Integrand size = 39, antiderivative size = 23 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {(d+e x)^6}{3 \left (a+b x+c x^2\right )^3} \] Output:
-1/3*(e*x+d)^6/(c*x^2+b*x+a)^3
Leaf count is larger than twice the leaf count of optimal. \(118\) vs. \(2(23)=46\).
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 5.13 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=\frac {3 c^2 e^6 x^4 (a+b x)+3 c e^6 x^2 (a+b x)^2+e^6 (a+b x)^3-c^3 d \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )}{3 c^3 (a+x (b+c x))^3} \] Input:
Integrate[((d + e*x)^5*(b*d - 2*a*e + (2*c*d - b*e)*x))/(a + b*x + c*x^2)^ 4,x]
Output:
(3*c^2*e^6*x^4*(a + b*x) + 3*c*e^6*x^2*(a + b*x)^2 + e^6*(a + b*x)^3 - c^3 *d*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e ^5*x^5))/(3*c^3*(a + x*(b + c*x))^3)
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1226}
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.
\(\displaystyle \int \frac {(d+e x)^5 (-2 a e+x (2 c d-b e)+b d)}{\left (a+b x+c x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 1226 |
\(\displaystyle -\frac {(d+e x)^6}{3 \left (a+b x+c x^2\right )^3}\) |
Input:
Int[((d + e*x)^5*(b*d - 2*a*e + (2*c*d - b*e)*x))/(a + b*x + c*x^2)^4,x]
Output:
-1/3*(d + e*x)^6/(a + b*x + c*x^2)^3
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[b*(e*f + d*g) - 2*(c*d*f + a*e*g), 0]
Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(21)=42\).
Time = 1.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 7.78
method | result | size |
risch | \(\frac {\frac {\left (b e -2 c d \right ) e^{5} x^{5}}{c}+\frac {e^{4} \left (a c \,e^{2}+b^{2} e^{2}-5 c^{2} d^{2}\right ) x^{4}}{c^{2}}+\frac {e^{3} \left (6 a b c \,e^{3}+b^{3} e^{3}-20 d^{3} c^{3}\right ) x^{3}}{3 c^{3}}+\frac {e^{2} \left (e^{4} a^{2} c +a \,b^{2} e^{4}-5 d^{4} c^{3}\right ) x^{2}}{c^{3}}+\frac {e \left (a^{2} b \,e^{5}-2 d^{5} c^{3}\right ) x}{c^{3}}+\frac {e^{6} a^{3}-d^{6} c^{3}}{3 c^{3}}}{\left (c \,x^{2}+b x +a \right )^{3}}\) | \(179\) |
norman | \(\frac {\frac {\left (b \,e^{6}-2 c d \,e^{5}\right ) x^{5}}{c}+\frac {\left (a^{2} b \,e^{6}-2 c^{3} d^{5} e \right ) x}{c^{3}}+\frac {\left (a c \,e^{6}+b^{2} e^{6}-5 c^{2} d^{2} e^{4}\right ) x^{4}}{c^{2}}+\frac {\left (a^{2} c \,e^{6}+a \,b^{2} e^{6}-5 c^{3} d^{4} e^{2}\right ) x^{2}}{c^{3}}+\frac {e^{6} a^{3}-d^{6} c^{3}}{3 c^{3}}+\frac {\left (6 a b c \,e^{6}+b^{3} e^{6}-20 e^{3} c^{3} d^{3}\right ) x^{3}}{3 c^{3}}}{\left (c \,x^{2}+b x +a \right )^{3}}\) | \(181\) |
