\(\int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx\) [1682]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=-\frac {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}} \]

[Out]

-4/9*(d*x+c)^(9/4)/(-a*d+b*c)/(b*x+a)^(9/4)

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} \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=-\frac {4 (c+d x)^{9/4}}{9 (a+b x)^{9/4} (b c-a d)} \]

[In]

Int[(c + d*x)^(5/4)/(a + b*x)^(13/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(9*(b*c - a*d)*(a + b*x)^(9/4))

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 {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=-\frac {4 (c+d x)^{9/4}}{9 (b c-a d) (a+b x)^{9/4}} \]

[In]

Integrate[(c + d*x)^(5/4)/(a + b*x)^(13/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(9*(b*c - a*d)*(a + b*x)^(9/4))

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {4 \left (d x +c \right )^{\frac {9}{4}}}{9 \left (b x +a \right )^{\frac {9}{4}} \left (a d -b c \right )}\)	\(27\)

[In]

int((d*x+c)^(5/4)/(b*x+a)^(13/4),x,method=_RETURNVERBOSE)

[Out]

4/9/(b*x+a)^(9/4)*(d*x+c)^(9/4)/(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 = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.25 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=-\frac {4 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{9 \, {\left (a^{3} b c - a^{4} d + {\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x\right )}} \]

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="fricas")

[Out]

-4/9*(d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^3*b*c - a^4*d + (b^4*c - a*b^3*d)*x^3 + 3*(a
*b^3*c - a^2*b^2*d)*x^2 + 3*(a^2*b^2*c - a^3*b*d)*x)

Sympy [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\left (a + b x\right )^{\frac {13}{4}}}\, dx \]

[In]

integrate((d*x+c)**(5/4)/(b*x+a)**(13/4),x)

[Out]

Integral((c + d*x)**(5/4)/(a + b*x)**(13/4), x)

Maxima [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {13}{4}}} \,d x } \]

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(13/4), x)

Giac [F]

\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {13}{4}}} \,d x } \]

[In]

integrate((d*x+c)^(5/4)/(b*x+a)^(13/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(13/4), x)

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.09 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx=\frac {4\,c^2\,{\left (c+d\,x\right )}^{1/4}+4\,d^2\,x^2\,{\left (c+d\,x\right )}^{1/4}+8\,c\,d\,x\,{\left (c+d\,x\right )}^{1/4}}{{\left (a+b\,x\right )}^{1/4}\,\left (9\,d\,a^3+18\,d\,a^2\,b\,x-9\,c\,a^2\,b+9\,d\,a\,b^2\,x^2-18\,c\,a\,b^2\,x-9\,c\,b^3\,x^2\right )} \]

[In]

int((c + d*x)^(5/4)/(a + b*x)^(13/4),x)

[Out]

(4*c^2*(c + d*x)^(1/4) + 4*d^2*x^2*(c + d*x)^(1/4) + 8*c*d*x*(c + d*x)^(1/4))/((a + b*x)^(1/4)*(9*a^3*d - 9*b^
3*c*x^2 - 9*a^2*b*c - 18*a*b^2*c*x + 18*a^2*b*d*x + 9*a*b^2*d*x^2))