Optimal. Leaf size=82 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac{13}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b^3 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0911788, 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 \[ \frac{6 \sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^2 \, _2F_1\left (-\frac{13}{6},\frac{1}{6};\frac{7}{6};-\frac{d (a+b x)}{b c-a d}\right )}{b^3 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(13/6)/(a + b*x)^(5/6),x]
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Rubi in Sympy [A] time = 13.572, size = 65, normalized size = 0.79 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{19}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{6}, \frac{19}{6} \\ \frac{25}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{19 \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(13/6)/(b*x+a)**(5/6),x)
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Mathematica [A] time = 0.277603, size = 141, normalized size = 1.72 \[ -\frac{3 \sqrt [6]{c+d x} \left (-d (a+b x) \left (91 a^2 d^2-26 a b d (8 c+d x)+b^2 \left (133 c^2+58 c d x+16 d^2 x^2\right )\right )-91 (b c-a d)^3 \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{112 b^3 d (a+b x)^{5/6}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(13/6)/(a + b*x)^(5/6),x]
[Out]
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Maple [F] time = 0.056, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{13}{6}}} \left ( bx+a \right ) ^{-{\frac{5}{6}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(13/6)/(b*x+a)^(5/6),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{13}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(13/6)/(b*x + a)^(5/6),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{5}{6}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(13/6)/(b*x + a)^(5/6),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((d*x+c)**(13/6)/(b*x+a)**(5/6),x)
[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((d*x + c)^(13/6)/(b*x + a)^(5/6),x, algorithm="giac")
[Out]