3.19.33 \(\int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx\) [1833]

Optimal. Leaf size=332 \[ -\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 b^{7/6}} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 332, 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 {\sqrt {3} \sqrt [6]{d} \text {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 )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \text {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 )}{b^{7/6}}-\frac {\sqrt [6]{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 )}{2 b^{7/6}}+\frac {\sqrt [6]{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 )}{2 b^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(1/6)/(a + b*x)^(7/6),x]

[Out]

(-6*(c + d*x)^(1/6))/(b*(a + b*x)^(1/6)) + (Sqrt[3]*d^(1/6)*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sq
rt[3]*b^(1/6)*(c + d*x)^(1/6))])/b^(7/6) - (Sqrt[3]*d^(1/6)*ArcTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sq
rt[3]*b^(1/6)*(c + d*x)^(1/6))])/b^(7/6) + (2*d^(1/6)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))/(b^(1/6)*(c + d*x)^(1/
6))])/b^(7/6) - (d^(1/6)*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*b^(7/6)) + (d^(1/6)*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*b^(7/6))

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

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Mathematica [A]
time = 0.28, size = 239, normalized size = 0.72 \begin {gather*} \frac {-\frac {6 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{a+b x}}-\sqrt {3} \sqrt [6]{d} \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 )+\sqrt {3} \sqrt [6]{d} \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 )+2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\sqrt [6]{d} \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 )}{b^{7/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(1/6)/(a + b*x)^(7/6),x]

[Out]

((-6*b^(1/6)*(c + d*x)^(1/6))/(a + b*x)^(1/6) - Sqrt[3]*d^(1/6)*ArcTan[(1 - (2*b^(1/6)*(c + d*x)^(1/6))/(d^(1/
6)*(a + b*x)^(1/6)))/Sqrt[3]] + Sqrt[3]*d^(1/6)*ArcTan[(1 + (2*b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/
6)))/Sqrt[3]] + 2*d^(1/6)*ArcTanh[(b^(1/6)*(c + d*x)^(1/6))/(d^(1/6)*(a + b*x)^(1/6))] + d^(1/6)*ArcTanh[(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))])/b^(7/6)

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/6)/(b*x+a)^(7/6),x)

[Out]

int((d*x+c)^(1/6)/(b*x+a)^(7/6),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((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(7/6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (241) = 482\).
time = 1.15, size = 663, normalized size = 2.00 \begin {gather*} -\frac {4 \, \sqrt {3} {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b^{6} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{7} x + a b^{6}\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} + \sqrt {3} {\left (b d x + a d\right )}}{3 \, {\left (b d x + a d\right )}}\right ) + 4 \, \sqrt {3} {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b^{6} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{7} x + a b^{6}\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} - \sqrt {3} {\left (b d x + a d\right )}}{3 \, {\left (b d x + a d\right )}}\right ) - {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )}}{b x + a}\right ) + {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )}}{b x + a}\right ) - 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + 12 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{2 \, {\left (b^{2} x + a b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/6)/(b*x+a)**(7/6),x)

[Out]

Integral((c + d*x)**(1/6)/(a + b*x)**(7/6), x)

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Giac [F]
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((d*x+c)^(1/6)/(b*x+a)^(7/6),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/6)/(b*x + a)^(7/6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(1/6)/(a + b*x)^(7/6),x)

[Out]

int((c + d*x)^(1/6)/(a + b*x)^(7/6), x)

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