3.1840 \(\int \frac{(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

[Out]

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

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Rubi [A]  time = 0.0874622, 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 \[ -\frac{6 (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},-\frac{1}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/6)/(a + b*x)^(7/6),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.1892, size = 68, normalized size = 0.94 \[ \frac{6 d \left (a + b x\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{11}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{6}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{11 \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/6)/(b*x+a)**(7/6),x)

[Out]

6*d*(a + b*x)**(5/6)*(c + d*x)**(11/6)*hyper((7/6, 11/6), (17/6,), b*(-c - d*x)/
(a*d - b*c))/(11*(d*(a + b*x)/(a*d - b*c))**(5/6)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.106784, size = 74, normalized size = 1.03 \[ \frac{6 (c+d x)^{5/6} \left (\sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-1\right )}{b \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/6)/(a + b*x)^(7/6),x]

[Out]

(6*(c + d*x)^(5/6)*(-1 + ((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)]))/(b*(a + b*x)^(1/6))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{5}{6}}} \left ( bx+a \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/6)/(b*x+a)^(7/6),x)

[Out]

int((d*x+c)^(5/6)/(b*x+a)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/6)/(b*x + a)^(7/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{5}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/6)/(b*x + a)^(7/6),x, algorithm="fricas")

[Out]

integral((d*x + c)^(5/6)/(b*x + a)^(7/6), 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)**(5/6)/(b*x+a)**(7/6),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)^(5/6)/(b*x + a)^(7/6),x, algorithm="giac")

[Out]

Timed out