3.1788 \(\int (a+b x)^{5/6} (c+d x)^{5/6} \, dx\)

Optimal. Leaf size=74 \[ \frac{6 (a+b x)^{11/6} (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

[Out]

(6*(a + b*x)^(11/6)*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, 11/6, 17/6, -((d*(a
+ b*x))/(b*c - a*d))])/(11*b*((b*(c + d*x))/(b*c - a*d))^(5/6))

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Rubi [A]  time = 0.0851136, antiderivative size = 74, 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 \[ \frac{6 (a+b x)^{11/6} (c+d x)^{5/6} \, _2F_1\left (-\frac{5}{6},\frac{11}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/6}} \]

Antiderivative was successfully verified.

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

[Out]

(6*(a + b*x)^(11/6)*(c + d*x)^(5/6)*Hypergeometric2F1[-5/6, 11/6, 17/6, -((d*(a
+ b*x))/(b*c - a*d))])/(11*b*((b*(c + d*x))/(b*c - a*d))^(5/6))

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Rubi in Sympy [A]  time = 15.3284, size = 61, normalized size = 0.82 \[ \frac{6 \left (a + b x\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{11}{6}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, \frac{11}{6} \\ \frac{17}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{11 d \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.230219, size = 108, normalized size = 1.46 \[ \frac{3 (c+d x)^{5/6} \left (d (a+b x) (a d+b (c+2 d x))-(b c-a d)^2 \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{16 b d^2 \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c + d*x)^(5/6)*(d*(a + b*x)*(a*d + b*(c + 2*d*x)) - (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)]))/(16*b*d^2*(a + b*x)^(1/6))

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{{\frac{5}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out