Integrand size = 15, antiderivative size = 58 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {(a+b x)^6}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^6}{42 (b c-a d)^2 (c+d x)^6} \]
Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(58)=116\).
Time = 0.04 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {6 a^5 d^5+5 a^4 b d^4 (c+7 d x)+4 a^3 b^2 d^3 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a^2 b^3 d^2 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+2 a b^4 d \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+b^5 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )}{42 d^6 (c+d x)^7} \]
-1/42*(6*a^5*d^5 + 5*a^4*b*d^4*(c + 7*d*x) + 4*a^3*b^2*d^3*(c^2 + 7*c*d*x + 21*d^2*x^2) + 3*a^2*b^3*d^2*(c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3 ) + 2*a*b^4*d*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*d^4*x^ 4) + b^5*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5))/(d^6*(c + d*x)^7)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b x)^5}{(c+d x)^8} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {b \int \frac {(a+b x)^5}{(c+d x)^7}dx}{7 (b c-a d)}+\frac {(a+b x)^6}{7 (c+d x)^7 (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {b (a+b x)^6}{42 (c+d x)^6 (b c-a d)^2}+\frac {(a+b x)^6}{7 (c+d x)^7 (b c-a d)}\) |
3.14.66.3.1 Defintions of rubi rules used
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(54)=108\).
Time = 0.23 (sec) , antiderivative size = 246, normalized size of antiderivative = 4.24
method | result | size |
risch | \(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 b^{4} \left (2 a d +b c \right ) x^{4}}{6 d^{2}}-\frac {5 b^{3} \left (3 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) x^{3}}{6 d^{3}}-\frac {b^{2} \left (4 a^{3} d^{3}+3 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{2}}{2 d^{4}}-\frac {b \left (5 a^{4} d^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{6 d^{5}}-\frac {6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6}}}{\left (d x +c \right )^{7}}\) | \(246\) |
default | \(-\frac {5 b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{6} \left (d x +c \right )^{4}}-\frac {b^{5}}{2 d^{6} \left (d x +c \right )^{2}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{6 d^{6} \left (d x +c \right )^{6}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{7 d^{6} \left (d x +c \right )^{7}}-\frac {5 b^{4} \left (a d -b c \right )}{3 d^{6} \left (d x +c \right )^{3}}-\frac {2 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{6} \left (d x +c \right )^{5}}\) | \(265\) |
norman | \(\frac {-\frac {b^{5} x^{5}}{2 d}-\frac {5 \left (2 a \,b^{4} d^{2}+b^{5} c d \right ) x^{4}}{6 d^{3}}-\frac {5 \left (3 a^{2} b^{3} d^{3}+2 a \,b^{4} c \,d^{2}+b^{5} c^{2} d \right ) x^{3}}{6 d^{4}}-\frac {\left (4 a^{3} b^{2} d^{4}+3 a^{2} c \,b^{3} d^{3}+2 a \,b^{4} c^{2} d^{2}+b^{5} c^{3} d \right ) x^{2}}{2 d^{5}}-\frac {\left (5 a^{4} b \,d^{5}+4 a^{3} b^{2} c \,d^{4}+3 a^{2} b^{3} c^{2} d^{3}+2 a \,b^{4} c^{3} d^{2}+b^{5} c^{4} d \right ) x}{6 d^{6}}-\frac {6 a^{5} d^{6}+5 a^{4} b c \,d^{5}+4 a^{3} b^{2} c^{2} d^{4}+3 a^{2} b^{3} c^{3} d^{3}+2 a \,b^{4} c^{4} d^{2}+b^{5} c^{5} d}{42 d^{7}}}{\left (d x +c \right )^{7}}\) | \(269\) |
