\(\int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx\) [1825]

Optimal result

Integrand size = 19, antiderivative size = 378 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx=\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}-\frac {5 (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}-\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{11/6} \sqrt [6]{d}} \]

[Out]

(b*x+a)^(1/6)*(d*x+c)^(5/6)/b+5/3*(-a*d+b*c)*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(11/6)/d^(
1/6)-5/12*(-a*d+b*c)*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^(11/6)/d^(1/6)+5/12*(-a*d+b*c)*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^(11/6)/d^(1/6)+5/6*(-a*d+b*c)*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))/b^(11/6)/d^(1/6)*3^(1/2)+5/6*(-a*d+b*c)*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))/b^(11/6)/d^(1/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {52, 65, 246, 216, 648, 632, 210, 642, 214} \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx=-\frac {5 (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}-\frac {5 (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{11/6} \sqrt [6]{d}}+\frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b} \]

[In]

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

[Out]

((a + b*x)^(1/6)*(c + d*x)^(5/6))/b - (5*(b*c - a*d)*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b
^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(11/6)*d^(1/6)) + (5*(b*c - a*d)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x
)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(11/6)*d^(1/6)) + (5*(b*c - a*d)*ArcTanh[(d^(1/6)*(a
+ b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(11/6)*d^(1/6)) - (5*(b*c - a*d)*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)])/(12*b^(11/6)*d^(1/6)) + (5*(b
*c - a*d)*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)])/(12*b^(11/6)*d^(1/6))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 216

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 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 {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{6 b} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d x^6}{b}}} \, dx,x,\sqrt [6]{a+b x}\right )}{b^2} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b^2} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {\sqrt [6]{b}-\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 )}{3 b^{11/6}}+\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {\sqrt [6]{b}+\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 )}{3 b^{11/6}}+\frac {(5 (b c-a d)) \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 )}{3 b^{5/3}} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}+\frac {(5 (b c-a d)) \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 )}{4 b^{5/3}}+\frac {(5 (b c-a d)) \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 )}{4 b^{5/3}}-\frac {(5 (b c-a d)) \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 )}{12 b^{11/6} \sqrt [6]{d}}+\frac {(5 (b c-a d)) \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 )}{12 b^{11/6} \sqrt [6]{d}} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}+\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}-\frac {5 (b c-a d) \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 )}{12 b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \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 )}{12 b^{11/6} \sqrt [6]{d}}+\frac {(5 (b c-a d)) \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 )}{2 b^{11/6} \sqrt [6]{d}}-\frac {(5 (b c-a d)) \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 )}{2 b^{11/6} \sqrt [6]{d}} \\ & = \frac {\sqrt [6]{a+b x} (c+d x)^{5/6}}{b}-\frac {5 (b c-a d) \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 )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \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 )}{2 \sqrt {3} b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{11/6} \sqrt [6]{d}}-\frac {5 (b c-a d) \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 )}{12 b^{11/6} \sqrt [6]{d}}+\frac {5 (b c-a d) \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 )}{12 b^{11/6} \sqrt [6]{d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.78 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx=\frac {6 b^{5/6} \sqrt [6]{d} \sqrt [6]{a+b x} (c+d x)^{5/6}+5 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-5 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+10 (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+5 (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 b^{11/6} \sqrt [6]{d}} \]

[In]

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

[Out]

(6*b^(5/6)*d^(1/6)*(a + b*x)^(1/6)*(c + d*x)^(5/6) + 5*Sqrt[3]*(b*c - a*d)*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))] - 5*Sqrt[3]*(b*c - a*d)*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))] + 10*(b*c - a*d)*ArcTanh[(b^(1/6)*(c +
d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + 5*(b*c - a*d)*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))])/(6*b^(11/6)*d^(1/6))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1455 vs. \(2 (280) = 560\).

Time = 0.26 (sec) , antiderivative size = 1455, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{5/6}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(5*(sqrt(-3)*b + b)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*
d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6)*log(-5/2*(2*(b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + (b^2*
d*x + b^2*c + sqrt(-3)*(b^2*d*x + b^2*c))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3
+ 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6))/(d*x + c)) - 5*(sqrt(-3)*b + b)*((b^6*c^6 - 6
*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d
))^(1/6)*log(-5/2*(2*(b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d*x + b^2*c + sqrt(-3)*(b^2*d*x + b^2*
c))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 +
a^6*d^6)/(b^11*d))^(1/6))/(d*x + c)) - 5*(sqrt(-3)*b - b)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20
*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6)*log(-5/2*(2*(b*c - a*d)*(b*x
+ a)^(1/6)*(d*x + c)^(5/6) + (b^2*d*x + b^2*c - sqrt(-3)*(b^2*d*x + b^2*c))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2
*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6))/(d*x + c))
+ 5*(sqrt(-3)*b - b)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4
- 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6)*log(-5/2*(2*(b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d*x
+ b^2*c - sqrt(-3)*(b^2*d*x + b^2*c))*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15
*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6))/(d*x + c)) + 10*b*((b^6*c^6 - 6*a*b^5*c^5*d + 15*
a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6)*log(-5*((
b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + (b^2*d*x + b^2*c)*((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2
- 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6))/(d*x + c)) - 10*b*((b^6*
c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/
(b^11*d))^(1/6)*log(-5*((b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d*x + b^2*c)*((b^6*c^6 - 6*a*b^5*c^
5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)/(b^11*d))^(1/6))
/(d*x + c)) + 12*(b*x + a)^(1/6)*(d*x + c)^(5/6))/b

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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