\(\int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx\) [1830]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {7776 b^3 \sqrt [6]{a+b x}}{1729 (b c-a d)^4 \sqrt [6]{c+d x}} \]

[Out]

6/19*(b*x+a)^(1/6)/(-a*d+b*c)/(d*x+c)^(19/6)+108/247*b*(b*x+a)^(1/6)/(-a*d+b*c)^2/(d*x+c)^(13/6)+1296/1729*b^2
*(b*x+a)^(1/6)/(-a*d+b*c)^3/(d*x+c)^(7/6)+7776/1729*b^3*(b*x+a)^(1/6)/(-a*d+b*c)^4/(d*x+c)^(1/6)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\frac {7776 b^3 \sqrt [6]{a+b x}}{1729 \sqrt [6]{c+d x} (b c-a d)^4}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (c+d x)^{7/6} (b c-a d)^3}+\frac {108 b \sqrt [6]{a+b x}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{19 (c+d x)^{19/6} (b c-a d)} \]

[In]

Int[1/((a + b*x)^(5/6)*(c + d*x)^(25/6)),x]

[Out]

(6*(a + b*x)^(1/6))/(19*(b*c - a*d)*(c + d*x)^(19/6)) + (108*b*(a + b*x)^(1/6))/(247*(b*c - a*d)^2*(c + d*x)^(
13/6)) + (1296*b^2*(a + b*x)^(1/6))/(1729*(b*c - a*d)^3*(c + d*x)^(7/6)) + (7776*b^3*(a + b*x)^(1/6))/(1729*(b
*c - a*d)^4*(c + d*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]

Rule 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {(18 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx}{19 (b c-a d)} \\ & = \frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {\left (216 b^2\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx}{247 (b c-a d)^2} \\ & = \frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {\left (1296 b^3\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{1729 (b c-a d)^3} \\ & = \frac {6 \sqrt [6]{a+b x}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {108 b \sqrt [6]{a+b x}}{247 (b c-a d)^2 (c+d x)^{13/6}}+\frac {1296 b^2 \sqrt [6]{a+b x}}{1729 (b c-a d)^3 (c+d x)^{7/6}}+\frac {7776 b^3 \sqrt [6]{a+b x}}{1729 (b c-a d)^4 \sqrt [6]{c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (-91 a^3 d^3+21 a^2 b d^2 (19 c+6 d x)-3 a b^2 d \left (247 c^2+228 c d x+72 d^2 x^2\right )+b^3 \left (1729 c^3+4446 c^2 d x+4104 c d^2 x^2+1296 d^3 x^3\right )\right )}{1729 (b c-a d)^4 (c+d x)^{19/6}} \]

[In]

Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(25/6)),x]

[Out]

(6*(a + b*x)^(1/6)*(-91*a^3*d^3 + 21*a^2*b*d^2*(19*c + 6*d*x) - 3*a*b^2*d*(247*c^2 + 228*c*d*x + 72*d^2*x^2) +
b^3*(1729*c^3 + 4446*c^2*d*x + 4104*c*d^2*x^2 + 1296*d^3*x^3)))/(1729*(b*c - a*d)^4*(c + d*x)^(19/6))

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26

method	result	size
gosper	\(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (-1296 d^{3} x^{3} b^{3}+216 x^{2} a \,b^{2} d^{3}-4104 x^{2} b^{3} c \,d^{2}-126 x \,a^{2} b \,d^{3}+684 x a \,b^{2} c \,d^{2}-4446 x \,b^{3} c^{2} d +91 a^{3} d^{3}-399 a^{2} b c \,d^{2}+741 a \,b^{2} c^{2} d -1729 b^{3} c^{3}\right )}{1729 \left (d x +c \right )^{\frac {19}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(171\)

[In]

int(1/(b*x+a)^(5/6)/(d*x+c)^(25/6),x,method=_RETURNVERBOSE)

[Out]

-6/1729*(b*x+a)^(1/6)*(-1296*b^3*d^3*x^3+216*a*b^2*d^3*x^2-4104*b^3*c*d^2*x^2-126*a^2*b*d^3*x+684*a*b^2*c*d^2*
x-4446*b^3*c^2*d*x+91*a^3*d^3-399*a^2*b*c*d^2+741*a*b^2*c^2*d-1729*b^3*c^3)/(d*x+c)^(19/6)/(a^4*d^4-4*a^3*b*c*
d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (112) = 224\).

Time = 0.24 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\frac {6 \, {\left (1296 \, b^{3} d^{3} x^{3} + 1729 \, b^{3} c^{3} - 741 \, a b^{2} c^{2} d + 399 \, a^{2} b c d^{2} - 91 \, a^{3} d^{3} + 216 \, {\left (19 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (247 \, b^{3} c^{2} d - 38 \, a b^{2} c d^{2} + 7 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{1729 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} + {\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}} \]

[In]

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(25/6),x, algorithm="fricas")

[Out]

6/1729*(1296*b^3*d^3*x^3 + 1729*b^3*c^3 - 741*a*b^2*c^2*d + 399*a^2*b*c*d^2 - 91*a^3*d^3 + 216*(19*b^3*c*d^2 -
a*b^2*d^3)*x^2 + 18*(247*b^3*c^2*d - 38*a*b^2*c*d^2 + 7*a^2*b*d^3)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^4*c^
8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4 + (b^4*c^4*d^4 - 4*a*b^3*c^3*d^5 + 6*a^2
*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*x^4 + 4*(b^4*c^5*d^3 - 4*a*b^3*c^4*d^4 + 6*a^2*b^2*c^3*d^5 - 4*a^3*b*c
^2*d^6 + a^4*c*d^7)*x^3 + 6*(b^4*c^6*d^2 - 4*a*b^3*c^5*d^3 + 6*a^2*b^2*c^4*d^4 - 4*a^3*b*c^3*d^5 + a^4*c^2*d^6
)*x^2 + 4*(b^4*c^7*d - 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^5*d^3 - 4*a^3*b*c^4*d^4 + a^4*c^3*d^5)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*x+a)**(5/6)/(d*x+c)**(25/6),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(25/6),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(25/6)), x)

Giac [F]

\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {25}{6}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(5/6)/(d*x+c)^(25/6),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(25/6)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{25/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{25/6}} \,d x \]

[In]

int(1/((a + b*x)^(5/6)*(c + d*x)^(25/6)),x)

[Out]

int(1/((a + b*x)^(5/6)*(c + d*x)^(25/6)), x)