3.18.80 \(\int \frac {(a+b x)^{5/6}}{(c+d x)^{5/6}} \, dx\) [1780]

Optimal. Leaf size=378 \[ \frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 \sqrt [6]{b} d^{11/6}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.40, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 13, 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) \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 \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}-\frac {5 (b c-a d) \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 )}{2 \sqrt {3} \sqrt [6]{b} d^{11/6}}+\frac {5 (b c-a d) \tan ^{-1}\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} \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}+\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(5/6)*(c + d*x)^(1/6))/d - (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^(1/6)*d^(11/6)) + (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^(1/6)*d^(11/6)) - (5*(b*c - a*d)*ArcTanh[(d^(1/6)*(a
+ b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(1/6)*d^(11/6)) + (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^(1/6)*d^(11/6)) - (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^(1/6)*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 \frac {(a+b x)^{5/6}}{(c+d x)^{5/6}} \, dx &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{6 d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \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 d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\frac {(5 (b c-a d)) \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 d}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{d}-\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 d^{5/3}}-\frac {(5 (b c-a d)) \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 )}{3 \sqrt [6]{b} d^{5/3}}-\frac {(5 (b c-a d)) \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 )}{3 \sqrt [6]{b} d^{5/3}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{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 \sqrt [6]{b} d^{11/6}}+\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 \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}+\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 d^{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 d^{5/3}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{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 \sqrt [6]{b} d^{11/6}}+\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 \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}+\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 \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}\\ &=\frac {(a+b x)^{5/6} \sqrt [6]{c+d x}}{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} \sqrt [6]{b} d^{11/6}}+\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} \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}+\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 \sqrt [6]{b} d^{11/6}}-\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 \sqrt [6]{b} d^{11/6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.03, size = 73, normalized size = 0.19 \begin {gather*} \frac {6 (a+b x)^{11/6} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {11}{6};\frac {17}{6};\frac {d (a+b x)}{-b c+a d}\right )}{11 b (c+d x)^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(a + b*x)^(11/6)*((b*(c + d*x))/(b*c - a*d))^(5/6)*Hypergeometric2F1[5/6, 11/6, 17/6, (d*(a + b*x))/(-(b*c)
+ a*d)])/(11*b*(c + d*x)^(5/6))

________________________________________________________________________________________

Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^(5/6),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2997 vs. \(2 (280) = 560\).
time = 0.40, size = 2997, normalized size = 7.93

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(20*sqrt(3)*d*((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*d^11))^(1/6)*arctan(1/3*(2*sqrt(3)*(b^2*c*d^9 - a*b*d^10)*(b*x + a)^(5/6)*(d*x +
c)^(1/6)*((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*d^11))^(5/6) + 2*sqrt(3)*(b^2*d^9*x + a*b*d^9)*sqrt(((b*c*d^2 - a*d^3)*(b*x + a)^(5/6)*(d*x
+ c)^(1/6)*((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*d^11))^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^4*x
+ a*d^4)*((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*d^11))^(1/3))/(b*x + a))*((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*d^11))^(5/6) + sqrt(3)*(a*b^6*c^6 - 6*a^2*b^5*c^5*d + 15*a
^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d^6 + (b^7*c^6 - 6*a*b^6*c^5*d
+ 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*x))/(a*b^6*c^6 -
6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d^6 + (b
^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*
b*d^6)*x)) + 20*sqrt(3)*d*((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*d^11))^(1/6)*arctan(1/3*(2*sqrt(3)*(b^2*c*d^9 - a*b*d^10)*(b*x + a)^(5/6)*(
d*x + c)^(1/6)*((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*d^11))^(5/6) + 2*sqrt(3)*(b^2*d^9*x + a*b*d^9)*sqrt(-((b*c*d^2 - a*d^3)*(b*x + a)^(5/6
)*(d*x + c)^(1/6)*((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*d^11))^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b
*d^4*x + a*d^4)*((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*d^11))^(1/3))/(b*x + a))*((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*d^11))^(5/6) - sqrt(3)*(a*b^6*c^6 - 6*a^2*b^5*c^5*d
+ 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d^6 + (b^7*c^6 - 6*a*b^6
*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*x))/(a*b^
6*c^6 - 6*a^2*b^5*c^5*d + 15*a^3*b^4*c^4*d^2 - 20*a^4*b^3*c^3*d^3 + 15*a^5*b^2*c^2*d^4 - 6*a^6*b*c*d^5 + a^7*d
^6 + (b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5
+ a^6*b*d^6)*x)) + 5*d*((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*d^11))^(1/6)*log(25*((b*c*d^2 - a*d^3)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((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*d^11))^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^4*x + a*d^4)*((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
*d^11))^(1/3))/(b*x + a)) - 5*d*((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*d^11))^(1/6)*log(-25*((b*c*d^2 - a*d^3)*(b*x + a)^(5/6)*(d*x + c)^(1/
6)*((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*d^11))^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b*d^4*x + a*d^4)
*((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*d^11))^(1/3))/(b*x + a)) + 10*d*((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*d^11))^(1/6)*log(-5*((b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^
(1/6) + (b*d^2*x + a*d^2)*((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*d^11))^(1/6))/(b*x + a)) - 10*d*((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*d^11))^(1/6)*log(-5*((b*c - a*d)*(
b*x + a)^(5/6)*(d*x + c)^(1/6) - (b*d^2*x + a*d^2)*((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*d^11))^(1/6))/(b*x + a)) - 12*(b*x + a)^(5/6)*(d*x
+ c)^(1/6))/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F] N/A
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)^(5/6),x)

[Out]

Could not integrate

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________