\(\int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx\) [1767]

Optimal result

Integrand size = 19, antiderivative size = 82 \[ \int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx=\frac {6 (b c-a d) (a+b x)^{7/6} \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {7}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 82, 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} \[ \int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx=\frac {6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {7}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]

[In]

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

[Out]

(6*(b*c - a*d)*(a + b*x)^(7/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-7/6, 7/6, 13/6, -((d*(a + b*x))/(b*c - a*d))
])/(7*b^2*((b*(c + d*x))/(b*c - a*d))^(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 c-a d) \sqrt [6]{c+d x}\right ) \int \sqrt [6]{a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{7/6} \, dx}{b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \\ & = \frac {6 (b c-a d) (a+b x)^{7/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {7}{6},\frac {7}{6};\frac {13}{6};-\frac {d (a+b x)}{b c-a d}\right )}{7 b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \sqrt [6]{a+b x} (c+d x)^{7/6} \, dx=\frac {6 (a+b x)^{7/6} (c+d x)^{7/6} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {7}{6},\frac {13}{6},\frac {d (a+b x)}{-b c+a d}\right )}{7 b \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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