\(\int \frac {1}{(d+e x)^2 (b x+c x^2)^3} \, dx\) [284]

Optimal result

Integrand size = 19, antiderivative size = 230 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d^2 x^2}+\frac {3 c d+2 b e}{b^4 d^3 x}+\frac {c^4}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac {c^4 (3 c d-5 b e)}{b^4 (c d-b e)^3 (b+c x)}-\frac {e^5}{d^3 (c d-b e)^3 (d+e x)}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^4}-\frac {3 c^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\frac {c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac {2 b e+3 c d}{b^4 d^3 x}+\frac {c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {1}{2 b^3 d^2 x^2}+\frac {3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac {3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac {e^5}{d^3 (d+e x) (c d-b e)^3} \]

[In]

Int[1/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

-1/2*1/(b^3*d^2*x^2) + (3*c*d + 2*b*e)/(b^4*d^3*x) + c^4/(2*b^3*(c*d - b*e)^2*(b + c*x)^2) + (c^4*(3*c*d - 5*b
*e))/(b^4*(c*d - b*e)^3*(b + c*x)) - e^5/(d^3*(c*d - b*e)^3*(d + e*x)) + (3*(2*c^2*d^2 + 2*b*c*d*e + b^2*e^2)*
Log[x])/(b^5*d^4) - (3*c^4*(2*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^4) + (3*e^5*(2*c
*d - b*e)*Log[d + e*x])/(d^4*(c*d - b*e)^4)

Rule 712

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^3 d^2 x^3}+\frac {-3 c d-2 b e}{b^4 d^3 x^2}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 d^4 x}-\frac {c^5}{b^3 (-c d+b e)^2 (b+c x)^3}-\frac {c^5 (-3 c d+5 b e)}{b^4 (-c d+b e)^3 (b+c x)^2}-\frac {3 c^5 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 (-c d+b e)^4 (b+c x)}+\frac {e^6}{d^3 (c d-b e)^3 (d+e x)^2}+\frac {3 e^6 (2 c d-b e)}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx \\ & = -\frac {1}{2 b^3 d^2 x^2}+\frac {3 c d+2 b e}{b^4 d^3 x}+\frac {c^4}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac {c^4 (3 c d-5 b e)}{b^4 (c d-b e)^3 (b+c x)}-\frac {e^5}{d^3 (c d-b e)^3 (d+e x)}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^4}-\frac {3 c^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d^2 x^2}+\frac {3 c d+2 b e}{b^4 d^3 x}+\frac {c^4}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac {c^4 (-3 c d+5 b e)}{b^4 (-c d+b e)^3 (b+c x)}-\frac {e^5}{d^3 (c d-b e)^3 (d+e x)}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^4}-\frac {3 c^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \]

[In]

Integrate[1/((d + e*x)^2*(b*x + c*x^2)^3),x]

[Out]

-1/2*1/(b^3*d^2*x^2) + (3*c*d + 2*b*e)/(b^4*d^3*x) + c^4/(2*b^3*(c*d - b*e)^2*(b + c*x)^2) + (c^4*(-3*c*d + 5*
b*e))/(b^4*(-(c*d) + b*e)^3*(b + c*x)) - e^5/(d^3*(c*d - b*e)^3*(d + e*x)) + (3*(2*c^2*d^2 + 2*b*c*d*e + b^2*e
^2)*Log[x])/(b^5*d^4) - (3*c^4*(2*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*Log[b + c*x])/(b^5*(c*d - b*e)^4) + (3*e^5*
(2*c*d - b*e)*Log[d + e*x])/(d^4*(c*d - b*e)^4)

