Integrand size = 15, antiderivative size = 117 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {(b c-a d)^4}{8 d^5 (c+d x)^8}+\frac {4 b (b c-a d)^3}{7 d^5 (c+d x)^7}-\frac {b^2 (b c-a d)^2}{d^5 (c+d x)^6}+\frac {4 b^3 (b c-a d)}{5 d^5 (c+d x)^5}-\frac {b^4}{4 d^5 (c+d x)^4} \] Output:
-1/8*(-a*d+b*c)^4/d^5/(d*x+c)^8+4/7*b*(-a*d+b*c)^3/d^5/(d*x+c)^7-b^2*(-a*d +b*c)^2/d^5/(d*x+c)^6+4/5*b^3*(-a*d+b*c)/d^5/(d*x+c)^5-1/4*b^4/d^5/(d*x+c) ^4
Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {35 a^4 d^4+20 a^3 b d^3 (c+8 d x)+10 a^2 b^2 d^2 \left (c^2+8 c d x+28 d^2 x^2\right )+4 a b^3 d \left (c^3+8 c^2 d x+28 c d^2 x^2+56 d^3 x^3\right )+b^4 \left (c^4+8 c^3 d x+28 c^2 d^2 x^2+56 c d^3 x^3+70 d^4 x^4\right )}{280 d^5 (c+d x)^8} \] Input:
Integrate[(a + b*x)^4/(c + d*x)^9,x]
Output:
-1/280*(35*a^4*d^4 + 20*a^3*b*d^3*(c + 8*d*x) + 10*a^2*b^2*d^2*(c^2 + 8*c* d*x + 28*d^2*x^2) + 4*a*b^3*d*(c^3 + 8*c^2*d*x + 28*c*d^2*x^2 + 56*d^3*x^3 ) + b^4*(c^4 + 8*c^3*d*x + 28*c^2*d^2*x^2 + 56*c*d^3*x^3 + 70*d^4*x^4))/(d ^5*(c + d*x)^8)
Time = 0.26 (sec) , antiderivative size = 117, 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 = {53, 2009}
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.
\(\displaystyle \int \frac {(a+b x)^4}{(c+d x)^9} \, dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (-\frac {4 b^3 (b c-a d)}{d^4 (c+d x)^6}+\frac {6 b^2 (b c-a d)^2}{d^4 (c+d x)^7}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)^8}+\frac {(a d-b c)^4}{d^4 (c+d x)^9}+\frac {b^4}{d^4 (c+d x)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 b^3 (b c-a d)}{5 d^5 (c+d x)^5}-\frac {b^2 (b c-a d)^2}{d^5 (c+d x)^6}+\frac {4 b (b c-a d)^3}{7 d^5 (c+d x)^7}-\frac {(b c-a d)^4}{8 d^5 (c+d x)^8}-\frac {b^4}{4 d^5 (c+d x)^4}\) |
Input:
Int[(a + b*x)^4/(c + d*x)^9,x]
Output:
-1/8*(b*c - a*d)^4/(d^5*(c + d*x)^8) + (4*b*(b*c - a*d)^3)/(7*d^5*(c + d*x )^7) - (b^2*(b*c - a*d)^2)/(d^5*(c + d*x)^6) + (4*b^3*(b*c - a*d))/(5*d^5* (c + d*x)^5) - b^4/(4*d^5*(c + d*x)^4)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.46
method | result | size |
risch | \(\frac {-\frac {b^{4} x^{4}}{4 d}-\frac {b^{3} \left (4 a d +b c \right ) x^{3}}{5 d^{2}}-\frac {b^{2} \left (10 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{10 d^{3}}-\frac {b \left (20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{35 d^{4}}-\frac {35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}}{280 d^{5}}}{\left (x d +c \right )^{8}}\) | \(171\) |
gosper | \(-\frac {70 d^{4} x^{4} b^{4}+224 a \,b^{3} d^{4} x^{3}+56 b^{4} c \,d^{3} x^{3}+280 a^{2} b^{2} d^{4} x^{2}+112 a \,b^{3} c \,d^{3} x^{2}+28 b^{4} c^{2} d^{2} x^{2}+160 a^{3} b \,d^{4} x +80 a^{2} b^{2} c \,d^{3} x +32 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}}{280 d^{5} \left (x d +c \right )^{8}}\) | \(185\) |
orering | \(-\frac {70 d^{4} x^{4} b^{4}+224 a \,b^{3} d^{4} x^{3}+56 b^{4} c \,d^{3} x^{3}+280 a^{2} b^{2} d^{4} x^{2}+112 a \,b^{3} c \,d^{3} x^{2}+28 b^{4} c^{2} d^{2} x^{2}+160 a^{3} b \,d^{4} x +80 a^{2} b^{2} c \,d^{3} x +32 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}}{280 d^{5} \left (x d +c \right )^{8}}\) | \(185\) |
