Integrand size = 29, antiderivative size = 86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3} \] Output:
-1/4*(-A*e+B*d)*(b*x+a)^3/e/(-a*e+b*d)/(e*x+d)^4+1/12*(A*b*e-4*B*a*e+3*B*b *d)*(b*x+a)^3/e/(-a*e+b*d)^2/(e*x+d)^3
Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^5,x]
Output:
-1/12*(a^2*e^2*(3*A*e + B*(d + 4*e*x)) + 2*a*b*e*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^2)) + b^2*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)))/(e^4*(d + e*x)^4)
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1184, 27, 87, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right ) (A+B x)}{(d+e x)^5} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^2 (a+b x)^2 (A+B x)}{(d+e x)^5}dx}{b^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-4 a B e+A b e+3 b B d) \int \frac {(a+b x)^2}{(d+e x)^4}dx}{4 e (b d-a e)}-\frac {(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac {(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^5,x]
Output:
-1/4*((B*d - A*e)*(a + b*x)^3)/(e*(b*d - a*e)*(d + e*x)^4) + ((3*b*B*d + A *b*e - 4*a*B*e)*(a + b*x)^3)/(12*e*(b*d - a*e)^2*(d + e*x)^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.73 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.77
method | result | size |
risch | \(\frac {-\frac {B \,b^{2} x^{3}}{e}-\frac {b \left (A b e +2 B a e +3 B b d \right ) x^{2}}{2 e^{2}}-\frac {\left (2 A a b \,e^{2}+A \,b^{2} d e +B \,e^{2} a^{2}+2 B a b d e +3 B \,b^{2} d^{2}\right ) x}{3 e^{3}}-\frac {3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 B \,b^{2} d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(152\) |
norman | \(\frac {-\frac {B \,b^{2} x^{3}}{e}-\frac {\left (A \,b^{2} e +2 B e a b +3 B \,b^{2} d \right ) x^{2}}{2 e^{2}}-\frac {\left (2 A a b \,e^{2}+A \,b^{2} d e +B \,e^{2} a^{2}+2 B a b d e +3 B \,b^{2} d^{2}\right ) x}{3 e^{3}}-\frac {3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 B \,b^{2} d^{3}}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(156\) |
default | \(-\frac {2 A a b \,e^{2}-2 A \,b^{2} d e +B \,e^{2} a^{2}-4 B a b d e +3 B \,b^{2} d^{2}}{3 e^{4} \left (e x +d \right )^{3}}-\frac {a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {B \,b^{2}}{e^{4} \left (e x +d \right )}-\frac {b \left (A b e +2 B a e -3 B b d \right )}{2 e^{4} \left (e x +d \right )^{2}}\) | \(166\) |
gosper | \(-\frac {12 B \,b^{2} x^{3} e^{3}+6 A \,b^{2} e^{3} x^{2}+12 B \,x^{2} a b \,e^{3}+18 B \,x^{2} b^{2} d \,e^{2}+8 A x a b \,e^{3}+4 A x \,b^{2} d \,e^{2}+4 B x \,a^{2} e^{3}+8 B x a b d \,e^{2}+12 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 B \,b^{2} d^{3}}{12 e^{4} \left (e x +d \right )^{4}}\) | \(167\) |
parallelrisch | \(-\frac {12 B \,b^{2} x^{3} e^{3}+6 A \,b^{2} e^{3} x^{2}+12 B \,x^{2} a b \,e^{3}+18 B \,x^{2} b^{2} d \,e^{2}+8 A x a b \,e^{3}+4 A x \,b^{2} d \,e^{2}+4 B x \,a^{2} e^{3}+8 B x a b d \,e^{2}+12 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 B \,b^{2} d^{3}}{12 e^{4} \left (e x +d \right )^{4}}\) | \(167\) |
orering | \(-\frac {\left (12 B \,b^{2} x^{3} e^{3}+6 A \,b^{2} e^{3} x^{2}+12 B \,x^{2} a b \,e^{3}+18 B \,x^{2} b^{2} d \,e^{2}+8 A x a b \,e^{3}+4 A x \,b^{2} d \,e^{2}+4 B x \,a^{2} e^{3}+8 B x a b d \,e^{2}+12 B x \,b^{2} d^{2} e +3 a^{2} A \,e^{3}+2 A a b d \,e^{2}+A \,b^{2} d^{2} e +B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e +3 B \,b^{2} d^{3}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}{12 e^{4} \left (b x +a \right )^{2} \left (e x +d \right )^{4}}\) | \(190\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x,method=_RETURNVERBOSE)
Output:
(-B/e*b^2*x^3-1/2*b*(A*b*e+2*B*a*e+3*B*b*d)/e^2*x^2-1/3*(2*A*a*b*e^2+A*b^2 *d*e+B*a^2*e^2+2*B*a*b*d*e+3*B*b^2*d^2)/e^3*x-1/12*(3*A*a^2*e^3+2*A*a*b*d* e^2+A*b^2*d^2*e+B*a^2*d*e^2+2*B*a*b*d^2*e+3*B*b^2*d^3)/e^4)/(e*x+d)^4
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (82) = 164\).
Time = 0.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.17 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="fricas")
Output:
-1/12*(12*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 3*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^ 2*e + (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)* x^2 + 4*(3*B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3)* x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (75) = 150\).
