3.1843 \(\int \frac{1}{(a+b x)^{7/6} (c+d x)^{13/6}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{6 b \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{13}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^2} \]

[Out]

(-6*b*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 13/6, 5/6, -((d*
(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*(a + b*x)^(1/6)*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0904365, antiderivative size = 80, 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 \[ -\frac{6 b \sqrt [6]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{6},\frac{13}{6};\frac{5}{6};-\frac{d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/6)*(c + d*x)^(13/6)),x]

[Out]

(-6*b*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 13/6, 5/6, -((d*
(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*(a + b*x)^(1/6)*(c + d*x)^(1/6))

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Rubi in Sympy [A]  time = 14.2498, size = 73, normalized size = 0.91 \[ - \frac{6 d \left (a + b x\right )^{\frac{5}{6}}{{}_{2}F_{1}\left (\begin{matrix} \frac{7}{6}, - \frac{7}{6} \\ - \frac{1}{6} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{7 \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{\frac{5}{6}} \left (c + d x\right )^{\frac{7}{6}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/6)/(d*x+c)**(13/6),x)

[Out]

-6*d*(a + b*x)**(5/6)*hyper((7/6, -7/6), (-1/6,), b*(-c - d*x)/(a*d - b*c))/(7*(
d*(a + b*x)/(a*d - b*c))**(5/6)*(c + d*x)**(7/6)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.404952, size = 138, normalized size = 1.72 \[ \frac{6 \left (5 a^2 d^2+64 b^2 (c+d x)^2 \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{b (c+d x)}{b c-a d}\right )-10 a b d (5 c+4 d x)-5 b^2 \left (7 c^2+24 c d x+16 d^2 x^2\right )\right )}{35 \sqrt [6]{a+b x} (c+d x)^{7/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(13/6)),x]

[Out]

(6*(5*a^2*d^2 - 10*a*b*d*(5*c + 4*d*x) - 5*b^2*(7*c^2 + 24*c*d*x + 16*d^2*x^2) +
64*b^2*((d*(a + b*x))/(-(b*c) + a*d))^(1/6)*(c + d*x)^2*Hypergeometric2F1[1/6,
5/6, 11/6, (b*(c + d*x))/(b*c - a*d)]))/(35*(b*c - a*d)^3*(a + b*x)^(1/6)*(c + d
*x)^(7/6))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{7}{6}}} \left ( dx+c \right ) ^{-{\frac{13}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/6)/(d*x+c)^(13/6),x)

[Out]

int(1/(b*x+a)^(7/6)/(d*x+c)^(13/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(13/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(13/6)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(13/6)),x, algorithm="fricas")

[Out]

integral(1/((b*d^2*x^3 + a*c^2 + (2*b*c*d + a*d^2)*x^2 + (b*c^2 + 2*a*c*d)*x)*(b
*x + a)^(1/6)*(d*x + c)^(1/6)), 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(1/(b*x+a)**(7/6)/(d*x+c)**(13/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(13/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(13/6)), x)