\(\int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx\) [1808]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\frac {6 (a+b x)^{5/6}}{5 (b c-a d) (c+d x)^{5/6}} \]

[Out]

6/5*(b*x+a)^(5/6)/(-a*d+b*c)/(d*x+c)^(5/6)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\frac {6 (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)} \]

[In]

Int[1/((a + b*x)^(1/6)*(c + d*x)^(11/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(5*(b*c - a*d)*(c + d*x)^(5/6))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{5/6}}{5 (b c-a d) (c+d x)^{5/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\frac {6 (a+b x)^{5/6}}{5 (b c-a d) (c+d x)^{5/6}} \]

[In]

Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(11/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(5*(b*c - a*d)*(c + d*x)^(5/6))

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
gosper	\(-\frac {6 \left (b x +a \right )^{\frac {5}{6}}}{5 \left (d x +c \right )^{\frac {5}{6}} \left (a d -b c \right )}\)	\(27\)

[In]

int(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x,method=_RETURNVERBOSE)

[Out]

-6/5*(b*x+a)^(5/6)/(d*x+c)^(5/6)/(a*d-b*c)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\frac {6 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{5 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x\right )}} \]

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\int \frac {1}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac {11}{6}}}\, dx \]

[In]

integrate(1/(b*x+a)**(1/6)/(d*x+c)**(11/6),x)

[Out]

Integral(1/((a + b*x)**(1/6)*(c + d*x)**(11/6)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(11/6)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(1/6)/(d*x+c)^(11/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(11/6)), x)

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx=-\frac {6\,{\left (a+b\,x\right )}^{5/6}}{\left (5\,a\,d-5\,b\,c\right )\,{\left (c+d\,x\right )}^{5/6}} \]

[In]

int(1/((a + b*x)^(1/6)*(c + d*x)^(11/6)),x)

[Out]

-(6*(a + b*x)^(5/6))/((5*a*d - 5*b*c)*(c + d*x)^(5/6))