\(\int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx\) [1831]

Optimal result

Integrand size = 19, antiderivative size = 449 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{144 b^{19/6}} \]

[Out]

91/12*d*(-a*d+b*c)*(b*x+a)^(5/6)*(d*x+c)^(1/6)/b^3+13/2*d*(b*x+a)^(5/6)*(d*x+c)^(7/6)/b^2-6*(d*x+c)^(13/6)/b/(
b*x+a)^(1/6)+91/36*d^(1/6)*(-a*d+b*c)^2*arctanh(d^(1/6)*(b*x+a)^(1/6)/b^(1/6)/(d*x+c)^(1/6))/b^(19/6)-91/144*d
^(1/6)*(-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^(19/6)+91/144*d^(1/6)*(-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^(19/6)-91/72*d^(1/6)*(-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^(19/6)*3^(1/2)-91/72*d^(1/6)*(-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^(19/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{24 \sqrt {3} b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{24 \sqrt {3} b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{36 b^{19/6}}-\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{144 b^{19/6}}+\frac {91 \sqrt [6]{d} (b c-a d)^2 \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{144 b^{19/6}}+\frac {91 d (a+b x)^{5/6} \sqrt [6]{c+d x} (b c-a d)}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}} \]

[In]

Int[(c + d*x)^(13/6)/(a + b*x)^(7/6),x]

[Out]

(91*d*(b*c - a*d)*(a + b*x)^(5/6)*(c + d*x)^(1/6))/(12*b^3) + (13*d*(a + b*x)^(5/6)*(c + d*x)^(7/6))/(2*b^2) -
(6*(c + d*x)^(13/6))/(b*(a + b*x)^(1/6)) + (91*d^(1/6)*(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^(19/6)) - (91*d^(1/6)*(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^(19/6)) + (91*d^(1/6)*(b*c - a
*d)^2*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/(36*b^(19/6)) - (91*d^(1/6)*(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^(19/6)) + (91*d^(1/6)*(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^(19/6))

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)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {(13 d) \int \frac {(c+d x)^{7/6}}{\sqrt [6]{a+b x}} \, dx}{b} \\ & = \frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {(91 d (b c-a d)) \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx}{12 b^2} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (b c-a d)^2\right ) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{72 b^3} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (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^4} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 d (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^4} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {\left (91 \sqrt [3]{d} (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^{19/6}}+\frac {\left (91 \sqrt [3]{d} (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^{19/6}}+\frac {\left (91 \sqrt [3]{d} (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^3} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (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^{19/6}}-\frac {\left (91 \sqrt [6]{d} (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^{19/6}}+\frac {\left (91 \sqrt [6]{d} (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^{19/6}}-\frac {\left (91 \sqrt [3]{d} (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^3}-\frac {\left (91 \sqrt [3]{d} (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^3} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (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^{19/6}}-\frac {91 \sqrt [6]{d} (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^{19/6}}+\frac {91 \sqrt [6]{d} (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^{19/6}}-\frac {\left (91 \sqrt [6]{d} (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^{19/6}}+\frac {\left (91 \sqrt [6]{d} (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^{19/6}} \\ & = \frac {91 d (b c-a d) (a+b x)^{5/6} \sqrt [6]{c+d x}}{12 b^3}+\frac {13 d (a+b x)^{5/6} (c+d x)^{7/6}}{2 b^2}-\frac {6 (c+d x)^{13/6}}{b \sqrt [6]{a+b x}}+\frac {91 \sqrt [6]{d} (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^{19/6}}-\frac {91 \sqrt [6]{d} (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^{19/6}}+\frac {91 \sqrt [6]{d} (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^{19/6}}-\frac {91 \sqrt [6]{d} (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^{19/6}}+\frac {91 \sqrt [6]{d} (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^{19/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x} \left (-91 a^2 d^2-13 a b d (-13 c+d x)+b^2 \left (-72 c^2+25 c d x+6 d^2 x^2\right )\right )}{\sqrt [6]{a+b x}}-91 \sqrt {3} \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+91 \sqrt {3} \sqrt [6]{d} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+182 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+91 \sqrt [6]{d} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{72 b^{19/6}} \]

[In]

