Optimal. Leaf size=82 \[ \frac{6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{7}{6};\frac{13}{6};-\frac{d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.0927567, 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 \[ \frac{6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{7}{6};\frac{13}{6};-\frac{d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(1/6)*(c + d*x)^(7/6),x]
[Out]
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Rubi in Sympy [A] time = 13.0374, size = 61, normalized size = 0.74 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{13}{6}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{13}{6} \\ \frac{19}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{13 d \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/6)*(d*x+c)**(7/6),x)
[Out]
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Mathematica [A] time = 0.261103, size = 142, normalized size = 1.73 \[ \frac{3 \sqrt [6]{c+d x} \left (-d (a+b x) \left (7 a^2 d^2-2 a b d (8 c+d x)+b^2 \left (-\left (7 c^2+30 c d x+16 d^2 x^2\right )\right )\right )-7 (b c-a d)^3 \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )\right )}{112 b^2 d^2 (a+b x)^{5/6}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(1/6)*(c + d*x)^(7/6),x]
[Out]
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int \sqrt [6]{bx+a} \left ( dx+c \right ) ^{{\frac{7}{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/6)*(d*x+c)^(7/6),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{7}{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/6)*(d*x + c)^(7/6),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{7}{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(1/6)*(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)**(1/6)*(d*x+c)**(7/6),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)^(7/6),x, algorithm="giac")
[Out]