3.1813 \(\int \frac{(c+d x)^{11/6}}{\sqrt [6]{a+b x}} \, dx\)

Optimal. Leaf size=82 \[ \frac{6 (a+b x)^{5/6} (c+d x)^{5/6} (b c-a d) \, _2F_1\left (-\frac{11}{6},\frac{5}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 b^2 \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

[Out]

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

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Rubi [A]  time = 0.100153, antiderivative size = 82, 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 \[ \frac{6 (a+b x)^{5/6} (c+d x)^{5/6} (b c-a d) \, _2F_1\left (-\frac{11}{6},\frac{5}{6};\frac{11}{6};-\frac{d (a+b x)}{b c-a d}\right )}{5 b^2 \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(11/6)/(a + b*x)^(1/6),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.7333, size = 65, normalized size = 0.79 \[ \frac{6 \left (a + b x\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{17}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{6}, \frac{17}{6} \\ \frac{23}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{17 \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(11/6)/(b*x+a)**(1/6),x)

[Out]

6*(a + b*x)**(5/6)*(c + d*x)**(17/6)*hyper((1/6, 17/6), (23/6,), b*(-c - d*x)/(a
*d - b*c))/(17*(d*(a + b*x)/(a*d - b*c))**(5/6)*(a*d - b*c))

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Mathematica [A]  time = 0.239431, size = 111, normalized size = 1.35 \[ \frac{3 (c+d x)^{5/6} \left (11 (b c-a d)^2 \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) (11 a d-21 b c-10 b d x)\right )}{80 b^2 d \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(11/6)/(a + b*x)^(1/6),x]

[Out]

(3*(c + d*x)^(5/6)*(-(d*(a + b*x)*(-21*b*c + 11*a*d - 10*b*d*x)) + 11*(b*c - a*d
)^2*((d*(a + b*x))/(-(b*c) + a*d))^(1/6)*Hypergeometric2F1[1/6, 5/6, 11/6, (b*(c
+ d*x))/(b*c - a*d)]))/(80*b^2*d*(a + b*x)^(1/6))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{11}{6}}}{\frac{1}{\sqrt [6]{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(11/6)/(b*x+a)^(1/6),x)

[Out]

int((d*x+c)^(11/6)/(b*x+a)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{11}{6}}}{{\left (b x + a\right )}^{\frac{1}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(11/6)/(b*x + a)^(1/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(11/6)/(b*x + a)^(1/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{11}{6}}}{{\left (b x + a\right )}^{\frac{1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(11/6)/(b*x + a)^(1/6),x, algorithm="fricas")

[Out]

integral((d*x + c)^(11/6)/(b*x + a)^(1/6), x)

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Sympy [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)**(11/6)/(b*x+a)**(1/6),x)

[Out]

Timed out

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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)^(11/6)/(b*x + a)^(1/6),x, algorithm="giac")

[Out]

Timed out