3.18.81 \(\int \frac {(a+b x)^{5/6}}{(c+d x)^{11/6}} \, dx\) [1781]

Optimal. Leaf size=334 \[ -\frac {6 (a+b x)^{5/6}}{5 d (c+d x)^{5/6}}+\frac {\sqrt {3} b^{5/6} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{11/6}}-\frac {\sqrt {3} b^{5/6} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{11/6}}+\frac {2 b^{5/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{11/6}}-\frac {b^{5/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{11/6}}+\frac {b^{5/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{11/6}} \]

[Out]

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

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Rubi [A]
time = 0.39, antiderivative size = 334, 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 = {49, 65, 338, 302, 648, 632, 210, 642, 214} \begin {gather*} -\frac {b^{5/6} \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^{11/6}}+\frac {b^{5/6} \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^{11/6}}+\frac {\sqrt {3} b^{5/6} \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 )}{d^{11/6}}-\frac {\sqrt {3} b^{5/6} \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 )}{d^{11/6}}+\frac {2 b^{5/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{11/6}}-\frac {6 (a+b x)^{5/6}}{5 d (c+d x)^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.30, size = 244, normalized size = 0.73 \begin {gather*} \frac {-\frac {6 d^{5/6} (a+b x)^{5/6}}{(c+d x)^{5/6}}-5 \sqrt {3} b^{5/6} \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} b^{5/6} \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 b^{5/6} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+5 b^{5/6} \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 )}{5 d^{11/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)^(11/6),x]')

[Out]

Timed out

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{6}}}{\left (d x +c \right )^{\frac {11}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/6)/(d*x+c)^(11/6),x)

[Out]

int((b*x+a)^(5/6)/(d*x+c)^(11/6),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)^(11/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(11/6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (241) = 482\).
time = 0.41, size = 755, normalized size = 2.26

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)^(11/6),x, algorithm="fricas")

[Out]

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

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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 {11}{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/6)/(d*x+c)**(11/6),x)

[Out]

Integral((a + b*x)**(5/6)/(c + d*x)**(11/6), x)

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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)^(11/6),x)

[Out]

Could not integrate

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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 )}^{11/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/6)/(c + d*x)^(11/6),x)

[Out]

int((a + b*x)^(5/6)/(c + d*x)^(11/6), x)

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