3.1844 \(\int \frac{1}{(a+b x)^{7/6} (c+d x)^{19/6}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{6 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^3} \]

[Out]

(-6*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 19/6, 5/6, -((
d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(a + b*x)^(1/6)*(c + d*x)^(1/6))

_______________________________________________________________________________________

Rubi [A]  time = 0.0915884, 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 \[ -\frac{6 b^2 \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/6)*(c + d*x)^(19/6)),x]

[Out]

(-6*b^2*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 19/6, 5/6, -((
d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(a + b*x)^(1/6)*(c + d*x)^(1/6))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.3448, size = 73, normalized size = 0.89 \[ - \frac{6 d \left (a + b x\right )^{\frac{5}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{6}, - \frac{13}{6} \\ - \frac{7}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{13 \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{13}{6}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/6)/(d*x+c)**(19/6),x)

[Out]

-6*d*(a + b*x)**(5/6)*hyper((7/6, -13/6), (-7/6,), b*(-c - d*x)/(a*d - b*c))/(13
*(d*(a + b*x)/(a*d - b*c))**(5/6)*(c + d*x)**(13/6)*(a*d - b*c)**2)

_______________________________________________________________________________________

Mathematica [B]  time = 0.414632, size = 179, normalized size = 2.18 \[ \frac{768 b^3 (c+d x)^3 \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-30 \left (a^3 d^3-a^2 b d^2 (5 c+2 d x)+a b^2 d \left (23 c^2+36 c d x+16 d^2 x^2\right )+b^3 \left (13 c^3+62 c^2 d x+80 c d^2 x^2+32 d^3 x^3\right )\right )}{65 \sqrt [6]{a+b x} (c+d x)^{13/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(19/6)),x]

[Out]

(-30*(a^3*d^3 - a^2*b*d^2*(5*c + 2*d*x) + a*b^2*d*(23*c^2 + 36*c*d*x + 16*d^2*x^
2) + b^3*(13*c^3 + 62*c^2*d*x + 80*c*d^2*x^2 + 32*d^3*x^3)) + 768*b^3*((d*(a + b
*x))/(-(b*c) + a*d))^(1/6)*(c + d*x)^3*Hypergeometric2F1[1/6, 5/6, 11/6, (b*(c +
d*x))/(b*c - a*d)])/(65*(b*c - a*d)^4*(a + b*x)^(1/6)*(c + d*x)^(13/6))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{19}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x)

[Out]

int(1/(b*x+a)^(7/6)/(d*x+c)^(19/6),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{19}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b d^{3} x^{4} + a c^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} x^{2} +{\left (b c^{3} + 3 \, a c^{2} d\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)),x, algorithm="fricas")

[Out]

integral(1/((b*d^3*x^4 + a*c^3 + (3*b*c*d^2 + a*d^3)*x^3 + 3*(b*c^2*d + a*c*d^2)
*x^2 + (b*c^3 + 3*a*c^2*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(1/(b*x+a)**(7/6)/(d*x+c)**(19/6),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{19}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(19/6)), x)