\(\int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx\) [1772]

Optimal result

Integrand size = 19, antiderivative size = 427 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}+\frac {5 (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{11/6} d^{7/6}} \]

[Out]

5/12*(-a*d+b*c)*(b*x+a)^(1/6)*(d*x+c)^(5/6)/b/d+1/2*(b*x+a)^(7/6)*(d*x+c)^(5/6)/b-5/36*(-a*d+b*c)^2*arctanh(d^
(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(11/6)/d^(7/6)+5/144*(-a*d+b*c)^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^(11/6)/d^(7/6)-5/144*(-a*d+b*c)^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^(11/6)/d^(7/6)-5/72*(-a*d+b*c
)^2*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^(7/6)*3^(1/2)-5/72
*(-a*d+b*c)^2*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^(7/6)*3^(
1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\frac {5 (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {5 (b c-a d)^2 \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{144 b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{144 b^{11/6} d^{7/6}}+\frac {5 \sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b} \]

[In]

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

[Out]

(5*(b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(5/6))/(12*b*d) + ((a + b*x)^(7/6)*(c + d*x)^(5/6))/(2*b) + (5*(b*c -
a*d)^2*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/(24*Sqrt[3]*b^(11/6
)*d^(7/6)) - (5*(b*c - a*d)^2*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))
])/(24*Sqrt[3]*b^(11/6)*d^(7/6)) - (5*(b*c - a*d)^2*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6)
)])/(36*b^(11/6)*d^(7/6)) + (5*(b*c - a*d)^2*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)])/(144*b^(11/6)*d^(7/6)) - (5*(b*c - a*d)^2*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)])/(144*b^(11/6)*d^(7/6))

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 {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}+\frac {(5 (b c-a d)) \int \frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}} \, dx}{12 b} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {\left (5 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{72 b d} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{12 b^2 d} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{12 b^2 d} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{36 b^{11/6} d}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{36 b^{11/6} d}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{36 b^{5/3} d} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {\left (5 (b c-a d)^2\right ) \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 )}{144 b^{11/6} d^{7/6}}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{144 b^{11/6} d^{7/6}}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{48 b^{5/3} d}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{48 b^{5/3} d} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}-\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {5 (b c-a d)^2 \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 )}{144 b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \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 )}{144 b^{11/6} d^{7/6}}-\frac {\left (5 (b c-a d)^2\right ) \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 )}{24 b^{11/6} d^{7/6}}+\frac {\left (5 (b c-a d)^2\right ) \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 )}{24 b^{11/6} d^{7/6}} \\ & = \frac {5 (b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}{12 b d}+\frac {(a+b x)^{7/6} (c+d x)^{5/6}}{2 b}+\frac {5 (b c-a d)^2 \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 )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \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 )}{24 \sqrt {3} b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{11/6} d^{7/6}}+\frac {5 (b c-a d)^2 \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 )}{144 b^{11/6} d^{7/6}}-\frac {5 (b c-a d)^2 \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 )}{144 b^{11/6} d^{7/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.74 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\frac {6 b^{5/6} \sqrt [6]{d} \sqrt [6]{a+b x} (c+d x)^{5/6} (5 b c+a d+6 b d x)-5 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+5 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-10 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )-5 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{72 b^{11/6} d^{7/6}} \]

[In]

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

[Out]

(6*b^(5/6)*d^(1/6)*(a + b*x)^(1/6)*(c + d*x)^(5/6)*(5*b*c + a*d + 6*b*d*x) - 5*Sqrt[3]*(b*c - a*d)^2*ArcTan[(S
qrt[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
)^2*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)^2*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] - 5*(b*c - a*d)^2*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))])/(72*b^(11/6)*d^(
7/6))

Maple [F]

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

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2599 vs. \(2 (321) = 642\).

Time = 0.28 (sec) , antiderivative size = 2599, normalized size of antiderivative = 6.09 \[ \int \sqrt [6]{a+b x} (c+d x)^{5/6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/144*(10*b*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d
^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d
^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6)*log(5*((b^2*c^2 - 2*a*b*c*d + a^2*
d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + (b^2*d^2*x + b^2*c*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10
*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5
*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*
d^7))^(1/6))/(d*x + c)) - 10*b*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6)*log(5*((b^2*c^2
- 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d^2*x + b^2*c*d)*((b^12*c^12 - 12*a*b^11*c^11*d
+ 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 +
a^12*d^12)/(b^11*d^7))^(1/6))/(d*x + c)) + 5*(sqrt(-3)*b*d + b*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^1
0*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b
^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/
(b^11*d^7))^(1/6)*log(5/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) + (b^2*d^2*x + b^
2*c*d + sqrt(-3)*(b^2*d^2*x + b^2*c*d))*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^
9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^
4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6))/(d*x + c
)) - 5*(sqrt(-3)*b*d + b*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*
a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*
a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6)*log(5/2*(2*(b^2*c^2 -
2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d^2*x + b^2*c*d + sqrt(-3)*(b^2*d^2*x + b^2*c*d))
*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b
^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b
^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6))/(d*x + c)) - 5*(sqrt(-3)*b*d - b*d)*((b^12*c^12
- 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 +
924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10
- 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6)*log(5/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(
d*x + c)^(5/6) + (b^2*d^2*x + b^2*c*d - sqrt(-3)*(b^2*d^2*x + b^2*c*d))*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^
2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*
a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d
^12)/(b^11*d^7))^(1/6))/(d*x + c)) + 5*(sqrt(-3)*b*d - b*d)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*
d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*
d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d
^7))^(1/6)*log(5/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6) - (b^2*d^2*x + b^2*c*d -
sqrt(-3)*(b^2*d^2*x + b^2*c*d))*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^11*d^7))^(1/6))/(d*x + c)) - 12
*(6*b*d*x + 5*b*c + a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(b*d)

Sympy [F]

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

[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]

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

[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]

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

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

[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)