\(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^3} \, dx\) [1121]

Optimal result

Integrand size = 24, antiderivative size = 232 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x}{e^5}-\frac {c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac {B c^2 x^3}{3 e^3}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{e^6} \]

[Out]

-(A*c*e*(-2*b*e+3*c*d)-B*(b^2*e^2-6*b*c*d*e+6*c^2*d^2))*x/e^5-1/2*c*(-A*c*e-2*B*b*e+3*B*c*d)*x^2/e^4+1/3*B*c^2
*x^3/e^3+1/2*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^2-d*(-b*e+c*d)*(B*d*(-3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))/e^
6/(e*x+d)+(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*ln(e*x+d)/e^6

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac {x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}-\frac {c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac {B c^2 x^3}{3 e^3} \]

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^3,x]

[Out]

-(((A*c*e*(3*c*d - 2*b*e) - B*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))*x)/e^5) - (c*(3*B*c*d - 2*b*B*e - A*c*e)*x^2)
/(2*e^4) + (B*c^2*x^3)/(3*e^3) + (d^2*(B*d - A*e)*(c*d - b*e)^2)/(2*e^6*(d + e*x)^2) - (d*(c*d - b*e)*(B*d*(5*
c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(e^6*(d + e*x)) + ((A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d
^2 - 12*b*c*d*e + 3*b^2*e^2))*Log[d + e*x])/e^6

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-A c e (3 c d-2 b e)+B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{e^5}+\frac {c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac {B c^2 x^2}{e^3}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^3}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^2}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)}\right ) \, dx \\ & = -\frac {\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x}{e^5}-\frac {c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac {B c^2 x^3}{3 e^3}+\frac {d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {6 e \left (A c e (-3 c d+2 b e)+B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x+3 c e^2 (-3 B c d+2 b B e+A c e) x^2+2 B c^2 e^3 x^3+\frac {3 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^2}-\frac {6 d (c d-b e) (B d (5 c d-3 b e)+2 A e (-2 c d+b e))}{d+e x}+6 \left (B d \left (-10 c^2 d^2+12 b c d e-3 b^2 e^2\right )+A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) \log (d+e x)}{6 e^6} \]

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^3,x]

[Out]

(6*e*(A*c*e*(-3*c*d + 2*b*e) + B*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2))*x + 3*c*e^2*(-3*B*c*d + 2*b*B*e + A*c*e)*x
^2 + 2*B*c^2*e^3*x^3 + (3*d^2*(B*d - A*e)*(c*d - b*e)^2)/(d + e*x)^2 - (6*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) +
2*A*e*(-2*c*d + b*e)))/(d + e*x) + 6*(B*d*(-10*c^2*d^2 + 12*b*c*d*e - 3*b^2*e^2) + A*e*(6*c^2*d^2 - 6*b*c*d*e
+ b^2*e^2))*Log[d + e*x])/(6*e^6)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.26