gosper | \(-\frac {21 x^{5} b^{5} d^{5}+70 x^{4} a \,b^{4} d^{5}+35 x^{4} b^{5} c \,d^{4}+105 x^{3} a^{2} b^{3} d^{5}+70 x^{3} a \,b^{4} c \,d^{4}+35 x^{3} b^{5} c^{2} d^{3}+84 x^{2} a^{3} b^{2} d^{5}+63 x^{2} a^{2} b^{3} c \,d^{4}+42 x^{2} a \,b^{4} c^{2} d^{3}+21 x^{2} b^{5} c^{3} d^{2}+35 x \,a^{4} b \,d^{5}+28 x \,a^{3} b^{2} c \,d^{4}+21 x \,a^{2} b^{3} c^{2} d^{3}+14 x a \,b^{4} c^{3} d^{2}+7 x \,b^{5} c^{4} d +6 a^{5} d^{5}+5 a^{4} b c \,d^{4}+4 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+2 a \,b^{4} c^{4} d +b^{5} c^{5}}{42 d^{6} \left (d x +c \right )^{7}}\) | \(272\) |
parallelrisch | \(\frac {-21 b^{5} x^{5} d^{6}-70 a \,b^{4} d^{6} x^{4}-35 b^{5} c \,d^{5} x^{4}-105 a^{2} b^{3} d^{6} x^{3}-70 a \,b^{4} c \,d^{5} x^{3}-35 b^{5} c^{2} d^{4} x^{3}-84 a^{3} b^{2} d^{6} x^{2}-63 a^{2} b^{3} c \,d^{5} x^{2}-42 a \,b^{4} c^{2} d^{4} x^{2}-21 b^{5} c^{3} d^{3} x^{2}-35 a^{4} b \,d^{6} x -28 a^{3} b^{2} c \,d^{5} x -21 a^{2} b^{3} c^{2} d^{4} x -14 a \,b^{4} c^{3} d^{3} x -7 b^{5} c^{4} d^{2} x -6 a^{5} d^{6}-5 a^{4} b c \,d^{5}-4 a^{3} b^{2} c^{2} d^{4}-3 a^{2} b^{3} c^{3} d^{3}-2 a \,b^{4} c^{4} d^{2}-b^{5} c^{5} d}{42 d^{7} \left (d x +c \right )^{7}}\) | \(278\) |
(-1/2*b^5/d*x^5-5/6*b^4/d^2*(2*a*d+b*c)*x^4-5/6*b^3/d^3*(3*a^2*d^2+2*a*b*c *d+b^2*c^2)*x^3-1/2*b^2/d^4*(4*a^3*d^3+3*a^2*b*c*d^2+2*a*b^2*c^2*d+b^3*c^3 )*x^2-1/6*b/d^5*(5*a^4*d^4+4*a^3*b*c*d^3+3*a^2*b^2*c^2*d^2+2*a*b^3*c^3*d+b ^4*c^4)*x-1/42/d^6*(6*a^5*d^5+5*a^4*b*c*d^4+4*a^3*b^2*c^2*d^3+3*a^2*b^3*c^ 3*d^2+2*a*b^4*c^4*d+b^5*c^5))/(d*x+c)^7
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 326, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
-1/42*(21*b^5*d^5*x^5 + b^5*c^5 + 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^ 3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + 6*a^5*d^5 + 35*(b^5*c*d^4 + 2*a*b^4*d^5)*x ^4 + 35*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 21*(b^5*c^3*d^ 2 + 2*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + 4*a^3*b^2*d^5)*x^2 + 7*(b^5*c^4*d + 2*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x)/ (d^13*x^7 + 7*c*d^12*x^6 + 21*c^2*d^11*x^5 + 35*c^3*d^10*x^4 + 35*c^4*d^9* x^3 + 21*c^5*d^8*x^2 + 7*c^6*d^7*x + c^7*d^6)
Timed out. \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 326, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5} + 35 \, {\left (b^{5} c d^{4} + 2 \, a b^{4} d^{5}\right )} x^{4} + 35 \, {\left (b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 21 \, {\left (b^{5} c^{3} d^{2} + 2 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + 4 \, a^{3} b^{2} d^{5}\right )} x^{2} + 7 \, {\left (b^{5} c^{4} d + 2 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x}{42 \, {\left (d^{13} x^{7} + 7 \, c d^{12} x^{6} + 21 \, c^{2} d^{11} x^{5} + 35 \, c^{3} d^{10} x^{4} + 35 \, c^{4} d^{9} x^{3} + 21 \, c^{5} d^{8} x^{2} + 7 \, c^{6} d^{7} x + c^{7} d^{6}\right )}} \]
