3.1741 \(\int \frac{\sqrt [6]{c+d x}}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=367 \[ \frac{\sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} b \sqrt{a+b x} \sqrt [3]{b c-a d} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{2 \sqrt [6]{c+d x}}{b \sqrt{a+b x}} \]

[Out]

(-2*(c + d*x)^(1/6))/(b*Sqrt[a + b*x]) + ((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]*EllipticF[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])/(3^(1/4)*b*(b*c - a*d)^(1/3)*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 = 0.531794, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{b c-a d}-\left (1-\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} b \sqrt{a+b x} \sqrt [3]{b c-a d} \sqrt{-\frac{\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt{3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac{2 \sqrt [6]{c+d x}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/6)/(a + b*x)^(3/2),x]

[Out]

(-2*(c + d*x)^(1/6))/(b*Sqrt[a + b*x]) + ((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]*EllipticF[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])/(3^(1/4)*b*(b*c - a*d)^(1/3)*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 = 20.4576, size = 318, normalized size = 0.87 \[ \frac{3^{\frac{3}{4}} \sqrt{\frac{b^{\frac{2}{3}} \left (c + d x\right )^{\frac{2}{3}} - \sqrt [3]{b} \sqrt [3]{c + d x} \sqrt [3]{a d - b c} + \left (a d - b c\right )^{\frac{2}{3}}}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [6]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{b} \left (- \sqrt{3} + 1\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}}{\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 b \sqrt{\frac{\sqrt [3]{b} \sqrt [3]{c + d x} \left (\sqrt [3]{b} \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )}{\left (\sqrt [3]{b} \left (1 + \sqrt{3}\right ) \sqrt [3]{c + d x} + \sqrt [3]{a d - b c}\right )^{2}}} \sqrt [3]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{2 \sqrt [6]{c + d x}}{b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/6)/(b*x+a)**(3/2),x)

[Out]

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)*(c + d*x)**(1/6)*(b**(1/3)*(c + d*x)**(1/3) + (a*d - b*c)**(1/3
))*elliptic_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)/(3*b*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)*(a*
d - b*c)**(1/3)*sqrt(a - b*c/d + b*(c + d*x)/d)) - 2*(c + d*x)**(1/6)/(b*sqrt(a
+ b*x))

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Mathematica [C]  time = 0.0883351, size = 74, normalized size = 0.2 \[ \frac{2 \sqrt [6]{c+d x} \left (\sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )-1\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/6)/(a + b*x)^(3/2),x]

[Out]

(2*(c + d*x)^(1/6)*(-1 + Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/
6, 1/2, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(b*Sqrt[a + b*x])

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{1\sqrt [6]{dx+c} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/6)/(b*x+a)^(3/2),x)

[Out]

int((d*x+c)^(1/6)/(b*x+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((d*x + c)^(1/6)/(b*x + a)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/6)/(b*x+a)**(3/2),x)

[Out]

Integral((c + d*x)**(1/6)/(a + b*x)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(3/2), x)