Integrand size = 29, antiderivative size = 92 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {b (b d-a e) (B d-A e) x}{e^3}-\frac {(B d-A e) (a+b x)^2}{2 e^2}+\frac {B (a+b x)^3}{3 b e}-\frac {(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4} \] Output:
b*(-a*e+b*d)*(-A*e+B*d)*x/e^3-1/2*(-A*e+B*d)*(b*x+a)^2/e^2+1/3*B*(b*x+a)^3 /b/e-(-a*e+b*d)^2*(-A*e+B*d)*ln(e*x+d)/e^4
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {e x \left (6 a^2 B e^2+6 a b e (-2 B d+2 A e+B e x)+b^2 \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )-6 (b d-a e)^2 (B d-A e) \log (d+e x)}{6 e^4} \] Input:
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x),x]
Output:
(e*x*(6*a^2*B*e^2 + 6*a*b*e*(-2*B*d + 2*A*e + B*e*x) + b^2*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) - 6*(b*d - a*e)^2*(B*d - A*e)*Log [d + e*x])/(6*e^4)
Time = 0.44 (sec) , antiderivative size = 92, 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, 86, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right ) (A+B x)}{d+e x} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int \frac {b^2 (a+b x)^2 (A+B x)}{d+e x}dx}{b^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(a+b x)^2 (A+B x)}{d+e x}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {(a e-b d)^2 (A e-B d)}{e^3 (d+e x)}-\frac {b (b d-a e) (A e-B d)}{e^3}+\frac {b (a+b x) (A e-B d)}{e^2}+\frac {B (a+b x)^2}{e}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4}+\frac {b x (b d-a e) (B d-A e)}{e^3}-\frac {(a+b x)^2 (B d-A e)}{2 e^2}+\frac {B (a+b x)^3}{3 b e}\) |
Input:
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x),x]
Output:
(b*(b*d - a*e)*(B*d - A*e)*x)/e^3 - ((B*d - A*e)*(a + b*x)^2)/(2*e^2) + (B *(a + b*x)^3)/(3*b*e) - ((b*d - a*e)^2*(B*d - A*e)*Log[d + e*x])/e^4
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.61
method | result | size |
norman | \(\frac {\left (2 A a b \,e^{2}-A \,b^{2} d e +B \,e^{2} a^{2}-2 B a b d e +B \,b^{2} d^{2}\right ) x}{e^{3}}+\frac {B \,b^{2} x^{3}}{3 e}+\frac {b \left (A b e +2 B a e -B b d \right ) x^{2}}{2 e^{2}}+\frac {\left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(148\) |
default | \(\frac {\frac {1}{3} x^{3} B \,b^{2} e^{2}+\frac {1}{2} A \,b^{2} e^{2} x^{2}+B a b \,e^{2} x^{2}-\frac {1}{2} B \,b^{2} d e \,x^{2}+2 A a b \,e^{2} x -A \,b^{2} d e x +B \,a^{2} e^{2} x -2 B a b d e x +B \,b^{2} d^{2} x}{e^{3}}+\frac {\left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -B \,b^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(161\) |
risch | \(\frac {B \,b^{2} x^{3}}{3 e}+\frac {A \,b^{2} x^{2}}{2 e}+\frac {B a b \,x^{2}}{e}-\frac {B \,b^{2} d \,x^{2}}{2 e^{2}}+\frac {2 A a b x}{e}-\frac {A \,b^{2} d x}{e^{2}}+\frac {B \,a^{2} x}{e}-\frac {2 B a b d x}{e^{2}}+\frac {B \,b^{2} d^{2} x}{e^{3}}+\frac {\ln \left (e x +d \right ) a^{2} A}{e}-\frac {2 \ln \left (e x +d \right ) A a b d}{e^{2}}+\frac {\ln \left (e x +d \right ) A \,b^{2} d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}+\frac {2 \ln \left (e x +d \right ) B a b \,d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,b^{2} d^{3}}{e^{4}}\) | \(197\) |
parallelrisch | \(\frac {2 B \,b^{2} x^{3} e^{3}+3 A \,b^{2} e^{3} x^{2}+6 B \,x^{2} a b \,e^{3}-3 B \,x^{2} b^{2} d \,e^{2}+6 A \ln \left (e x +d \right ) a^{2} e^{3}-12 A \ln \left (e x +d \right ) a b d \,e^{2}+6 A \ln \left (e x +d \right ) b^{2} d^{2} e +12 A x a b \,e^{3}-6 A x \,b^{2} d \,e^{2}-6 B \ln \left (e x +d \right ) a^{2} d \,e^{2}+12 B \ln \left (e x +d \right ) a b \,d^{2} e -6 B \ln \left (e x +d \right ) b^{2} d^{3}+6 B x \,a^{2} e^{3}-12 B x a b d \,e^{2}+6 B x \,b^{2} d^{2} e}{6 e^{4}}\) | \(198\) |
Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
(2*A*a*b*e^2-A*b^2*d*e+B*a^2*e^2-2*B*a*b*d*e+B*b^2*d^2)/e^3*x+1/3*B/e*b^2* x^3+1/2*b/e^2*(A*b*e+2*B*a*e-B*b*d)*x^2+(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-B*b^2*d^3)/e^4*ln(e*x+d)