default | \(-\frac {b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{5} \left (x d +c \right )^{6}}-\frac {4 b^{3} \left (a d -b c \right )}{5 d^{5} \left (x d +c \right )^{5}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{7 d^{5} \left (x d +c \right )^{7}}-\frac {b^{4}}{4 d^{5} \left (x d +c \right )^{4}}-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{8 d^{5} \left (x d +c \right )^{8}}\) | \(186\) |
parallelrisch | \(\frac {-70 b^{4} x^{4} d^{7}-224 a \,b^{3} d^{7} x^{3}-56 b^{4} c \,d^{6} x^{3}-280 a^{2} b^{2} d^{7} x^{2}-112 a \,b^{3} c \,d^{6} x^{2}-28 b^{4} c^{2} d^{5} x^{2}-160 a^{3} b \,d^{7} x -80 a^{2} b^{2} c \,d^{6} x -32 a \,b^{3} c^{2} d^{5} x -8 b^{4} c^{3} d^{4} x -35 a^{4} d^{7}-20 a^{3} b c \,d^{6}-10 a^{2} b^{2} c^{2} d^{5}-4 a \,b^{3} c^{3} d^{4}-b^{4} c^{4} d^{3}}{280 d^{8} \left (x d +c \right )^{8}}\) | \(193\) |
norman | \(\frac {-\frac {b^{4} x^{4}}{4 d}-\frac {\left (4 a \,b^{3} d^{4}+b^{4} c \,d^{3}\right ) x^{3}}{5 d^{5}}-\frac {\left (10 a^{2} b^{2} d^{5}+4 a \,b^{3} c \,d^{4}+b^{4} c^{2} d^{3}\right ) x^{2}}{10 d^{6}}-\frac {\left (20 a^{3} b \,d^{6}+10 a^{2} b^{2} c \,d^{5}+4 a \,b^{3} c^{2} d^{4}+b^{4} c^{3} d^{3}\right ) x}{35 d^{7}}-\frac {35 a^{4} d^{7}+20 a^{3} b c \,d^{6}+10 a^{2} b^{2} c^{2} d^{5}+4 a \,b^{3} c^{3} d^{4}+b^{4} c^{4} d^{3}}{280 d^{8}}}{\left (x d +c \right )^{8}}\) | \(197\) |
Input:
int((b*x+a)^4/(d*x+c)^9,x,method=_RETURNVERBOSE)
Output:
(-1/4*b^4/d*x^4-1/5*b^3/d^2*(4*a*d+b*c)*x^3-1/10*b^2/d^3*(10*a^2*d^2+4*a*b *c*d+b^2*c^2)*x^2-1/35*b/d^4*(20*a^3*d^3+10*a^2*b*c*d^2+4*a*b^2*c^2*d+b^3* c^3)*x-1/280/d^5*(35*a^4*d^4+20*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2+4*a*b^3*c^3 *d+b^4*c^4))/(d*x+c)^8
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).
Time = 0.11 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 56 \, {\left (b^{4} c d^{3} + 4 \, a b^{3} d^{4}\right )} x^{3} + 28 \, {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d + 4 \, a b^{3} c^{2} d^{2} + 10 \, a^{2} b^{2} c d^{3} + 20 \, a^{3} b d^{4}\right )} x}{280 \, {\left (d^{13} x^{8} + 8 \, c d^{12} x^{7} + 28 \, c^{2} d^{11} x^{6} + 56 \, c^{3} d^{10} x^{5} + 70 \, c^{4} d^{9} x^{4} + 56 \, c^{5} d^{8} x^{3} + 28 \, c^{6} d^{7} x^{2} + 8 \, c^{7} d^{6} x + c^{8} d^{5}\right )}} \] Input:
integrate((b*x+a)^4/(d*x+c)^9,x, algorithm="fricas")
Output:
-1/280*(70*b^4*d^4*x^4 + b^4*c^4 + 4*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 20 *a^3*b*c*d^3 + 35*a^4*d^4 + 56*(b^4*c*d^3 + 4*a*b^3*d^4)*x^3 + 28*(b^4*c^2 *d^2 + 4*a*b^3*c*d^3 + 10*a^2*b^2*d^4)*x^2 + 8*(b^4*c^3*d + 4*a*b^3*c^2*d^ 2 + 10*a^2*b^2*c*d^3 + 20*a^3*b*d^4)*x)/(d^13*x^8 + 8*c*d^12*x^7 + 28*c^2* d^11*x^6 + 56*c^3*d^10*x^5 + 70*c^4*d^9*x^4 + 56*c^5*d^8*x^3 + 28*c^6*d^7* x^2 + 8*c^7*d^6*x + c^8*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (104) = 208\).
