3.1794 \(\int (a+b x)^{7/6} (c+d x)^{7/6} \, dx\)

Optimal. Leaf size=82 \[ \frac{6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

(6*(b*c - a*d)*(a + b*x)^(13/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 13/6, 19
/6, -((d*(a + b*x))/(b*c - a*d))])/(13*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6))

_______________________________________________________________________________________

Rubi [A]  time = 0.096567, 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 \[ \frac{6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d) \, _2F_1\left (-\frac{7}{6},\frac{13}{6};\frac{19}{6};-\frac{d (a+b x)}{b c-a d}\right )}{13 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/6)*(c + d*x)^(7/6),x]

[Out]

(6*(b*c - a*d)*(a + b*x)^(13/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 13/6, 19
/6, -((d*(a + b*x))/(b*c - a*d))])/(13*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.7937, size = 70, normalized size = 0.85 \[ \frac{6 \sqrt [6]{a + b x} \left (c + d x\right )^{\frac{13}{6}} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{6}, \frac{13}{6} \\ \frac{19}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{13 d^{2} \sqrt [6]{\frac{d \left (a + b x\right )}{a d - b c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/6)*(d*x+c)**(7/6),x)

[Out]

6*(a + b*x)**(1/6)*(c + d*x)**(13/6)*(a*d - b*c)*hyper((-7/6, 13/6), (19/6,), b*
(-c - d*x)/(a*d - b*c))/(13*d**2*(d*(a + b*x)/(a*d - b*c))**(1/6))

_______________________________________________________________________________________

Mathematica [B]  time = 0.346107, size = 183, normalized size = 2.23 \[ \frac{3 \sqrt [6]{c+d x} \left (7 (b c-a d)^4 \left (\frac{d (a+b x)}{a d-b c}\right )^{5/6} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )-d (a+b x) \left (7 a^3 d^3-a^2 b d^2 (23 c+2 d x)-a b^2 d \left (23 c^2+92 c d x+48 d^2 x^2\right )+b^3 \left (7 c^3-2 c^2 d x-48 c d^2 x^2-32 d^3 x^3\right )\right )\right )}{320 b^2 d^3 (a+b x)^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/6)*(c + d*x)^(7/6),x]

[Out]

(3*(c + d*x)^(1/6)*(-(d*(a + b*x)*(7*a^3*d^3 - a^2*b*d^2*(23*c + 2*d*x) - a*b^2*
d*(23*c^2 + 92*c*d*x + 48*d^2*x^2) + b^3*(7*c^3 - 2*c^2*d*x - 48*c*d^2*x^2 - 32*
d^3*x^3))) + 7*(b*c - a*d)^4*((d*(a + b*x))/(-(b*c) + a*d))^(5/6)*Hypergeometric
2F1[1/6, 5/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(320*b^2*d^3*(a + b*x)^(5/6))

_______________________________________________________________________________________

Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{7}{6}}} \left ( dx+c \right ) ^{{\frac{7}{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/6)*(d*x+c)^(7/6),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{7}{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)*(d*x + c)^(7/6),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)*(d*x + c)^(7/6),x, algorithm="fricas")

[Out]

integral((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^(1/6)*(d*x + c)^(1/6), x)

_______________________________________________________________________________________

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)**(7/6)*(d*x+c)**(7/6),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(7/6)*(d*x + c)^(7/6),x, algorithm="giac")

[Out]

Timed out