3.1821 \(\int \frac{\sqrt [6]{c+d x}}{(a+b x)^{5/6}} \, dx\)

Optimal. Leaf size=72 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

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

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Rubi [A]  time = 0.0848262, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(1/6)/(a + b*x)^(5/6),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.4542, size = 65, normalized size = 0.9 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{7}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{7}{6} \\ \frac{13}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{7 \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/6)/(b*x+a)**(5/6),x)

[Out]

6*(a + b*x)**(1/6)*(c + d*x)**(7/6)*hyper((5/6, 7/6), (13/6,), b*(-c - d*x)/(a*d
- b*c))/(7*(d*(a + b*x)/(a*d - b*c))**(1/6)*(a*d - b*c))

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Mathematica [A]  time = 0.145655, size = 91, normalized size = 1.26 \[ \frac{3 \sqrt [6]{c+d x} \left ((b c-a d) \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{b d (a+b x)^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(1/6)/(a + b*x)^(5/6),x]

[Out]

(3*(c + d*x)^(1/6)*(d*(a + b*x) + (b*c - a*d)*((d*(a + b*x))/(-(b*c) + a*d))^(5/
6)*Hypergeometric2F1[1/6, 5/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(b*d*(a + b*x)^
(5/6))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1\sqrt [6]{dx+c} \left ( bx+a \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/6)/(b*x+a)^(5/6),x)

[Out]

int((d*x+c)^(1/6)/(b*x+a)^(5/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(5/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(5/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(1/6)/(b*x + a)^(5/6),x, algorithm="fricas")

[Out]

integral((d*x + c)^(1/6)/(b*x + a)^(5/6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac{5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/6)/(b*x+a)**(5/6),x)

[Out]

Integral((c + d*x)**(1/6)/(a + b*x)**(5/6), 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((d*x + c)^(1/6)/(b*x + a)^(5/6),x, algorithm="giac")

[Out]

Timed out