Time = 9.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=\frac {- 35 a^{4} d^{4} - 20 a^{3} b c d^{3} - 10 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4} - 70 b^{4} d^{4} x^{4} + x^{3} \left (- 224 a b^{3} d^{4} - 56 b^{4} c d^{3}\right ) + x^{2} \left (- 280 a^{2} b^{2} d^{4} - 112 a b^{3} c d^{3} - 28 b^{4} c^{2} d^{2}\right ) + x \left (- 160 a^{3} b d^{4} - 80 a^{2} b^{2} c d^{3} - 32 a b^{3} c^{2} d^{2} - 8 b^{4} c^{3} d\right )}{280 c^{8} d^{5} + 2240 c^{7} d^{6} x + 7840 c^{6} d^{7} x^{2} + 15680 c^{5} d^{8} x^{3} + 19600 c^{4} d^{9} x^{4} + 15680 c^{3} d^{10} x^{5} + 7840 c^{2} d^{11} x^{6} + 2240 c d^{12} x^{7} + 280 d^{13} x^{8}} \] Input:
integrate((b*x+a)**4/(d*x+c)**9,x)
Output:
(-35*a**4*d**4 - 20*a**3*b*c*d**3 - 10*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3 *d - b**4*c**4 - 70*b**4*d**4*x**4 + x**3*(-224*a*b**3*d**4 - 56*b**4*c*d* *3) + x**2*(-280*a**2*b**2*d**4 - 112*a*b**3*c*d**3 - 28*b**4*c**2*d**2) + x*(-160*a**3*b*d**4 - 80*a**2*b**2*c*d**3 - 32*a*b**3*c**2*d**2 - 8*b**4* c**3*d))/(280*c**8*d**5 + 2240*c**7*d**6*x + 7840*c**6*d**7*x**2 + 15680*c **5*d**8*x**3 + 19600*c**4*d**9*x**4 + 15680*c**3*d**10*x**5 + 7840*c**2*d **11*x**6 + 2240*c*d**12*x**7 + 280*d**13*x**8)
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).
Time = 0.04 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 56 \, {\left (b^{4} c d^{3} + 4 \, a b^{3} d^{4}\right )} x^{3} + 28 \, {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d + 4 \, a b^{3} c^{2} d^{2} + 10 \, a^{2} b^{2} c d^{3} + 20 \, a^{3} b d^{4}\right )} x}{280 \, {\left (d^{13} x^{8} + 8 \, c d^{12} x^{7} + 28 \, c^{2} d^{11} x^{6} + 56 \, c^{3} d^{10} x^{5} + 70 \, c^{4} d^{9} x^{4} + 56 \, c^{5} d^{8} x^{3} + 28 \, c^{6} d^{7} x^{2} + 8 \, c^{7} d^{6} x + c^{8} d^{5}\right )}} \] Input:
integrate((b*x+a)^4/(d*x+c)^9,x, algorithm="maxima")
Output:
-1/280*(70*b^4*d^4*x^4 + b^4*c^4 + 4*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 20 *a^3*b*c*d^3 + 35*a^4*d^4 + 56*(b^4*c*d^3 + 4*a*b^3*d^4)*x^3 + 28*(b^4*c^2 *d^2 + 4*a*b^3*c*d^3 + 10*a^2*b^2*d^4)*x^2 + 8*(b^4*c^3*d + 4*a*b^3*c^2*d^ 2 + 10*a^2*b^2*c*d^3 + 20*a^3*b*d^4)*x)/(d^13*x^8 + 8*c*d^12*x^7 + 28*c^2* d^11*x^6 + 56*c^3*d^10*x^5 + 70*c^4*d^9*x^4 + 56*c^5*d^8*x^3 + 28*c^6*d^7* x^2 + 8*c^7*d^6*x + c^8*d^5)
Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + 56 \, b^{4} c d^{3} x^{3} + 224 \, a b^{3} d^{4} x^{3} + 28 \, b^{4} c^{2} d^{2} x^{2} + 112 \, a b^{3} c d^{3} x^{2} + 280 \, a^{2} b^{2} d^{4} x^{2} + 8 \, b^{4} c^{3} d x + 32 \, a b^{3} c^{2} d^{2} x + 80 \, a^{2} b^{2} c d^{3} x + 160 \, a^{3} b d^{4} x + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}}{280 \, {\left (d x + c\right )}^{8} d^{5}} \] Input:
integrate((b*x+a)^4/(d*x+c)^9,x, algorithm="giac")
Output:
-1/280*(70*b^4*d^4*x^4 + 56*b^4*c*d^3*x^3 + 224*a*b^3*d^4*x^3 + 28*b^4*c^2 *d^2*x^2 + 112*a*b^3*c*d^3*x^2 + 280*a^2*b^2*d^4*x^2 + 8*b^4*c^3*d*x + 32* a*b^3*c^2*d^2*x + 80*a^2*b^2*c*d^3*x + 160*a^3*b*d^4*x + b^4*c^4 + 4*a*b^3 *c^3*d + 10*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 + 35*a^4*d^4)/((d*x + c)^8*d^ 5)
Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=-\frac {\frac {35\,a^4\,d^4+20\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4}{280\,d^5}+\frac {b^4\,x^4}{4\,d}+\frac {b^3\,x^3\,\left (4\,a\,d+b\,c\right )}{5\,d^2}+\frac {b\,x\,\left (20\,a^3\,d^3+10\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{35\,d^4}+\frac {b^2\,x^2\,\left (10\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{10\,d^3}}{c^8+8\,c^7\,d\,x+28\,c^6\,d^2\,x^2+56\,c^5\,d^3\,x^3+70\,c^4\,d^4\,x^4+56\,c^3\,d^5\,x^5+28\,c^2\,d^6\,x^6+8\,c\,d^7\,x^7+d^8\,x^8} \] Input:
int((a + b*x)^4/(c + d*x)^9,x)
Output:
-((35*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 20*a^3*b*c* d^3)/(280*d^5) + (b^4*x^4)/(4*d) + (b^3*x^3*(4*a*d + b*c))/(5*d^2) + (b*x* (20*a^3*d^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2))/(35*d^4) + (b^2*x ^2*(10*a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/(10*d^3))/(c^8 + d^8*x^8 + 8*c*d^7* x^7 + 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^7*d*x)
Time = 0.16 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x)^4}{(c+d x)^9} \, dx=\frac {-70 b^{4} d^{4} x^{4}-224 a \,b^{3} d^{4} x^{3}-56 b^{4} c \,d^{3} x^{3}-280 a^{2} b^{2} d^{4} x^{2}-112 a \,b^{3} c \,d^{3} x^{2}-28 b^{4} c^{2} d^{2} x^{2}-160 a^{3} b \,d^{4} x -80 a^{2} b^{2} c \,d^{3} x -32 a \,b^{3} c^{2} d^{2} x -8 b^{4} c^{3} d x -35 a^{4} d^{4}-20 a^{3} b c \,d^{3}-10 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}}{280 d^{5} \left (d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}\right )} \] Input:
int((b*x+a)^4/(d*x+c)^9,x)
Output:
( - 35*a**4*d**4 - 20*a**3*b*c*d**3 - 160*a**3*b*d**4*x - 10*a**2*b**2*c** 2*d**2 - 80*a**2*b**2*c*d**3*x - 280*a**2*b**2*d**4*x**2 - 4*a*b**3*c**3*d - 32*a*b**3*c**2*d**2*x - 112*a*b**3*c*d**3*x**2 - 224*a*b**3*d**4*x**3 - b**4*c**4 - 8*b**4*c**3*d*x - 28*b**4*c**2*d**2*x**2 - 56*b**4*c*d**3*x** 3 - 70*b**4*d**4*x**4)/(280*d**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))