Integrate[(c + d*x)^(13/6)/(a + b*x)^(7/6),x]

[Out]

((6*b^(1/6)*(c + d*x)^(1/6)*(-91*a^2*d^2 - 13*a*b*d*(-13*c + d*x) + b^2*(-72*c^2 + 25*c*d*x + 6*d^2*x^2)))/(a
+ b*x)^(1/6) - 91*Sqrt[3]*d^(1/6)*(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))] + 91*Sqrt[3]*d^(1/6)*(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))] + 182*d^(1/6)*(b*c - a*d)^2*ArcTanh[(b^(1/6)*(c + d*x)
^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + 91*d^(1/6)*(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^(19/6))

Maple [F]

\[\int \frac {\left (d x +c \right )^{\frac {13}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]

[In]

int((d*x+c)^(13/6)/(b*x+a)^(7/6),x)

[Out]

int((d*x+c)^(13/6)/(b*x+a)^(7/6),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2686 vs. \(2 (339) = 678\).

Time = 0.28 (sec) , antiderivative size = 2686, normalized size of antiderivative = 5.98 \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="fricas")

[Out]

1/144*(91*(b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3
- 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8
+ 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/
6)*log(91/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^4*x + a*b^3 + sqrt(-3)*(b^
4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^
8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^
3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(b*x + a)) - 91*(b^4*x + a*b^3 + sq
rt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*
a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*
a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(91/2*(2*(b^2*c^2 - 2*a
*b*c*d + a^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^4*x + a*b^3 + sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d -
12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 +
924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11
- 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(b*x + a)) + 91*(b^4*x + a*b^3 - sqrt(-3)*(b^4*x + a*b^3))*((b^1
2*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7
*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^
2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(91/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(
5/6)*(d*x + c)^(1/6) + (b^4*x + a*b^3 - sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*
b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^
7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^
13)/b^19)^(1/6))/(b*x + a)) - 91*(b^4*x + a*b^3 - sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2
+ 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^
7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12
+ a^12*d^13)/b^19)^(1/6)*log(91/2*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^4*x
+ a*b^3 - sqrt(-3)*(b^4*x + a*b^3))*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c
^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c
^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(b*x + a)) +
182*(b^4*x + a*b^3)*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4
*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9
*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(91*((b^2*c^2 - 2*a*b*c*d
+ a^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^4*x + a*b^3)*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*
c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5
*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b
^19)^(1/6))/(b*x + a)) - 182*(b^4*x + a*b^3)*((b^12*c^12*d - 12*a*b^11*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a
^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a
^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11*b*c*d^12 + a^12*d^13)/b^19)^(1/6)*log(9
1*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - (b^4*x + a*b^3)*((b^12*c^12*d - 12*a*b^11
*c^11*d^2 + 66*a^2*b^10*c^10*d^3 - 220*a^3*b^9*c^9*d^4 + 495*a^4*b^8*c^8*d^5 - 792*a^5*b^7*c^7*d^6 + 924*a^6*b
^6*c^6*d^7 - 792*a^7*b^5*c^5*d^8 + 495*a^8*b^4*c^4*d^9 - 220*a^9*b^3*c^3*d^10 + 66*a^10*b^2*c^2*d^11 - 12*a^11
*b*c*d^12 + a^12*d^13)/b^19)^(1/6))/(b*x + a)) + 12*(6*b^2*d^2*x^2 - 72*b^2*c^2 + 169*a*b*c*d - 91*a^2*d^2 + (
25*b^2*c*d - 13*a*b*d^2)*x)*(b*x + a)^(5/6)*(d*x + c)^(1/6))/(b^4*x + a*b^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**(13/6)/(b*x+a)**(7/6),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {13}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]

[In]

integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(13/6)/(b*x + a)^(7/6), x)

Giac [F]

\[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {13}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]

[In]

integrate((d*x+c)^(13/6)/(b*x+a)^(7/6),x, algorithm="giac")

[Out]

integrate((d*x + c)^(13/6)/(b*x + a)^(7/6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{13/6}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{13/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \]

[In]

int((c + d*x)^(13/6)/(a + b*x)^(7/6),x)

[Out]

int((c + d*x)^(13/6)/(a + b*x)^(7/6), x)