method	result	size
norman	\(\frac {\frac {d^{2} \left (3 A \,b^{2} e^{3}-18 A b c d \,e^{2}+18 A \,c^{2} d^{2} e -9 B \,b^{2} d \,e^{2}+36 B b c \,d^{2} e -30 B \,c^{2} d^{3}\right )}{2 e^{6}}+\frac {\left (6 A b c \,e^{2}-6 A \,c^{2} d e +3 B \,b^{2} e^{2}-12 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {B \,c^{2} x^{5}}{3 e}+\frac {c \left (3 A c e +6 B b e -5 B c d \right ) x^{4}}{6 e^{2}}+\frac {2 d \left (A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right ) x}{e^{5}}}{\left (e x +d \right )^{2}}+\frac {\left (A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\)	\(292\)
default	\(\frac {\frac {1}{3} B \,c^{2} x^{3} e^{2}+\frac {1}{2} A \,c^{2} e^{2} x^{2}+B b c \,e^{2} x^{2}-\frac {3}{2} B \,c^{2} d e \,x^{2}+2 A b c \,e^{2} x -3 A \,c^{2} d e x +B \,b^{2} e^{2} x -6 B b c d e x +6 B \,c^{2} d^{2} x}{e^{5}}+\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{e^{6} \left (e x +d \right )}+\frac {\left (A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{2 e^{6} \left (e x +d \right )^{2}}\)	\(302\)
risch	\(\frac {B \,c^{2} x^{3}}{3 e^{3}}+\frac {A \,c^{2} x^{2}}{2 e^{3}}+\frac {B b c \,x^{2}}{e^{3}}-\frac {3 B \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {2 A b c x}{e^{3}}-\frac {3 A \,c^{2} d x}{e^{4}}+\frac {b^{2} B x}{e^{3}}-\frac {6 B b c d x}{e^{4}}+\frac {6 B \,c^{2} d^{2} x}{e^{5}}+\frac {\left (2 A \,b^{2} d \,e^{3}-6 A b c \,d^{2} e^{2}+4 A \,c^{2} d^{3} e -3 B \,b^{2} d^{2} e^{2}+8 B b c \,d^{3} e -5 B \,c^{2} d^{4}\right ) x +\frac {d^{2} \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+7 A \,c^{2} d^{2} e -5 B \,b^{2} d \,e^{2}+14 B b c \,d^{2} e -9 B \,c^{2} d^{3}\right )}{2 e}}{e^{5} \left (e x +d \right )^{2}}+\frac {\ln \left (e x +d \right ) A \,b^{2}}{e^{3}}-\frac {6 \ln \left (e x +d \right ) A b c d}{e^{4}}+\frac {6 \ln \left (e x +d \right ) A \,c^{2} d^{2}}{e^{5}}-\frac {3 \ln \left (e x +d \right ) B \,b^{2} d}{e^{4}}+\frac {12 \ln \left (e x +d \right ) B b c \,d^{2}}{e^{5}}-\frac {10 \ln \left (e x +d \right ) B \,c^{2} d^{3}}{e^{6}}\)	\(340\)
parallelrisch	\(\frac {-90 B \,c^{2} d^{5}-120 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e +12 A \ln \left (e x +d \right ) x \,b^{2} d \,e^{4}+72 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{2}-36 B \ln \left (e x +d \right ) x \,b^{2} d^{2} e^{3}+36 A \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{3}-18 B \ln \left (e x +d \right ) x^{2} b^{2} d \,e^{4}-60 B \ln \left (e x +d \right ) x^{2} c^{2} d^{3} e^{2}-60 B \ln \left (e x +d \right ) c^{2} d^{5}+3 A \,x^{4} c^{2} e^{5}+6 B \,x^{3} b^{2} e^{5}+2 B \,x^{5} c^{2} e^{5}-36 A \ln \left (e x +d \right ) b c \,d^{3} e^{2}+72 B \ln \left (e x +d \right ) b c \,d^{4} e +144 B x b c \,d^{3} e^{2}-72 A x b c \,d^{2} e^{3}-24 B \,x^{3} b c d \,e^{4}+9 A \,b^{2} d^{2} e^{3}+54 A \,c^{2} d^{4} e -27 B \,b^{2} d^{3} e^{2}-54 A b c \,d^{3} e^{2}+108 B b c \,d^{4} e +72 B \ln \left (e x +d \right ) x^{2} b c \,d^{2} e^{3}-36 A \ln \left (e x +d \right ) x^{2} b c d \,e^{4}-72 A \ln \left (e x +d \right ) x b c \,d^{2} e^{3}+144 B \ln \left (e x +d \right ) x b c \,d^{3} e^{2}+12 A \,x^{3} b c \,e^{5}-12 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} c^{2} d^{2} e^{3}+12 A x \,b^{2} d \,e^{4}+72 A x \,c^{2} d^{3} e^{2}-36 B x \,b^{2} d^{2} e^{3}-120 B x \,c^{2} d^{4} e +6 A \ln \left (e x +d \right ) b^{2} d^{2} e^{3}+36 A \ln \left (e x +d \right ) c^{2} d^{4} e -18 B \ln \left (e x +d \right ) b^{2} d^{3} e^{2}+6 B \,x^{4} b c \,e^{5}-5 B \,x^{4} c^{2} d \,e^{4}+6 A \ln \left (e x +d \right ) x^{2} b^{2} e^{5}}{6 e^{6} \left (e x +d \right )^{2}}\)	\(584\)

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

(1/2*d^2*(3*A*b^2*e^3-18*A*b*c*d*e^2+18*A*c^2*d^2*e-9*B*b^2*d*e^2+36*B*b*c*d^2*e-30*B*c^2*d^3)/e^6+1/3*(6*A*b*
c*e^2-6*A*c^2*d*e+3*B*b^2*e^2-12*B*b*c*d*e+10*B*c^2*d^2)/e^3*x^3+1/3*B*c^2*x^5/e+1/6*c*(3*A*c*e+6*B*b*e-5*B*c*
d)/e^2*x^4+2*d*(A*b^2*e^3-6*A*b*c*d*e^2+6*A*c^2*d^2*e-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)/e^5*x)/(e*x+d
)^2+1/e^6*(A*b^2*e^3-6*A*b*c*d*e^2+6*A*c^2*d^2*e-3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (227) = 454\).