Time = 5.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.59 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=\frac {- 3 A a^{2} e^{3} - 2 A a b d e^{2} - A b^{2} d^{2} e - B a^{2} d e^{2} - 2 B a b d^{2} e - 3 B b^{2} d^{3} - 12 B b^{2} e^{3} x^{3} + x^{2} \left (- 6 A b^{2} e^{3} - 12 B a b e^{3} - 18 B b^{2} d e^{2}\right ) + x \left (- 8 A a b e^{3} - 4 A b^{2} d e^{2} - 4 B a^{2} e^{3} - 8 B a b d e^{2} - 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**5,x)
Output:
(-3*A*a**2*e**3 - 2*A*a*b*d*e**2 - A*b**2*d**2*e - B*a**2*d*e**2 - 2*B*a*b *d**2*e - 3*B*b**2*d**3 - 12*B*b**2*e**3*x**3 + x**2*(-6*A*b**2*e**3 - 12* B*a*b*e**3 - 18*B*b**2*d*e**2) + x*(-8*A*a*b*e**3 - 4*A*b**2*d*e**2 - 4*B* a**2*e**3 - 8*B*a*b*d*e**2 - 12*B*b**2*d**2*e))/(12*d**4*e**4 + 48*d**3*e* *5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (82) = 164\).
Time = 0.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.17 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="maxima")
Output:
-1/12*(12*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 3*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^ 2*e + (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)* x^2 + 4*(3*B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3)* x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (82) = 164\).
Time = 0.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.70 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {\frac {12 \, B a b}{{\left (e x + d\right )}^{2}} + \frac {6 \, A b^{2}}{{\left (e x + d\right )}^{2}} - \frac {16 \, B a b d}{{\left (e x + d\right )}^{3}} - \frac {8 \, A b^{2} d}{{\left (e x + d\right )}^{3}} + \frac {6 \, B a b d^{2}}{{\left (e x + d\right )}^{4}} + \frac {3 \, A b^{2} d^{2}}{{\left (e x + d\right )}^{4}} + \frac {12 \, B b^{2}}{{\left (e x + d\right )} e} - \frac {18 \, B b^{2} d}{{\left (e x + d\right )}^{2} e} + \frac {12 \, B b^{2} d^{2}}{{\left (e x + d\right )}^{3} e} - \frac {3 \, B b^{2} d^{3}}{{\left (e x + d\right )}^{4} e} + \frac {4 \, B a^{2} e}{{\left (e x + d\right )}^{3}} + \frac {8 \, A a b e}{{\left (e x + d\right )}^{3}} - \frac {3 \, B a^{2} d e}{{\left (e x + d\right )}^{4}} - \frac {6 \, A a b d e}{{\left (e x + d\right )}^{4}} + \frac {3 \, A a^{2} e^{2}}{{\left (e x + d\right )}^{4}}}{12 \, e^{3}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="giac")
Output:
-1/12*(12*B*a*b/(e*x + d)^2 + 6*A*b^2/(e*x + d)^2 - 16*B*a*b*d/(e*x + d)^3 - 8*A*b^2*d/(e*x + d)^3 + 6*B*a*b*d^2/(e*x + d)^4 + 3*A*b^2*d^2/(e*x + d) ^4 + 12*B*b^2/((e*x + d)*e) - 18*B*b^2*d/((e*x + d)^2*e) + 12*B*b^2*d^2/(( e*x + d)^3*e) - 3*B*b^2*d^3/((e*x + d)^4*e) + 4*B*a^2*e/(e*x + d)^3 + 8*A* a*b*e/(e*x + d)^3 - 3*B*a^2*d*e/(e*x + d)^4 - 6*A*a*b*d*e/(e*x + d)^4 + 3* A*a^2*e^2/(e*x + d)^4)/e^3
Time = 11.85 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.14 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=-\frac {\frac {B\,a^2\,d\,e^2+3\,A\,a^2\,e^3+2\,B\,a\,b\,d^2\,e+2\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+A\,b^2\,d^2\,e}{12\,e^4}+\frac {x\,\left (B\,a^2\,e^2+2\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{3\,e^3}+\frac {b\,x^2\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{2\,e^2}+\frac {B\,b^2\,x^3}{e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x)^5,x)
Output:
-((3*A*a^2*e^3 + 3*B*b^2*d^3 + A*b^2*d^2*e + B*a^2*d*e^2 + 2*A*a*b*d*e^2 + 2*B*a*b*d^2*e)/(12*e^4) + (x*(B*a^2*e^2 + 3*B*b^2*d^2 + 2*A*a*b*e^2 + A*b ^2*d*e + 2*B*a*b*d*e))/(3*e^3) + (b*x^2*(A*b*e + 2*B*a*e + 3*B*b*d))/(2*e^ 2) + (B*b^2*x^3)/e)/(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e *x)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx=\frac {b^{3} e^{3} x^{4}-6 a \,b^{2} d \,e^{2} x^{2}-4 a^{2} b d \,e^{2} x -4 a \,b^{2} d^{2} e x -a^{3} d \,e^{2}-a^{2} b \,d^{2} e -a \,b^{2} d^{3}}{4 d \,e^{3} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x)
Output:
( - a**3*d*e**2 - a**2*b*d**2*e - 4*a**2*b*d*e**2*x - a*b**2*d**3 - 4*a*b* *2*d**2*e*x - 6*a*b**2*d*e**2*x**2 + b**3*e**3*x**4)/(4*d*e**3*(d**4 + 4*d **3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))