default | \(-\frac {-\frac {\left (b e -2 c d \right ) e^{5} x^{5}}{c}-\frac {e^{4} \left (a c \,e^{2}+b^{2} e^{2}-5 c^{2} d^{2}\right ) x^{4}}{c^{2}}-\frac {e^{3} \left (6 a b c \,e^{3}+b^{3} e^{3}-20 d^{3} c^{3}\right ) x^{3}}{3 c^{3}}-\frac {e^{2} \left (e^{4} a^{2} c +a \,b^{2} e^{4}-5 d^{4} c^{3}\right ) x^{2}}{c^{3}}-\frac {e \left (a^{2} b \,e^{5}-2 d^{5} c^{3}\right ) x}{c^{3}}-\frac {e^{6} a^{3}-d^{6} c^{3}}{3 c^{3}}}{\left (c \,x^{2}+b x +a \right )^{3}}\) | \(184\) |
gosper | \(\frac {3 b \,c^{2} e^{6} x^{5}-6 c^{3} d \,e^{5} x^{5}+3 a \,c^{2} e^{6} x^{4}+3 b^{2} c \,e^{6} x^{4}-15 c^{3} d^{2} e^{4} x^{4}+6 a b c \,e^{6} x^{3}+b^{3} e^{6} x^{3}-20 c^{3} d^{3} e^{3} x^{3}+3 a^{2} c \,e^{6} x^{2}+3 a \,b^{2} e^{6} x^{2}-15 c^{3} d^{4} e^{2} x^{2}+3 a^{2} b \,e^{6} x -6 c^{3} d^{5} e x +e^{6} a^{3}-d^{6} c^{3}}{3 c^{3} \left (c \,x^{2}+b x +a \right )^{3}}\) | \(189\) |
parallelrisch | \(\frac {3 b \,c^{2} e^{6} x^{5}-6 c^{3} d \,e^{5} x^{5}+3 a \,c^{2} e^{6} x^{4}+3 b^{2} c \,e^{6} x^{4}-15 c^{3} d^{2} e^{4} x^{4}+6 a b c \,e^{6} x^{3}+b^{3} e^{6} x^{3}-20 c^{3} d^{3} e^{3} x^{3}+3 a^{2} c \,e^{6} x^{2}+3 a \,b^{2} e^{6} x^{2}-15 c^{3} d^{4} e^{2} x^{2}+3 a^{2} b \,e^{6} x -6 c^{3} d^{5} e x +e^{6} a^{3}-d^{6} c^{3}}{3 c^{3} \left (c \,x^{2}+b x +a \right )^{3}}\) | \(189\) |
orering | \(-\frac {\left (3 b \,c^{2} e^{6} x^{5}-6 c^{3} d \,e^{5} x^{5}+3 a \,c^{2} e^{6} x^{4}+3 b^{2} c \,e^{6} x^{4}-15 c^{3} d^{2} e^{4} x^{4}+6 a b c \,e^{6} x^{3}+b^{3} e^{6} x^{3}-20 c^{3} d^{3} e^{3} x^{3}+3 a^{2} c \,e^{6} x^{2}+3 a \,b^{2} e^{6} x^{2}-15 c^{3} d^{4} e^{2} x^{2}+3 a^{2} b \,e^{6} x -6 c^{3} d^{5} e x +e^{6} a^{3}-d^{6} c^{3}\right ) \left (b d -2 a e +\left (-b e +2 c d \right ) x \right )}{3 \left (c \,x^{2}+b x +a \right )^{3} c^{3} \left (b e x -2 c d x +2 a e -b d \right )}\) | \(228\) |
Input:
int((e*x+d)^5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x^2+b*x+a)^4,x,method=_RETURNV ERBOSE)
Output:
((b*e-2*c*d)*e^5/c*x^5+e^4*(a*c*e^2+b^2*e^2-5*c^2*d^2)/c^2*x^4+1/3*e^3*(6* a*b*c*e^3+b^3*e^3-20*c^3*d^3)/c^3*x^3+e^2*(a^2*c*e^4+a*b^2*e^4-5*c^3*d^4)/ c^3*x^2+e*(a^2*b*e^5-2*c^3*d^5)/c^3*x+1/3*(a^3*e^6-c^3*d^6)/c^3)/(c*x^2+b* x+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (21) = 42\).
