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