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