\(\int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx\) [1800]

Optimal result

Integrand size = 19, antiderivative size = 358 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {2 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{13/6}}+\frac {b^{7/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{13/6}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 358, 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 = {49, 65, 246, 216, 648, 632, 210, 642, 214} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=-\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{d^{13/6}}+\frac {2 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 d^{13/6}}+\frac {b^{7/6} \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 d^{13/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}} \]

[In]

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

[Out]

(-6*(a + b*x)^(7/6))/(7*d*(c + d*x)^(7/6)) - (6*b*(a + b*x)^(1/6))/(d^2*(c + d*x)^(1/6)) - (Sqrt[3]*b^(7/6)*Ar
cTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/d^(13/6) + (Sqrt[3]*b^(7/6)*A
rcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])/d^(13/6) + (2*b^(7/6)*ArcTan
h[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/6))])/d^(13/6) - (b^(7/6)*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)])/(2*d^(13/6)) + (b^(7/6)*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)])/(2*d^(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 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 {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}+\frac {b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx}{d} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {b^2 \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {(6 b) \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 )}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {(6 b) \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 )}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {\left (2 b^{7/6}\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 )}{d^2}+\frac {\left (2 b^{7/6}\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 )}{d^2}+\frac {\left (2 b^{4/3}\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 )}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 )}{2 d^{13/6}}+\frac {b^{7/6} \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 )}{2 d^{13/6}}+\frac {\left (3 b^{4/3}\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 )}{2 d^2}+\frac {\left (3 b^{4/3}\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 )}{2 d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 )}{2 d^{13/6}}+\frac {b^{7/6} \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 )}{2 d^{13/6}}+\frac {\left (3 b^{7/6}\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 )}{d^{13/6}}-\frac {\left (3 b^{7/6}\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 )}{d^{13/6}} \\ & = -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 )}{2 d^{13/6}}+\frac {b^{7/6} \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 )}{2 d^{13/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\frac {-\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x} (7 b c+a d+8 b d x)}{(c+d x)^{7/6}}+7 \sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-7 \sqrt {3} b^{7/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+14 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+7 b^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{7 d^{13/6}} \]

[In]

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

[Out]

((-6*d^(1/6)*(a + b*x)^(1/6)*(7*b*c + a*d + 8*b*d*x))/(c + d*x)^(7/6) + 7*Sqrt[3]*b^(7/6)*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))] - 7*Sqrt[3]*b^(7/6)*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*b^(7/6)*ArcTanh[(b^(1/
6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + 7*b^(7/6)*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))])/(7*d^(13/6))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (259) = 518\).

Time = 0.25 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx=\frac {7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2} + \sqrt {-3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b + {\left (d^{3} x + c d^{2} + \sqrt {-3} {\left (d^{3} x + c d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) - 7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2} + \sqrt {-3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b - {\left (d^{3} x + c d^{2} + \sqrt {-3} {\left (d^{3} x + c d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) + 7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2} - \sqrt {-3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b + {\left (d^{3} x + c d^{2} - \sqrt {-3} {\left (d^{3} x + c d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) - 7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2} - \sqrt {-3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {2 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b - {\left (d^{3} x + c d^{2} - \sqrt {-3} {\left (d^{3} x + c d^{2}\right )}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) + 14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) - 14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) - 12 \, {\left (8 \, b d x + 7 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]

[In]

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

[Out]

1/14*(7*(d^4*x^2 + 2*c*d^3*x + c^2*d^2 + sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1/6)*log((2*(b*
x + a)^(1/6)*(d*x + c)^(5/6)*b + (d^3*x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) - 7*(
d^4*x^2 + 2*c*d^3*x + c^2*d^2 + sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1/6)*log((2*(b*x + a)^(1
/6)*(d*x + c)^(5/6)*b - (d^3*x + c*d^2 + sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) + 7*(d^4*x^2 +
2*c*d^3*x + c^2*d^2 - sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1/6)*log((2*(b*x + a)^(1/6)*(d*x
+ c)^(5/6)*b + (d^3*x + c*d^2 - sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) - 7*(d^4*x^2 + 2*c*d^3*
x + c^2*d^2 - sqrt(-3)*(d^4*x^2 + 2*c*d^3*x + c^2*d^2))*(b^7/d^13)^(1/6)*log((2*(b*x + a)^(1/6)*(d*x + c)^(5/6
)*b - (d^3*x + c*d^2 - sqrt(-3)*(d^3*x + c*d^2))*(b^7/d^13)^(1/6))/(d*x + c)) + 14*(d^4*x^2 + 2*c*d^3*x + c^2*
d^2)*(b^7/d^13)^(1/6)*log(((b*x + a)^(1/6)*(d*x + c)^(5/6)*b + (d^3*x + c*d^2)*(b^7/d^13)^(1/6))/(d*x + c)) -
14*(d^4*x^2 + 2*c*d^3*x + c^2*d^2)*(b^7/d^13)^(1/6)*log(((b*x + a)^(1/6)*(d*x + c)^(5/6)*b - (d^3*x + c*d^2)*(
b^7/d^13)^(1/6))/(d*x + c)) - 12*(8*b*d*x + 7*b*c + a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6))/(d^4*x^2 + 2*c*d^3*x
+ c^2*d^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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