3.1791 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{13/6}} \, dx\)

Optimal. Leaf size=82 \[ \frac{6 b (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{11}{6},\frac{13}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)^2} \]

[Out]

(6*b*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[11/6,
13/6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)^2*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0883566, antiderivative size = 82, 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 \[ \frac{6 b (a+b x)^{11/6} \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{11}{6},\frac{13}{6};\frac{17}{6};-\frac{d (a+b x)}{b c-a d}\right )}{11 \sqrt [6]{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/6)/(c + d*x)^(13/6),x]

[Out]

(6*b*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[11/6,
13/6, 17/6, -((d*(a + b*x))/(b*c - a*d))])/(11*(b*c - a*d)^2*(c + d*x)^(1/6))

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Rubi in Sympy [A]  time = 12.8603, size = 66, normalized size = 0.8 \[ - \frac{6 \left (a + b x\right )^{\frac{5}{6}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, - \frac{7}{6} \\ - \frac{1}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{7 d \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/6)/(d*x+c)**(13/6),x)

[Out]

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

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Mathematica [A]  time = 0.22573, size = 117, normalized size = 1.43 \[ \frac{24 b^2 (c+d x)^2 \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-6 d (a+b x) (a d+4 b c+5 b d x)}{7 d^2 \sqrt [6]{a+b x} (c+d x)^{7/6} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/6)/(c + d*x)^(13/6),x]

[Out]

(-6*d*(a + b*x)*(4*b*c + a*d + 5*b*d*x) + 24*b^2*((d*(a + b*x))/(-(b*c) + a*d))^
(1/6)*(c + d*x)^2*Hypergeometric2F1[1/6, 5/6, 11/6, (b*(c + d*x))/(b*c - a*d)])/
(7*d^2*(-(b*c) + a*d)*(a + b*x)^(1/6)*(c + d*x)^(7/6))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{13}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/6)/(d*x+c)^(13/6),x)

[Out]

int((b*x+a)^(5/6)/(d*x+c)^(13/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(13/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(13/6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (d x + c\right )}^{\frac{1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(13/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(5/6)/((d^2*x^2 + 2*c*d*x + c^2)*(d*x + c)^(1/6)), 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)**(13/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(13/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(13/6), x)