\(\int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx\) [1834]

Optimal result

Integrand size = 19, antiderivative size = 30 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=-\frac {6 \sqrt [6]{c+d x}}{(b c-a d) \sqrt [6]{a+b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, 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} \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=-\frac {6 \sqrt [6]{c+d x}}{\sqrt [6]{a+b x} (b c-a d)} \]

[In]

Int[1/((a + b*x)^(7/6)*(c + d*x)^(5/6)),x]

[Out]

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

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 \sqrt [6]{c+d x}}{(b c-a d) \sqrt [6]{a+b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=-\frac {6 \sqrt [6]{c+d x}}{(b c-a d) \sqrt [6]{a+b x}} \]

[In]

Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(5/6)),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90

method	result	size
gosper	\(\frac {6 \left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {1}{6}} \left (a d -b c \right )}\)	\(27\)

[In]

int(1/(b*x+a)^(7/6)/(d*x+c)^(5/6),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=-\frac {6 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x} \]

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(5/6),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{6}} \left (c + d x\right )^{\frac {5}{6}}}\, dx \]

[In]

integrate(1/(b*x+a)**(7/6)/(d*x+c)**(5/6),x)

[Out]

Integral(1/((a + b*x)**(7/6)*(c + d*x)**(5/6)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {5}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(5/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(5/6)), x)

Giac [F]

\[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {5}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(7/6)/(d*x+c)^(5/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(5/6)), x)

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{7/6} (c+d x)^{5/6}} \, dx=\frac {6\,{\left (c+d\,x\right )}^{1/6}}{\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^{1/6}} \]

[In]

int(1/((a + b*x)^(7/6)*(c + d*x)^(5/6)),x)

[Out]

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