\(\int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx\) [1832]

Optimal result

Integrand size = 19, antiderivative size = 403 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{13/6}} \]

[Out]

7*d*(b*x+a)^(5/6)*(d*x+c)^(1/6)/b^2-6*(d*x+c)^(7/6)/b/(b*x+a)^(1/6)+7/3*d^(1/6)*(-a*d+b*c)*arctanh(d^(1/6)*(b*
x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(13/6)-7/12*d^(1/6)*(-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^(13/6)+7/12*d^(1/6)*(-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^(13/6)-7/6*d^(1/6)*(-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^(13/6)*3^(1/2)-7/6*d^(1/6)*(-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^(13/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {49, 52, 65, 338, 302, 648, 632, 210, 642, 214} \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}} \]

[In]

Int[(c + d*x)^(7/6)/(a + b*x)^(7/6),x]

[Out]

(7*d*(a + b*x)^(5/6)*(c + d*x)^(1/6))/b^2 - (6*(c + d*x)^(7/6))/(b*(a + b*x)^(1/6)) + (7*d^(1/6)*(b*c - a*d)*A
rcTan[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^(13/6)) - (7*d^
(1/6)*(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^(13/6)) + (7*d^(1/6)*(b*c - a*d)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(13/6)
) - (7*d^(1/6)*(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^(13/6)) + (7*d^(1/6)*(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^(13/6))

Rule 49

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

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*} \text {integral}& = -\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d) \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx}{b} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (b c-a d)) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{6 b^2} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (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^3} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (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^3} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\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 )}{3 b^{13/6}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\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 )}{3 b^{13/6}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\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 )}{3 b^2} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (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^{13/6}}-\frac {\left (7 \sqrt [6]{d} (b c-a d)\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 )}{12 b^{13/6}}+\frac {\left (7 \sqrt [6]{d} (b c-a d)\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 )}{12 b^{13/6}}-\frac {\left (7 \sqrt [3]{d} (b c-a d)\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 )}{4 b^2}-\frac {\left (7 \sqrt [3]{d} (b c-a d)\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 )}{4 b^2} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (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^{13/6}}-\frac {7 \sqrt [6]{d} (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^{13/6}}+\frac {7 \sqrt [6]{d} (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^{13/6}}-\frac {\left (7 \sqrt [6]{d} (b c-a d)\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 )}{2 b^{13/6}}+\frac {\left (7 \sqrt [6]{d} (b c-a d)\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 )}{2 b^{13/6}} \\ & = \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (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^{13/6}}-\frac {7 \sqrt [6]{d} (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^{13/6}}+\frac {7 \sqrt [6]{d} (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^{13/6}}-\frac {7 \sqrt [6]{d} (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^{13/6}}+\frac {7 \sqrt [6]{d} (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^{13/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x} (-6 b c+7 a d+b d x)}{\sqrt [6]{a+b x}}+7 \sqrt {3} \sqrt [6]{d} (-b c+a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+7 \sqrt {3} \sqrt [6]{d} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+14 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+7 \sqrt [6]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 b^{13/6}} \]

[In]

Integrate[(c + d*x)^(7/6)/(a + b*x)^(7/6),x]

[Out]

((6*b^(1/6)*(c + d*x)^(1/6)*(-6*b*c + 7*a*d + b*d*x))/(a + b*x)^(1/6) + 7*Sqrt[3]*d^(1/6)*(-(b*c) + a*d)*ArcTa
n[(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))] + 7*Sqrt[3]*d^(1/6
)*(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))]
+ 14*d^(1/6)*(b*c - a*d)*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + 7*d^(1/6)*(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^(13/6))

Maple [F]

\[\int \frac {\left (d x +c \right )^{\frac {7}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]

[In]

int((d*x+c)^(7/6)/(b*x+a)^(7/6),x)

[Out]

int((d*x+c)^(7/6)/(b*x+a)^(7/6),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (301) = 602\).

Time = 0.28 (sec) , antiderivative size = 1562, normalized size of antiderivative = 3.88 \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(7/6)/(b*x+a)^(7/6),x, algorithm="fricas")

[Out]

1/12*(7*(b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3
*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/
6)*(d*x + c)^(1/6) + (b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4
*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) - 7*(b^3*x +
a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 1
5*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/
6) - (b^3*x + a*b^2 + sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^
3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) + 7*(b^3*x + a*b^2 - sqrt(-3
)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^
5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^3*x + a*
b^2 - sqrt(-3)*(b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a
^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) - 7*(b^3*x + a*b^2 - sqrt(-3)*(b^3*x + a*b^2
))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^
6 + a^6*d^7)/b^13)^(1/6)*log(-7/2*(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^3*x + a*b^2 - sqrt(-3)*(
b^3*x + a*b^2))*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 -
6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) + 14*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*
b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7*((b*c - a*
d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a
^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) - 14*(b^3*x + a*b^2)*((
b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a
^6*d^7)/b^13)^(1/6)*log(-7*((b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^
5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6
))/(b*x + a)) + 12*(b*d*x - 6*b*c + 7*a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b^3*x + a*b^2)

Sympy [F]

\[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\int \frac {\left (c + d x\right )^{\frac {7}{6}}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \]

[In]

integrate((d*x+c)**(7/6)/(b*x+a)**(7/6),x)

[Out]

Integral((c + d*x)**(7/6)/(a + b*x)**(7/6), x)

Maxima [F]

\[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {7}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]

[In]

integrate((d*x+c)^(7/6)/(b*x+a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(7/6)/(b*x + a)^(7/6), x)

Giac [F]

\[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {7}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]

[In]

integrate((d*x+c)^(7/6)/(b*x+a)^(7/6),x, algorithm="giac")

[Out]

integrate((d*x + c)^(7/6)/(b*x + a)^(7/6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{7/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \]

[In]

int((c + d*x)^(7/6)/(a + b*x)^(7/6),x)

[Out]

int((c + d*x)^(7/6)/(a + b*x)^(7/6), x)