-1/42*(21*b^5*d^5*x^5 + b^5*c^5 + 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^ 3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 + 6*a^5*d^5 + 35*(b^5*c*d^4 + 2*a*b^4*d^5)*x ^4 + 35*(b^5*c^2*d^3 + 2*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 21*(b^5*c^3*d^ 2 + 2*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 + 4*a^3*b^2*d^5)*x^2 + 7*(b^5*c^4*d + 2*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x)/ (d^13*x^7 + 7*c*d^12*x^6 + 21*c^2*d^11*x^5 + 35*c^3*d^10*x^4 + 35*c^4*d^9* x^3 + 21*c^5*d^8*x^2 + 7*c^6*d^7*x + c^7*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 4.67 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=-\frac {21 \, b^{5} d^{5} x^{5} + 35 \, b^{5} c d^{4} x^{4} + 70 \, a b^{4} d^{5} x^{4} + 35 \, b^{5} c^{2} d^{3} x^{3} + 70 \, a b^{4} c d^{4} x^{3} + 105 \, a^{2} b^{3} d^{5} x^{3} + 21 \, b^{5} c^{3} d^{2} x^{2} + 42 \, a b^{4} c^{2} d^{3} x^{2} + 63 \, a^{2} b^{3} c d^{4} x^{2} + 84 \, a^{3} b^{2} d^{5} x^{2} + 7 \, b^{5} c^{4} d x + 14 \, a b^{4} c^{3} d^{2} x + 21 \, a^{2} b^{3} c^{2} d^{3} x + 28 \, a^{3} b^{2} c d^{4} x + 35 \, a^{4} b d^{5} x + b^{5} c^{5} + 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + 4 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + 6 \, a^{5} d^{5}}{42 \, {\left (d x + c\right )}^{7} d^{6}} \]
-1/42*(21*b^5*d^5*x^5 + 35*b^5*c*d^4*x^4 + 70*a*b^4*d^5*x^4 + 35*b^5*c^2*d ^3*x^3 + 70*a*b^4*c*d^4*x^3 + 105*a^2*b^3*d^5*x^3 + 21*b^5*c^3*d^2*x^2 + 4 2*a*b^4*c^2*d^3*x^2 + 63*a^2*b^3*c*d^4*x^2 + 84*a^3*b^2*d^5*x^2 + 7*b^5*c^ 4*d*x + 14*a*b^4*c^3*d^2*x + 21*a^2*b^3*c^2*d^3*x + 28*a^3*b^2*c*d^4*x + 3 5*a^4*b*d^5*x + b^5*c^5 + 2*a*b^4*c^4*d + 3*a^2*b^3*c^3*d^2 + 4*a^3*b^2*c^ 2*d^3 + 5*a^4*b*c*d^4 + 6*a^5*d^5)/((d*x + c)^7*d^6)
Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {{\left (a+b\,x\right )}^6\,\left (7\,b\,c-6\,a\,d+b\,d\,x\right )}{42\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^7} \]
Time = 0.00 (sec) , antiderivative size = 338, normalized size of antiderivative = 5.83 \[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx=\frac {-21 b^{5} d^{5} x^{5}-70 a \,b^{4} d^{5} x^{4}-35 b^{5} c \,d^{4} x^{4}-105 a^{2} b^{3} d^{5} x^{3}-70 a \,b^{4} c \,d^{4} x^{3}-35 b^{5} c^{2} d^{3} x^{3}-84 a^{3} b^{2} d^{5} x^{2}-63 a^{2} b^{3} c \,d^{4} x^{2}-42 a \,b^{4} c^{2} d^{3} x^{2}-21 b^{5} c^{3} d^{2} x^{2}-35 a^{4} b \,d^{5} x -28 a^{3} b^{2} c \,d^{4} x -21 a^{2} b^{3} c^{2} d^{3} x -14 a \,b^{4} c^{3} d^{2} x -7 b^{5} c^{4} d x -6 a^{5} d^{5}-5 a^{4} b c \,d^{4}-4 a^{3} b^{2} c^{2} d^{3}-3 a^{2} b^{3} c^{3} d^{2}-2 a \,b^{4} c^{4} d -b^{5} c^{5}}{42 d^{6} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )} \]
int((a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x**3 + 5*a*b**4* x**4 + b**5*x**5)/(c**8 + 8*c**7*d*x + 28*c**6*d**2*x**2 + 56*c**5*d**3*x* *3 + 70*c**4*d**4*x**4 + 56*c**3*d**5*x**5 + 28*c**2*d**6*x**6 + 8*c*d**7* x**7 + d**8*x**8),x)
( - 6*a**5*d**5 - 5*a**4*b*c*d**4 - 35*a**4*b*d**5*x - 4*a**3*b**2*c**2*d* *3 - 28*a**3*b**2*c*d**4*x - 84*a**3*b**2*d**5*x**2 - 3*a**2*b**3*c**3*d** 2 - 21*a**2*b**3*c**2*d**3*x - 63*a**2*b**3*c*d**4*x**2 - 105*a**2*b**3*d* *5*x**3 - 2*a*b**4*c**4*d - 14*a*b**4*c**3*d**2*x - 42*a*b**4*c**2*d**3*x* *2 - 70*a*b**4*c*d**4*x**3 - 70*a*b**4*d**5*x**4 - b**5*c**5 - 7*b**5*c**4 *d*x - 21*b**5*c**3*d**2*x**2 - 35*b**5*c**2*d**3*x**3 - 35*b**5*c*d**4*x* *4 - 21*b**5*d**5*x**5)/(42*d**6*(c**7 + 7*c**6*d*x + 21*c**5*d**2*x**2 + 35*c**4*d**3*x**3 + 35*c**3*d**4*x**4 + 21*c**2*d**5*x**5 + 7*c*d**6*x**6 + d**7*x**7))