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {1}{2 b^{3} d^{2} x^{2}}-\frac {-2 b e -3 c d}{b^{4} d^{3} x}+\frac {\left (3 b^{2} e^{2}+6 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{4}}+\frac {c^{4}}{2 \left (b e -c d \right )^{2} b^{3} \left (c x +b \right )^{2}}+\frac {c^{4} \left (5 b e -3 c d \right )}{\left (b e -c d \right )^{3} b^{4} \left (c x +b \right )}-\frac {3 c^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{5}}+\frac {e^{5}}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )}-\frac {3 e^{5} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{d^{4} \left (b e -c d \right )^{4}}\)	\(226\)
norman	\(\frac {\frac {\left (-3 b^{6} e^{6}+2 b^{4} d^{2} e^{4} c^{2}+6 b^{3} c^{3} d^{3} e^{3}-20 b \,c^{5} d^{5} e +12 c^{6} d^{6}\right ) x^{3}}{d^{4} b^{4} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {1}{2 b d}+\frac {\left (3 b e +4 c d \right ) x}{2 b^{2} d^{2}}+\frac {c \left (-12 b^{6} e^{6}+20 b^{4} d^{2} e^{4} c^{2}+9 b^{3} c^{3} d^{3} e^{3}-39 b^{2} c^{4} d^{4} e^{2}-8 b \,c^{5} d^{5} e +18 c^{6} d^{6}\right ) x^{4}}{2 d^{4} b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {e \,c^{2} \left (-6 b^{5} e^{5}+13 b^{3} c^{2} d^{2} e^{3}+b^{2} c^{3} d^{3} e^{2}-32 b \,c^{4} d^{4} e +18 c^{5} d^{5}\right ) x^{5}}{2 d^{4} b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}}{\left (e x +d \right ) x^{2} \left (c x +b \right )^{2}}+\frac {3 \left (b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{4}}-\frac {3 c^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {3 e^{5} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}\)	\(569\)
risch	\(\frac {\frac {3 c^{2} e \left (b^{4} e^{4}-b^{3} c d \,e^{3}-b^{2} c^{2} d^{2} e^{2}+4 b \,c^{3} d^{3} e -2 c^{4} d^{4}\right ) x^{4}}{b^{4} d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {3 c \left (4 b^{5} e^{5}-3 b^{4} c d \,e^{4}-5 b^{3} c^{2} d^{2} e^{3}+10 b^{2} c^{3} d^{3} e^{2}+2 b \,c^{4} d^{4} e -4 c^{5} d^{5}\right ) x^{3}}{2 b^{4} d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\left (6 b^{5} e^{5}-13 b^{3} c^{2} d^{2} e^{3}-b^{2} c^{3} d^{3} e^{2}+32 b \,c^{4} d^{4} e -18 c^{5} d^{5}\right ) x^{2}}{2 b^{3} d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\left (3 b e +4 c d \right ) x}{2 b^{2} d^{2}}-\frac {1}{2 b d}}{\left (e x +d \right ) x^{2} \left (c x +b \right )^{2}}-\frac {15 c^{4} \ln \left (c x +b \right ) e^{2}}{b^{3} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}+\frac {18 c^{5} \ln \left (c x +b \right ) d e}{b^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {6 c^{6} \ln \left (c x +b \right ) d^{2}}{b^{5} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {3 e^{6} \ln \left (-e x -d \right ) b}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}+\frac {6 e^{5} \ln \left (-e x -d \right ) c}{d^{3} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}+\frac {3 \ln \left (-x \right ) e^{2}}{b^{3} d^{4}}+\frac {6 \ln \left (-x \right ) c e}{b^{4} d^{3}}+\frac {6 \ln \left (-x \right ) c^{2}}{b^{5} d^{2}}\)	\(754\)
parallelrisch	\(\text {Expression too large to display}\)	\(1421\)

[In]

int(1/(e*x+d)^2/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (226) = 452\).