Time = 0.07 (sec) , antiderivative size = 254, normalized size of antiderivative = 11.04 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {c^{3} d^{6} - a^{3} e^{6} + 3 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (5 \, c^{3} d^{2} e^{4} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + {\left (20 \, c^{3} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (5 \, c^{3} d^{4} e^{2} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - a^{2} b e^{6}\right )} x}{3 \, {\left (c^{6} x^{6} + 3 \, b c^{5} x^{5} + 3 \, a^{2} b c^{3} x + a^{3} c^{3} + 3 \, {\left (b^{2} c^{4} + a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} + 6 \, a b c^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2}\right )}} \] Input:
integrate((e*x+d)^5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x^2+b*x+a)^4,x, algorith m="fricas")
Output:
-1/3*(c^3*d^6 - a^3*e^6 + 3*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 3*(5*c^3*d^2*e ^4 - (b^2*c + a*c^2)*e^6)*x^4 + (20*c^3*d^3*e^3 - (b^3 + 6*a*b*c)*e^6)*x^3 + 3*(5*c^3*d^4*e^2 - (a*b^2 + a^2*c)*e^6)*x^2 + 3*(2*c^3*d^5*e - a^2*b*e^ 6)*x)/(c^6*x^6 + 3*b*c^5*x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^ 5)*x^4 + (b^3*c^3 + 6*a*b*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*x^2)
Timed out. \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x**2+b*x+a)**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (21) = 42\).
Time = 0.05 (sec) , antiderivative size = 254, normalized size of antiderivative = 11.04 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {c^{3} d^{6} - a^{3} e^{6} + 3 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (5 \, c^{3} d^{2} e^{4} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + {\left (20 \, c^{3} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (5 \, c^{3} d^{4} e^{2} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - a^{2} b e^{6}\right )} x}{3 \, {\left (c^{6} x^{6} + 3 \, b c^{5} x^{5} + 3 \, a^{2} b c^{3} x + a^{3} c^{3} + 3 \, {\left (b^{2} c^{4} + a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} + 6 \, a b c^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2}\right )}} \] Input:
integrate((e*x+d)^5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x^2+b*x+a)^4,x, algorith m="maxima")
Output:
-1/3*(c^3*d^6 - a^3*e^6 + 3*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 3*(5*c^3*d^2*e ^4 - (b^2*c + a*c^2)*e^6)*x^4 + (20*c^3*d^3*e^3 - (b^3 + 6*a*b*c)*e^6)*x^3 + 3*(5*c^3*d^4*e^2 - (a*b^2 + a^2*c)*e^6)*x^2 + 3*(2*c^3*d^5*e - a^2*b*e^ 6)*x)/(c^6*x^6 + 3*b*c^5*x^5 + 3*a^2*b*c^3*x + a^3*c^3 + 3*(b^2*c^4 + a*c^ 5)*x^4 + (b^3*c^3 + 6*a*b*c^4)*x^3 + 3*(a*b^2*c^3 + a^2*c^4)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (21) = 42\).
