3.1797 \(\int \frac{(a+b x)^{7/6}}{(c+d x)^{11/6}} \, dx\)

Optimal. Leaf size=81 \[ \frac{6 (a+b x)^{13/6} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{11}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 (c+d x)^{5/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(13/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[11/6, 13
/6, 19/6, -((d*(a + b*x))/(b*c - a*d))])/(13*(b*c - a*d)*(c + d*x)^(5/6))

_______________________________________________________________________________________

Rubi [A]  time = 0.0871387, antiderivative size = 81, 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 \[ \frac{6 (a+b x)^{13/6} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{11}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 (c+d x)^{5/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/6)/(c + d*x)^(11/6),x]

[Out]

(6*(a + b*x)^(13/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[11/6, 13
/6, 19/6, -((d*(a + b*x))/(b*c - a*d))])/(13*(b*c - a*d)*(c + d*x)^(5/6))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.4812, size = 73, normalized size = 0.9 \[ - \frac{6 \sqrt [6]{a + b x} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{6}, - \frac{5}{6} \\ \frac{1}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{5 d^{2} \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}} \left (c + d x\right )^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/6)/(d*x+c)**(11/6),x)

[Out]

-6*(a + b*x)**(1/6)*(a*d - b*c)*hyper((-7/6, -5/6), (1/6,), b*(-c - d*x)/(a*d -
b*c))/(5*d**2*(d*(a + b*x)/(a*d - b*c))**(1/6)*(c + d*x)**(5/6))

_______________________________________________________________________________________

Mathematica [A]  time = 0.347527, size = 99, normalized size = 1.22 \[ \frac{3 \sqrt [6]{a+b x} \sqrt [6]{c+d x} \left (\frac{7 b \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [6]{\frac{d (a+b x)}{a d-b c}}}+\frac{-2 a d+7 b c+5 b d x}{c+d x}\right )}{5 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/6)/(c + d*x)^(11/6),x]

[Out]

(3*(a + b*x)^(1/6)*(c + d*x)^(1/6)*((7*b*c - 2*a*d + 5*b*d*x)/(c + d*x) + (7*b*H
ypergeometric2F1[1/6, 5/6, 7/6, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*
c) + a*d))^(1/6)))/(5*d^2)

_______________________________________________________________________________________

Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{11}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/6)/(d*x+c)^(11/6),x)

[Out]

int((b*x+a)^(7/6)/(d*x+c)^(11/6),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{6}}}{{\left (d x + c\right )}^{\frac{11}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)/(d*x + c)^(11/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(11/6), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{7}{6}}}{{\left (d x + c\right )}^{\frac{11}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)/(d*x + c)^(11/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(7/6)/(d*x + c)^(11/6), x)

_______________________________________________________________________________________

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((b*x+a)**(7/6)/(d*x+c)**(11/6),x)

[Out]

Timed out

_______________________________________________________________________________________

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((b*x + a)^(7/6)/(d*x + c)^(11/6),x, algorithm="giac")

[Out]

Timed out