Time = 0.28 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} + 9 \, A b^{2} d^{2} e^{3} + 21 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - {\left (5 \, B c^{2} d e^{4} - 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (10 \, B c^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 3 \, {\left (21 \, B c^{2} d^{3} e^{2} - 11 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (B c^{2} d^{4} e + 2 \, A b^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 6 \, {\left (10 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + {\left (10 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 2 \, {\left (10 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/6*(2*B*c^2*e^5*x^5 - 27*B*c^2*d^5 + 9*A*b^2*d^2*e^3 + 21*(2*B*b*c + A*c^2)*d^4*e - 15*(B*b^2 + 2*A*b*c)*d^3*
e^2 - (5*B*c^2*d*e^4 - 3*(2*B*b*c + A*c^2)*e^5)*x^4 + 2*(10*B*c^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b
^2 + 2*A*b*c)*e^5)*x^3 + 3*(21*B*c^2*d^3*e^2 - 11*(2*B*b*c + A*c^2)*d^2*e^3 + 4*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 +
6*(B*c^2*d^4*e + 2*A*b^2*d*e^4 + (2*B*b*c + A*c^2)*d^3*e^2 - 2*(B*b^2 + 2*A*b*c)*d^2*e^3)*x - 6*(10*B*c^2*d^5
- A*b^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + (10*B*c^2*d^3*e^2 - A*b^2*e^5 - 6
*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 2*(10*B*c^2*d^4*e - A*b^2*d*e^4 - 6*(2*B*b*c + A
*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*log(e*x + d))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)

Sympy [A] (verification not implemented)

Time = 1.66 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {B c^{2} x^{3}}{3 e^{3}} + x^{2} \left (\frac {A c^{2}}{2 e^{3}} + \frac {B b c}{e^{3}} - \frac {3 B c^{2} d}{2 e^{4}}\right ) + x \left (\frac {2 A b c}{e^{3}} - \frac {3 A c^{2} d}{e^{4}} + \frac {B b^{2}}{e^{3}} - \frac {6 B b c d}{e^{4}} + \frac {6 B c^{2} d^{2}}{e^{5}}\right ) + \frac {3 A b^{2} d^{2} e^{3} - 10 A b c d^{3} e^{2} + 7 A c^{2} d^{4} e - 5 B b^{2} d^{3} e^{2} + 14 B b c d^{4} e - 9 B c^{2} d^{5} + x \left (4 A b^{2} d e^{4} - 12 A b c d^{2} e^{3} + 8 A c^{2} d^{3} e^{2} - 6 B b^{2} d^{2} e^{3} + 16 B b c d^{3} e^{2} - 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} - \frac {\left (- A b^{2} e^{3} + 6 A b c d e^{2} - 6 A c^{2} d^{2} e + 3 B b^{2} d e^{2} - 12 B b c d^{2} e + 10 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{6}} \]

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**3,x)

[Out]

B*c**2*x**3/(3*e**3) + x**2*(A*c**2/(2*e**3) + B*b*c/e**3 - 3*B*c**2*d/(2*e**4)) + x*(2*A*b*c/e**3 - 3*A*c**2*
d/e**4 + B*b**2/e**3 - 6*B*b*c*d/e**4 + 6*B*c**2*d**2/e**5) + (3*A*b**2*d**2*e**3 - 10*A*b*c*d**3*e**2 + 7*A*c
**2*d**4*e - 5*B*b**2*d**3*e**2 + 14*B*b*c*d**4*e - 9*B*c**2*d**5 + x*(4*A*b**2*d*e**4 - 12*A*b*c*d**2*e**3 +
8*A*c**2*d**3*e**2 - 6*B*b**2*d**2*e**3 + 16*B*b*c*d**3*e**2 - 10*B*c**2*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x +
2*e**8*x**2) - (-A*b**2*e**3 + 6*A*b*c*d*e**2 - 6*A*c**2*d**2*e + 3*B*b**2*d*e**2 - 12*B*b*c*d**2*e + 10*B*c**
2*d**3)*log(d + e*x)/e**6

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {9 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 7 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 2 \, {\left (5 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, B c^{2} e^{2} x^{3} - 3 \, {\left (3 \, B c^{2} d e - {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (6 \, B c^{2} d^{2} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac {{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

