3.1822 \(\int \frac{1}{(a+b x)^{5/6} (c+d x)^{5/6}} \, dx\)

Optimal. Leaf size=72 \[ \frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b (c+d x)^{5/6}} \]

[Out]

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

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Rubi [A]  time = 0.0879739, 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 \[ \frac{6 \sqrt [6]{a+b x} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b (c+d x)^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7338, size = 63, normalized size = 0.88 \[ \frac{6 \sqrt [6]{a + b x} \sqrt [6]{c + d x}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{1}{6} \\ \frac{7}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{\sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0550848, size = 71, normalized size = 0.99 \[ \frac{6 \sqrt [6]{c+d x} \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )}{d (a+b x)^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out