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{3} x^{3} - 3 \, {\left (B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \, {\left (B b^{2} d^{2} e - {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x - 6 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d),x, algorithm="fricas")
Output:
1/6*(2*B*b^2*e^3*x^3 - 3*(B*b^2*d*e^2 - (2*B*a*b + A*b^2)*e^3)*x^2 + 6*(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 - 6*(B*b^2* d^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)*log(e *x + d))/e^4
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {B b^{2} x^{3}}{3 e} + x^{2} \left (\frac {A b^{2}}{2 e} + \frac {B a b}{e} - \frac {B b^{2} d}{2 e^{2}}\right ) + x \left (\frac {2 A a b}{e} - \frac {A b^{2} d}{e^{2}} + \frac {B a^{2}}{e} - \frac {2 B a b d}{e^{2}} + \frac {B b^{2} d^{2}}{e^{3}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{4}} \] Input:
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d),x)
Output:
B*b**2*x**3/(3*e) + x**2*(A*b**2/(2*e) + B*a*b/e - B*b**2*d/(2*e**2)) + x* (2*A*a*b/e - A*b**2*d/e**2 + B*a**2/e - 2*B*a*b*d/e**2 + B*b**2*d**2/e**3) - (-A*e + B*d)*(a*e - b*d)**2*log(d + e*x)/e**4
Time = 0.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{2} x^{3} - 3 \, {\left (B b^{2} d e - {\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (B b^{2} d^{2} - {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac {{\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d),x, algorithm="maxima")
Output:
1/6*(2*B*b^2*e^2*x^3 - 3*(B*b^2*d*e - (2*B*a*b + A*b^2)*e^2)*x^2 + 6*(B*b^ 2*d^2 - (2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*x)/e^3 - (B*b^2*d^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)*log(e*x + d)/e^4
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {2 \, B b^{2} e^{2} x^{3} - 3 \, B b^{2} d e x^{2} + 6 \, B a b e^{2} x^{2} + 3 \, A b^{2} e^{2} x^{2} + 6 \, B b^{2} d^{2} x - 12 \, B a b d e x - 6 \, A b^{2} d e x + 6 \, B a^{2} e^{2} x + 12 \, A a b e^{2} x}{6 \, e^{3}} - \frac {{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{4}} \] Input:
integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d),x, algorithm="giac")
Output:
1/6*(2*B*b^2*e^2*x^3 - 3*B*b^2*d*e*x^2 + 6*B*a*b*e^2*x^2 + 3*A*b^2*e^2*x^2 + 6*B*b^2*d^2*x - 12*B*a*b*d*e*x - 6*A*b^2*d*e*x + 6*B*a^2*e^2*x + 12*A*a *b*e^2*x)/e^3 - (B*b^2*d^3 - 2*B*a*b*d^2*e - A*b^2*d^2*e + B*a^2*d*e^2 + 2 *A*a*b*d*e^2 - A*a^2*e^3)*log(abs(e*x + d))/e^4
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=x\,\left (\frac {B\,a^2+2\,A\,b\,a}{e}-\frac {d\,\left (\frac {A\,b^2+2\,B\,a\,b}{e}-\frac {B\,b^2\,d}{e^2}\right )}{e}\right )+x^2\,\left (\frac {A\,b^2+2\,B\,a\,b}{2\,e}-\frac {B\,b^2\,d}{2\,e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^2\,d\,e^2+A\,a^2\,e^3+2\,B\,a\,b\,d^2\,e-2\,A\,a\,b\,d\,e^2-B\,b^2\,d^3+A\,b^2\,d^2\,e\right )}{e^4}+\frac {B\,b^2\,x^3}{3\,e} \] Input:
int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x),x)
Output:
x*((B*a^2 + 2*A*a*b)/e - (d*((A*b^2 + 2*B*a*b)/e - (B*b^2*d)/e^2))/e) + x^ 2*((A*b^2 + 2*B*a*b)/(2*e) - (B*b^2*d)/(2*e^2)) + (log(d + e*x)*(A*a^2*e^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)) /e^4 + (B*b^2*x^3)/(3*e)
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx=\frac {6 \,\mathrm {log}\left (e x +d \right ) a^{3} e^{3}-18 \,\mathrm {log}\left (e x +d \right ) a^{2} b d \,e^{2}+18 \,\mathrm {log}\left (e x +d \right ) a \,b^{2} d^{2} e -6 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{3}+18 a^{2} b \,e^{3} x -18 a \,b^{2} d \,e^{2} x +9 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e x -3 b^{3} d \,e^{2} x^{2}+2 b^{3} e^{3} x^{3}}{6 e^{4}} \] Input:
int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d),x)
Output:
(6*log(d + e*x)*a**3*e**3 - 18*log(d + e*x)*a**2*b*d*e**2 + 18*log(d + e*x )*a*b**2*d**2*e - 6*log(d + e*x)*b**3*d**3 + 18*a**2*b*e**3*x - 18*a*b**2* d*e**2*x + 9*a*b**2*e**3*x**2 + 6*b**3*d**2*e*x - 3*b**3*d*e**2*x**2 + 2*b **3*e**3*x**3)/(6*e**4)