3.18.79 \(\int (a+b x)^{5/6} \sqrt [6]{c+d x} \, dx\) [1779]

Optimal. Leaf size=427 \[ \frac {(b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b d}+\frac {(a+b x)^{11/6} \sqrt [6]{c+d x}}{2 b}-\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{7/6} d^{11/6}}+\frac {5 (b c-a d)^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{7/6} d^{11/6}}-\frac {5 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{7/6} d^{11/6}}+\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{7/6} d^{11/6}}-\frac {5 (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{7/6} d^{11/6}} \]

[Out]

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

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Rubi [A]
time = 0.42, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {52, 65, 338, 302, 648, 632, 210, 642, 214} \begin {gather*} -\frac {5 (b c-a d)^2 \text {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^{7/6} d^{11/6}}+\frac {5 (b c-a d)^2 \text {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^{7/6} d^{11/6}}+\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^{7/6} d^{11/6}}-\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^{7/6} d^{11/6}}-\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^{7/6} d^{11/6}}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x} (b c-a d)}{12 b d}+\frac {(a+b x)^{11/6} \sqrt [6]{c+d x}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*(a + b*x)^(5/6)*(c + d*x)^(1/6))/(12*b*d) + ((a + b*x)^(11/6)*(c + d*x)^(1/6))/(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^(7/6)*
d^(11/6)) + (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^(7/6)*d^(11/6)) - (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^(7/6)*d^(11/6)) + (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^(7/6)*d^(11/6)) - (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^(7/6)*d^(11/6))

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

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Mathematica [A]
time = 0.89, size = 267, normalized size = 0.63 \begin {gather*} \frac {(b c-a d)^2 \left (\frac {6 \sqrt [6]{b} d^{5/6} (a+b x)^{5/6} \sqrt [6]{c+d x} (5 a d+b (c+6 d x))}{(b c-a d)^2}+5 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )-5 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )-10 \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )-5 \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )\right )}{72 b^{7/6} d^{11/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*((6*b^(1/6)*d^(5/6)*(a + b*x)^(5/6)*(c + d*x)^(1/6)*(5*a*d + b*(c + 6*d*x)))/(b*c - a*d)^2 + 5*
Sqrt[3]*ArcTan[(1 - (2*b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6)))/Sqrt[3]] - 5*Sqrt[3]*ArcTan[(1 + (2
*b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6)))/Sqrt[3]] - 10*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*
(a + b*x)^(1/6))] - 5*ArcTanh[(b^(1/6)*d^(1/6)*(a + b*x)^(1/6)*(c + d*x)^(1/6))/(d^(1/3)*(a + b*x)^(1/3) + b^(
1/3)*(c + d*x)^(1/3))]))/(72*b^(7/6)*d^(11/6))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {5}{6}} \left (d x +c \right )^{\frac {1}{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5633 vs. \(2 (321) = 642\).
time = 1.10, size = 5633, normalized size = 13.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(20*sqrt(3)*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^7*d^11))^(1/6)*arctan(-1/3*(2*sqrt(3)*(b^8
*c^2*d^9 - 2*a*b^7*c*d^10 + a^2*b^6*d^11)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((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 - 79
2*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^7*d^11))^(5/6) - 2*sqrt(3)*(b^7*d^9*x + a*b^6*d^9)*sqrt(((b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a^2*b*d^4)*(
b*x + a)^(5/6)*(d*x + c)^(1/6)*((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 + 4
95*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 - 2
20*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^7*d^11))^(1/6) + (b^4*c^4 - 4*a*b
^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b^3*d^4*x + a*b^2*d
^4)*((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^1
0*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^7*d^11))^(1/3))/(b*x + a))*((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^7*d^11))^(5/6) + sqrt(3)*(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c
^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c
^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12 + (b^13*c^12 - 12*a*b^12*c
^11*d + 66*a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*
c^6*d^6 - 792*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2
*c*d^11 + a^12*b*d^12)*x))/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 49
5*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 22
0*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12 + (b^13*c^12 - 12*a*b^12*c^11*d + 66*
a^2*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 7
92*a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a
^12*b*d^12)*x)) + 20*sqrt(3)*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^7*d^11))^(1/6)*arctan(-1/3*(2*s
qrt(3)*(b^8*c^2*d^9 - 2*a*b^7*c*d^10 + a^2*b^6*d^11)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((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^7*d^11))^(5/6) - 2*sqrt(3)*(b^7*d^9*x + a*b^6*d^9)*sqrt(-((b^3*c^2*d^2 - 2*a*b^2*c*d^3 +
a^2*b*d^4)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((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^7*d^11))^(1/6) - (b^4
*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b^3*d^4
*x + a*b^2*d^4)*((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^7*d^11))^(1/3))/(b*x + a))*((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^7*d^11))^(5/6) - sqrt(3)*(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 2
20*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 4
95*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12 + (b^13*c^12 -
12*a*b^12*c^11*d + 66*a^2*b^11*c^10*d^2 - 220*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{\frac {5}{6}} \sqrt [6]{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{1/6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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