\(\int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx\) [1801]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{13/6}}{13 (b c-a d) (c+d x)^{13/6}} \]

[Out]

6/13*(b*x+a)^(13/6)/(-a*d+b*c)/(d*x+c)^(13/6)

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} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{13/6}}{13 (c+d x)^{13/6} (b c-a d)} \]

[In]

Int[(a + b*x)^(7/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(13/6))/(13*(b*c - a*d)*(c + d*x)^(13/6))

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)^{13/6}}{13 (b c-a d) (c+d x)^{13/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\frac {6 (a+b x)^{13/6}}{13 (b c-a d) (c+d x)^{13/6}} \]

[In]

Integrate[(a + b*x)^(7/6)/(c + d*x)^(19/6),x]

[Out]

(6*(a + b*x)^(13/6))/(13*(b*c - a*d)*(c + d*x)^(13/6))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
gosper	\(-\frac {6 \left (b x +a \right )^{\frac {13}{6}}}{13 \left (d x +c \right )^{\frac {13}{6}} \left (a d -b c \right )}\)	\(27\)

[In]

int((b*x+a)^(7/6)/(d*x+c)^(19/6),x,method=_RETURNVERBOSE)

[Out]

-6/13*(b*x+a)^(13/6)/(d*x+c)^(13/6)/(a*d-b*c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (26) = 52\).

Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.25 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\frac {6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{13 \, {\left (b c^{4} - a c^{3} d + {\left (b c d^{3} - a d^{4}\right )} x^{3} + 3 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} x^{2} + 3 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} x\right )}} \]

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="fricas")

[Out]

6/13*(b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b*c^4 - a*c^3*d + (b*c*d^3 - a*d^4)*x^3 + 3*(b
*c^2*d^2 - a*c*d^3)*x^2 + 3*(b*c^3*d - a*c^2*d^2)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**(7/6)/(d*x+c)**(19/6),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(19/6), x)

Giac [F]

\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]

[In]

integrate((b*x+a)^(7/6)/(d*x+c)^(19/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(19/6), x)

Mupad [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 199, normalized size of antiderivative = 6.22 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx=-\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {6\,a^2\,{\left (a+b\,x\right )}^{1/6}}{13\,a\,d^4-13\,b\,c\,d^3}+\frac {6\,b^2\,x^2\,{\left (a+b\,x\right )}^{1/6}}{13\,a\,d^4-13\,b\,c\,d^3}+\frac {12\,a\,b\,x\,{\left (a+b\,x\right )}^{1/6}}{13\,a\,d^4-13\,b\,c\,d^3}\right )}{x^3-\frac {13\,b\,c^4-13\,a\,c^3\,d}{13\,a\,d^4-13\,b\,c\,d^3}+\frac {39\,c\,d^2\,x^2\,\left (a\,d-b\,c\right )}{13\,a\,d^4-13\,b\,c\,d^3}+\frac {39\,c^2\,d\,x\,\left (a\,d-b\,c\right )}{13\,a\,d^4-13\,b\,c\,d^3}} \]

[In]

int((a + b*x)^(7/6)/(c + d*x)^(19/6),x)

[Out]

-((c + d*x)^(5/6)*((6*a^2*(a + b*x)^(1/6))/(13*a*d^4 - 13*b*c*d^3) + (6*b^2*x^2*(a + b*x)^(1/6))/(13*a*d^4 - 1
3*b*c*d^3) + (12*a*b*x*(a + b*x)^(1/6))/(13*a*d^4 - 13*b*c*d^3)))/(x^3 - (13*b*c^4 - 13*a*c^3*d)/(13*a*d^4 - 1
3*b*c*d^3) + (39*c*d^2*x^2*(a*d - b*c))/(13*a*d^4 - 13*b*c*d^3) + (39*c^2*d*x*(a*d - b*c))/(13*a*d^4 - 13*b*c*
d^3))