-1/2*(9*B*c^2*d^5 - 3*A*b^2*d^2*e^3 - 7*(2*B*b*c + A*c^2)*d^4*e + 5*(B*b^2 + 2*A*b*c)*d^3*e^2 + 2*(5*B*c^2*d^4
*e - 2*A*b^2*d*e^4 - 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(e^8*x^2 + 2*d*e^7*x + d^2*
e^6) + 1/6*(2*B*c^2*e^2*x^3 - 3*(3*B*c^2*d*e - (2*B*b*c + A*c^2)*e^2)*x^2 + 6*(6*B*c^2*d^2 - 3*(2*B*b*c + A*c^
2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*x)/e^5 - (10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*
A*b*c)*d*e^2)*log(e*x + d)/e^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, A b c d e^{2} - A b^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} - \frac {9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \, {\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 2 \, A b^{2} d e^{4}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{6}} + \frac {2 \, B c^{2} e^{6} x^{3} - 9 \, B c^{2} d e^{5} x^{2} + 6 \, B b c e^{6} x^{2} + 3 \, A c^{2} e^{6} x^{2} + 36 \, B c^{2} d^{2} e^{4} x - 36 \, B b c d e^{5} x - 18 \, A c^{2} d e^{5} x + 6 \, B b^{2} e^{6} x + 12 \, A b c e^{6} x}{6 \, e^{9}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

-(10*B*c^2*d^3 - 12*B*b*c*d^2*e - 6*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 - A*b^2*e^3)*log(abs(e*x + d))
/e^6 - 1/2*(9*B*c^2*d^5 - 14*B*b*c*d^4*e - 7*A*c^2*d^4*e + 5*B*b^2*d^3*e^2 + 10*A*b*c*d^3*e^2 - 3*A*b^2*d^2*e^
3 + 2*(5*B*c^2*d^4*e - 8*B*b*c*d^3*e^2 - 4*A*c^2*d^3*e^2 + 3*B*b^2*d^2*e^3 + 6*A*b*c*d^2*e^3 - 2*A*b^2*d*e^4)*
x)/((e*x + d)^2*e^6) + 1/6*(2*B*c^2*e^6*x^3 - 9*B*c^2*d*e^5*x^2 + 6*B*b*c*e^6*x^2 + 3*A*c^2*e^6*x^2 + 36*B*c^2
*d^2*e^4*x - 36*B*b*c*d*e^5*x - 18*A*c^2*d*e^5*x + 6*B*b^2*e^6*x + 12*A*b*c*e^6*x)/e^9

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx=x^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{2\,e^3}-\frac {3\,B\,c^2\,d}{2\,e^4}\right )-x\,\left (\frac {3\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^3}-\frac {3\,B\,c^2\,d}{e^4}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{e^3}+\frac {3\,B\,c^2\,d^2}{e^5}\right )-\frac {\frac {5\,B\,b^2\,d^3\,e^2-3\,A\,b^2\,d^2\,e^3-14\,B\,b\,c\,d^4\,e+10\,A\,b\,c\,d^3\,e^2+9\,B\,c^2\,d^5-7\,A\,c^2\,d^4\,e}{2\,e}+x\,\left (3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (-3\,B\,b^2\,d\,e^2+A\,b^2\,e^3+12\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2-10\,B\,c^2\,d^3+6\,A\,c^2\,d^2\,e\right )}{e^6}+\frac {B\,c^2\,x^3}{3\,e^3} \]

[In]

int(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^3,x)

[Out]

x^2*((A*c^2 + 2*B*b*c)/(2*e^3) - (3*B*c^2*d)/(2*e^4)) - x*((3*d*((A*c^2 + 2*B*b*c)/e^3 - (3*B*c^2*d)/e^4))/e -
(B*b^2 + 2*A*b*c)/e^3 + (3*B*c^2*d^2)/e^5) - ((9*B*c^2*d^5 - 7*A*c^2*d^4*e - 3*A*b^2*d^2*e^3 + 5*B*b^2*d^3*e^
2 - 14*B*b*c*d^4*e + 10*A*b*c*d^3*e^2)/(2*e) + x*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*A*c^2*d^3*e + 3*B*b^2*d^2*e^
2 - 8*B*b*c*d^3*e + 6*A*b*c*d^2*e^2))/(d^2*e^5 + e^7*x^2 + 2*d*e^6*x) + (log(d + e*x)*(A*b^2*e^3 - 10*B*c^2*d^
3 + 6*A*c^2*d^2*e - 3*B*b^2*d*e^2 - 6*A*b*c*d*e^2 + 12*B*b*c*d^2*e))/e^6 + (B*c^2*x^3)/(3*e^3)