3.1768 \(\int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx\)

Optimal. Leaf size=82 \[ \frac{6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{7}{6};\frac{13}{6};-\frac{d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

(6*(b*c - a*d)*(a + b*x)^(7/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 7/6, 13/6
, -((d*(a + b*x))/(b*c - a*d))])/(7*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6))

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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]

(6*(b*c - a*d)*(a + b*x)^(7/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 7/6, 13/6
, -((d*(a + b*x))/(b*c - a*d))])/(7*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6))

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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]

6*(a + b*x)**(1/6)*(c + d*x)**(13/6)*hyper((-1/6, 13/6), (19/6,), b*(-c - d*x)/(
a*d - b*c))/(13*d*(d*(a + b*x)/(a*d - b*c))**(1/6))

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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]

(3*(c + d*x)^(1/6)*(-(d*(a + b*x)*(7*a^2*d^2 - 2*a*b*d*(8*c + d*x) - b^2*(7*c^2
+ 30*c*d*x + 16*d^2*x^2))) - 7*(b*c - a*d)^3*((d*(a + b*x))/(-(b*c) + a*d))^(5/6
)*Hypergeometric2F1[1/6, 5/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(112*b^2*d^2*(a
+ b*x)^(5/6))

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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]

int((b*x+a)^(1/6)*(d*x+c)^(7/6),x)

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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]

integrate((b*x + a)^(1/6)*(d*x + c)^(7/6), x)

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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]

integral((b*x + a)^(1/6)*(d*x + c)^(7/6), x)

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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]

Timed 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]

Timed out