3.1790 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{7/6}} \, dx\)

Optimal. Leaf size=81 \[ \frac{6 (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{7}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[7/6, 11/
6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0874021, 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 \[ \frac{6 (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{7}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[7/6, 11/
6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)*(c + d*x)^(1/6))

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Rubi in Sympy [A]  time = 13.0401, size = 63, normalized size = 0.78 \[ - \frac{6 \left (a + b x\right )^{\frac{5}{6}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \sqrt [6]{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.154923, size = 87, normalized size = 1.07 \[ \frac{6 b (c+d x) \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-6 d (a+b x)}{d^2 \sqrt [6]{a+b x} \sqrt [6]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*d*(a + b*x) + 6*b*((d*(a + b*x))/(-(b*c) + a*d))^(1/6)*(c + d*x)*Hypergeomet
ric2F1[1/6, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)])/(d^2*(a + b*x)^(1/6)*(c + d*x
)^(1/6))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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