Time = 0.16 (sec) , antiderivative size = 189, normalized size of antiderivative = 8.22 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {6 \, c^{3} d e^{5} x^{5} - 3 \, b c^{2} e^{6} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} - 3 \, b^{2} c e^{6} x^{4} - 3 \, a c^{2} e^{6} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} - b^{3} e^{6} x^{3} - 6 \, a b c e^{6} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} - 3 \, a b^{2} e^{6} x^{2} - 3 \, a^{2} c e^{6} x^{2} + 6 \, c^{3} d^{5} e x - 3 \, a^{2} b e^{6} x + c^{3} d^{6} - a^{3} e^{6}}{3 \, {\left (c x^{2} + b x + a\right )}^{3} c^{3}} \] Input:
integrate((e*x+d)^5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x^2+b*x+a)^4,x, algorith m="giac")
Output:
-1/3*(6*c^3*d*e^5*x^5 - 3*b*c^2*e^6*x^5 + 15*c^3*d^2*e^4*x^4 - 3*b^2*c*e^6 *x^4 - 3*a*c^2*e^6*x^4 + 20*c^3*d^3*e^3*x^3 - b^3*e^6*x^3 - 6*a*b*c*e^6*x^ 3 + 15*c^3*d^4*e^2*x^2 - 3*a*b^2*e^6*x^2 - 3*a^2*c*e^6*x^2 + 6*c^3*d^5*e*x - 3*a^2*b*e^6*x + c^3*d^6 - a^3*e^6)/((c*x^2 + b*x + a)^3*c^3)
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 10.61 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=\frac {\frac {a^3\,e^6-c^3\,d^6}{3\,c^3}+\frac {x^3\,\left (b^3\,e^6+6\,a\,b\,c\,e^6-20\,c^3\,d^3\,e^3\right )}{3\,c^3}+\frac {x\,\left (a^2\,b\,e^6-2\,c^3\,d^5\,e\right )}{c^3}+\frac {x^2\,\left (a^2\,c\,e^6+a\,b^2\,e^6-5\,c^3\,d^4\,e^2\right )}{c^3}+\frac {x^5\,\left (b\,e^6-2\,c\,d\,e^5\right )}{c}+\frac {x^4\,\left (b^2\,e^6-5\,c^2\,d^2\,e^4+a\,c\,e^6\right )}{c^2}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x} \] Input:
int(-((d + e*x)^5*(2*a*e - b*d + x*(b*e - 2*c*d)))/(a + b*x + c*x^2)^4,x)
Output:
((a^3*e^6 - c^3*d^6)/(3*c^3) + (x^3*(b^3*e^6 - 20*c^3*d^3*e^3 + 6*a*b*c*e^ 6))/(3*c^3) + (x*(a^2*b*e^6 - 2*c^3*d^5*e))/c^3 + (x^2*(a*b^2*e^6 + a^2*c* e^6 - 5*c^3*d^4*e^2))/c^3 + (x^5*(b*e^6 - 2*c*d*e^5))/c + (x^4*(b^2*e^6 - 5*c^2*d^2*e^4 + a*c*e^6))/c^2)/(x^2*(3*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 3 *b^2*c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x)
Time = 0.17 (sec) , antiderivative size = 274, normalized size of antiderivative = 11.91 \[ \int \frac {(d+e x)^5 (b d-2 a e+(2 c d-b e) x)}{\left (a+b x+c x^2\right )^4} \, dx=\frac {-b \,c^{2} e^{6} x^{6}+2 c^{3} d \,e^{5} x^{6}+6 a \,c^{2} d \,e^{5} x^{4}+6 b^{2} c d \,e^{5} x^{4}-15 b \,c^{2} d^{2} e^{4} x^{4}+12 a b c d \,e^{5} x^{3}+2 b^{3} d \,e^{5} x^{3}-20 b \,c^{2} d^{3} e^{3} x^{3}+6 a^{2} c d \,e^{5} x^{2}+6 a \,b^{2} d \,e^{5} x^{2}-15 b \,c^{2} d^{4} e^{2} x^{2}+6 a^{2} b d \,e^{5} x -6 b \,c^{2} d^{5} e x +2 a^{3} d \,e^{5}-b \,c^{2} d^{6}}{3 b \,c^{2} \left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:
int((e*x+d)^5*(b*d-2*a*e+(-b*e+2*c*d)*x)/(c*x^2+b*x+a)^4,x)
Output:
(2*a**3*d*e**5 + 6*a**2*b*d*e**5*x + 6*a**2*c*d*e**5*x**2 + 6*a*b**2*d*e** 5*x**2 + 12*a*b*c*d*e**5*x**3 + 6*a*c**2*d*e**5*x**4 + 2*b**3*d*e**5*x**3 + 6*b**2*c*d*e**5*x**4 - b*c**2*d**6 - 6*b*c**2*d**5*e*x - 15*b*c**2*d**4* e**2*x**2 - 20*b*c**2*d**3*e**3*x**3 - 15*b*c**2*d**2*e**4*x**4 - b*c**2*e **6*x**6 + 2*c**3*d*e**5*x**6)/(3*b*c**2*(a**3 + 3*a**2*b*x + 3*a**2*c*x** 2 + 3*a*b**2*x**2 + 6*a*b*c*x**3 + 3*a*c**2*x**4 + b**3*x**3 + 3*b**2*c*x* *4 + 3*b*c**2*x**5 + c**3*x**6))