\(\int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx\) [1769]

Optimal result

Integrand size = 19, antiderivative size = 74 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {7}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}} \]

[Out]

6/7*(b*x+a)^(7/6)*(b*(d*x+c)/(-a*d+b*c))^(5/6)*hypergeom([5/6, 7/6],[13/6],-d*(b*x+a)/(-a*d+b*c))/b/(d*x+c)^(5
/6)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, 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} \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {7}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}} \]

[In]

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

[Out]

(6*(a + b*x)^(7/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[5/6, 7/6, 13/6, -((d*(a + b*x))/(b*c -
a*d))])/(7*b*(c + d*x)^(5/6))

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 (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \int \frac {\sqrt [6]{a+b x}}{\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{5/6}} \, dx}{(c+d x)^{5/6}} \\ & = \frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b (c+d x)^{5/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\frac {6 (a+b x)^{7/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {7}{6},\frac {13}{6},\frac {d (a+b x)}{-b c+a d}\right )}{7 b (c+d x)^{5/6}} \]

[In]

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

[Out]

(6*(a + b*x)^(7/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[5/6, 7/6, 13/6, (d*(a + b*x))/(-(b*c) +
a*d)])/(7*b*(c + d*x)^(5/6))

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {5}{6}}}d x\]

[In]

int((b*x+a)^(1/6)/(d*x+c)^(5/6),x)

[Out]

int((b*x+a)^(1/6)/(d*x+c)^(5/6),x)

Fricas [F]

\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {5}{6}}} \,d x } \]

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(5/6),x, algorithm="fricas")

[Out]

integral((b*x + a)^(1/6)/(d*x + c)^(5/6), x)

Sympy [F]

\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\int \frac {\sqrt [6]{a + b x}}{\left (c + d x\right )^{\frac {5}{6}}}\, dx \]

[In]

integrate((b*x+a)**(1/6)/(d*x+c)**(5/6),x)

[Out]

Integral((a + b*x)**(1/6)/(c + d*x)**(5/6), x)

Maxima [F]

\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {5}{6}}} \,d x } \]

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(5/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(5/6), x)

Giac [F]

\[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}}}{{\left (d x + c\right )}^{\frac {5}{6}}} \,d x } \]

[In]

integrate((b*x+a)^(1/6)/(d*x+c)^(5/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/6)/(d*x + c)^(5/6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{5/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/6}}{{\left (c+d\,x\right )}^{5/6}} \,d x \]

[In]

int((a + b*x)^(1/6)/(c + d*x)^(5/6),x)

[Out]

int((a + b*x)^(1/6)/(c + d*x)^(5/6), x)