Optimal. Leaf size=74 \[ \frac{6 (a+b x)^{7/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{7}{6};\frac{13}{6};-\frac{d (a+b x)}{b c-a d}\right )}{7 b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0839888, 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 \[ \frac{6 (a+b x)^{7/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},\frac{7}{6};\frac{13}{6};-\frac{d (a+b x)}{b c-a d}\right )}{7 b \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(1/6)*(c + d*x)^(1/6),x]
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Rubi in Sympy [A] time = 15.0169, size = 61, normalized size = 0.82 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{7}{6}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{7}{6} \\ \frac{13}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{7 d \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/6)*(d*x+c)**(1/6),x)
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Mathematica [A] time = 0.187683, size = 108, normalized size = 1.46 \[ \frac{3 \sqrt [6]{c+d x} \left (d (a+b x) (a d+b (c+2 d x))-(b c-a d)^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{8 b d^2 (a+b x)^{5/6}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(1/6)*(c + d*x)^(1/6),x]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int \sqrt [6]{bx+a}\sqrt [6]{dx+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/6)*(d*x+c)^(1/6),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/6)*(d*x + c)^(1/6),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/6)*(d*x + c)^(1/6),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt [6]{a + b x} \sqrt [6]{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/6)*(d*x+c)**(1/6),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((b*x + a)^(1/6)*(d*x + c)^(1/6),x, algorithm="giac")
[Out]