\(\int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx\) [1841]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=-\frac {6 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {7}{6},\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) \sqrt [6]{a+b x} \sqrt [6]{c+d x}} \]

[Out]

-6*(b*(d*x+c)/(-a*d+b*c))^(1/6)*hypergeom([-1/6, 7/6],[5/6],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)/(b*x+a)^(1/6)/(d
*x+c)^(1/6)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {72, 71} \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=-\frac {6 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {7}{6},\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{\sqrt [6]{a+b x} \sqrt [6]{c+d x} (b c-a d)} \]

[In]

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

[Out]

(-6*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-1/6, 7/6, 5/6, -((d*(a + b*x))/(b*c - a*d))])/((b*c -
a*d)*(a + b*x)^(1/6)*(c + d*x)^(1/6))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{(a+b x)^{7/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{7/6}} \, dx}{(b c-a d) \sqrt [6]{c+d x}} \\ & = -\frac {6 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {1}{6},\frac {7}{6};\frac {5}{6};-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) \sqrt [6]{a+b x} \sqrt [6]{c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=-\frac {6 \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {7}{6},\frac {5}{6},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [6]{a+b x} (c+d x)^{7/6}} \]

[In]

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

[Out]

(-6*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[-1/6, 7/6, 5/6, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(a +
b*x)^(1/6)*(c + d*x)^(7/6))

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {7}{6}} \left (d x +c \right )^{\frac {7}{6}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]

[In]

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

[Out]

integral((b*x + a)^(5/6)*(d*x + c)^(5/6)/(b^2*d^2*x^4 + a^2*c^2 + 2*(b^2*c*d + a*b*d^2)*x^3 + (b^2*c^2 + 4*a*b
*c*d + a^2*d^2)*x^2 + 2*(a*b*c^2 + a^2*c*d)*x), x)

Sympy [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{6}} \left (c + d x\right )^{\frac {7}{6}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{7/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{7/6}} \,d x \]

[In]

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

[Out]

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