\(\int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx\) [1807]

Optimal result

Integrand size = 19, antiderivative size = 309 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}} \]

[Out]

2*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(1/6)/d^(5/6)-1/2*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/(d
*x+c)^(1/3)-b^(1/6)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))/b^(1/6)/d^(5/6)+1/2*ln(b^(1/3)+d^(1/3)*(b*x+a)^(1/3)/
(d*x+c)^(1/3)+b^(1/6)*d^(1/6)*(b*x+a)^(1/6)/(d*x+c)^(1/6))/b^(1/6)/d^(5/6)-arctan(-1/3*3^(1/2)+2/3*d^(1/6)*(b*
x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6)*3^(1/2))*3^(1/2)/b^(1/6)/d^(5/6)-arctan(1/3*3^(1/2)+2/3*d^(1/6)*(b*x+a)^(1/6)
/b^(1/6)/(d*x+c)^(1/6)*3^(1/2))*3^(1/2)/b^(1/6)/d^(5/6)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {65, 338, 302, 648, 632, 210, 642, 214} \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}} \]

[In]

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

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6))
- (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)
) + (2*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(b^(1/6)*d^(5/6)) - Log[b^(1/3) + (d^(1/3
)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c + d*x)^(1/6)]/(2*b^(1/6)*d^(5/6)) +
Log[b^(1/3) + (d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c + d*x)^(1/6)]/(
2*b^(1/6)*d^(5/6))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 302

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-
a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]
*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s
^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))*Int[1/(r^2 - s^2*x^2), x] + Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (
n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {6 \text {Subst}\left (\int \frac {x^4}{\left (c-\frac {a d}{b}+\frac {d x^6}{b}\right )^{5/6}} \, dx,x,\sqrt [6]{a+b x}\right )}{b} \\ & = \frac {6 \text {Subst}\left (\int \frac {x^4}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{d^{2/3}}+\frac {2 \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}-\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{2/3}}+\frac {2 \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}+\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{2/3}} \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\text {Subst}\left (\int \frac {\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{2/3}} \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}} \\ & = \frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx=\frac {\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}-\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+\arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{\sqrt [6]{b} d^{5/6}} \]

[In]

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

[Out]

(Sqrt[3]*(ArcTan[(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))/(2*d^(1/6)*(a + b*x)^(1/6) - b^(1/6)*(c + d*x)^(1/6))] + Ar
cTan[(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))/(2*d^(1/6)*(a + b*x)^(1/6) + b^(1/6)*(c + d*x)^(1/6))]) + 2*ArcTanh[(b^
(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6)
) + (b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))])/(b^(1/6)*d^(5/6))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx=\frac {1}{2} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + a d + \sqrt {-3} {\left (b d x + a d\right )}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - \frac {1}{2} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + a d + \sqrt {-3} {\left (b d x + a d\right )}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - \frac {1}{2} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + a d - \sqrt {-3} {\left (b d x + a d\right )}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + \frac {1}{2} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + a d - \sqrt {-3} {\left (b d x + a d\right )}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) \]

[In]

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

[Out]

1/2*(sqrt(-3) + 1)*(1/(b*d^5))^(1/6)*log(((b*d*x + a*d + sqrt(-3)*(b*d*x + a*d))*(1/(b*d^5))^(1/6) + 2*(b*x +
a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) - 1/2*(sqrt(-3) + 1)*(1/(b*d^5))^(1/6)*log(-((b*d*x + a*d + sqrt(-3)*(b*d
*x + a*d))*(1/(b*d^5))^(1/6) - 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) - 1/2*(sqrt(-3) - 1)*(1/(b*d^5))^
(1/6)*log(((b*d*x + a*d - sqrt(-3)*(b*d*x + a*d))*(1/(b*d^5))^(1/6) + 2*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x
+ a)) + 1/2*(sqrt(-3) - 1)*(1/(b*d^5))^(1/6)*log(-((b*d*x + a*d - sqrt(-3)*(b*d*x + a*d))*(1/(b*d^5))^(1/6) -
2*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) + (1/(b*d^5))^(1/6)*log(((b*d*x + a*d)*(1/(b*d^5))^(1/6) + (b*x
+ a)^(5/6)*(d*x + c)^(1/6))/(b*x + a)) - (1/(b*d^5))^(1/6)*log(-((b*d*x + a*d)*(1/(b*d^5))^(1/6) - (b*x + a)^(
5/6)*(d*x + c)^(1/6))/(b*x + a))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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