3)),x]

[Out]

(3*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*(c + d*x)^(2/3)*Hypergeometric2F1[1/2, 2/3
, 5/3, (b*(c + d*x))/(b*c - a*d)])/(2*d*Sqrt[a + b*x])

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(d*x+c)^(1/3),x)

[Out]

int(1/(b*x+a)^(1/2)/(d*x+c)^(1/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + a)*(d*x + c)^(1/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(1/3)), x)