Time = 40.40 (sec) , antiderivative size = 1305, normalized size of antiderivative = 5.67 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*c^4*d^7 - 4*b^5*c^3*d^6*e + 6*b^6*c^2*d^5*e^2 - 4*b^7*c*d^4*e^3 + b^8*d^3*e^4 - 6*(2*b*c^7*d^6*e - 6
*b^2*c^6*d^5*e^2 + 5*b^3*c^5*d^4*e^3 - 2*b^5*c^3*d^2*e^5 + b^6*c^2*d*e^6)*x^4 - 3*(4*b*c^7*d^7 - 6*b^2*c^6*d^6
*e - 8*b^3*c^5*d^5*e^2 + 15*b^4*c^4*d^4*e^3 - 2*b^5*c^3*d^3*e^4 - 7*b^6*c^2*d^2*e^5 + 4*b^7*c*d*e^6)*x^3 - (18
*b^2*c^6*d^7 - 50*b^3*c^5*d^6*e + 33*b^4*c^4*d^5*e^2 + 12*b^5*c^3*d^4*e^3 - 13*b^6*c^2*d^3*e^4 - 6*b^7*c*d^2*e
^5 + 6*b^8*d*e^6)*x^2 - (4*b^3*c^5*d^7 - 13*b^4*c^4*d^6*e + 12*b^5*c^3*d^5*e^2 + 2*b^6*c^2*d^4*e^3 - 8*b^7*c*d
^3*e^4 + 3*b^8*d^2*e^5)*x + 6*((2*c^8*d^6*e - 6*b*c^7*d^5*e^2 + 5*b^2*c^6*d^4*e^3)*x^5 + (2*c^8*d^7 - 2*b*c^7*
d^6*e - 7*b^2*c^6*d^5*e^2 + 10*b^3*c^5*d^4*e^3)*x^4 + (4*b*c^7*d^7 - 10*b^2*c^6*d^6*e + 4*b^3*c^5*d^5*e^2 + 5*
b^4*c^4*d^4*e^3)*x^3 + (2*b^2*c^6*d^7 - 6*b^3*c^5*d^6*e + 5*b^4*c^4*d^5*e^2)*x^2)*log(c*x + b) - 6*((2*b^5*c^3
*d*e^6 - b^6*c^2*e^7)*x^5 + (2*b^5*c^3*d^2*e^5 + 3*b^6*c^2*d*e^6 - 2*b^7*c*e^7)*x^4 + (4*b^6*c^2*d^2*e^5 - b^8
*e^7)*x^3 + (2*b^7*c*d^2*e^5 - b^8*d*e^6)*x^2)*log(e*x + d) - 6*((2*c^8*d^6*e - 6*b*c^7*d^5*e^2 + 5*b^2*c^6*d^
4*e^3 - 2*b^5*c^3*d*e^6 + b^6*c^2*e^7)*x^5 + (2*c^8*d^7 - 2*b*c^7*d^6*e - 7*b^2*c^6*d^5*e^2 + 10*b^3*c^5*d^4*e
^3 - 2*b^5*c^3*d^2*e^5 - 3*b^6*c^2*d*e^6 + 2*b^7*c*e^7)*x^4 + (4*b*c^7*d^7 - 10*b^2*c^6*d^6*e + 4*b^3*c^5*d^5*
e^2 + 5*b^4*c^4*d^4*e^3 - 4*b^6*c^2*d^2*e^5 + b^8*e^7)*x^3 + (2*b^2*c^6*d^7 - 6*b^3*c^5*d^6*e + 5*b^4*c^4*d^5*
e^2 - 2*b^7*c*d^2*e^5 + b^8*d*e^6)*x^2)*log(x))/((b^5*c^6*d^8*e - 4*b^6*c^5*d^7*e^2 + 6*b^7*c^4*d^6*e^3 - 4*b^
8*c^3*d^5*e^4 + b^9*c^2*d^4*e^5)*x^5 + (b^5*c^6*d^9 - 2*b^6*c^5*d^8*e - 2*b^7*c^4*d^7*e^2 + 8*b^8*c^3*d^6*e^3
- 7*b^9*c^2*d^5*e^4 + 2*b^10*c*d^4*e^5)*x^4 + (2*b^6*c^5*d^9 - 7*b^7*c^4*d^8*e + 8*b^8*c^3*d^7*e^2 - 2*b^9*c^2
*d^6*e^3 - 2*b^10*c*d^5*e^4 + b^11*d^4*e^5)*x^3 + (b^7*c^4*d^9 - 4*b^8*c^3*d^8*e + 6*b^9*c^2*d^7*e^2 - 4*b^10*
c*d^6*e^3 + b^11*d^5*e^4)*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (226) = 452\).

Time = 0.24 (sec) , antiderivative size = 752, normalized size of antiderivative = 3.27 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (2 \, c^{6} d^{2} - 6 \, b c^{5} d e + 5 \, b^{2} c^{4} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} + \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 6 \, {\left (2 \, c^{6} d^{4} e - 4 \, b c^{5} d^{3} e^{2} + b^{2} c^{4} d^{2} e^{3} + b^{3} c^{3} d e^{4} - b^{4} c^{2} e^{5}\right )} x^{4} - 3 \, {\left (4 \, c^{6} d^{5} - 2 \, b c^{5} d^{4} e - 10 \, b^{2} c^{4} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} + 3 \, b^{4} c^{2} d e^{4} - 4 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 32 \, b^{2} c^{4} d^{4} e + b^{3} c^{3} d^{3} e^{2} + 13 \, b^{4} c^{2} d^{2} e^{3} - 6 \, b^{6} e^{5}\right )} x^{2} - {\left (4 \, b^{2} c^{4} d^{5} - 9 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 5 \, b^{5} c d^{2} e^{3} - 3 \, b^{6} d e^{4}\right )} x}{2 \, {\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} + {\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} + {\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} + {\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac {3 \, {\left (2 \, c^{2} d^{2} + 2 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{4}} \]

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-3*(2*c^6*d^2 - 6*b*c^5*d*e + 5*b^2*c^4*e^2)*log(c*x + b)/(b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 -
4*b^8*c*d*e^3 + b^9*e^4) + 3*(2*c*d*e^5 - b*e^6)*log(e*x + d)/(c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 -
4*b^3*c*d^5*e^3 + b^4*d^4*e^4) - 1/2*(b^3*c^3*d^5 - 3*b^4*c^2*d^4*e + 3*b^5*c*d^3*e^2 - b^6*d^2*e^3 - 6*(2*c^6
*d^4*e - 4*b*c^5*d^3*e^2 + b^2*c^4*d^2*e^3 + b^3*c^3*d*e^4 - b^4*c^2*e^5)*x^4 - 3*(4*c^6*d^5 - 2*b*c^5*d^4*e -
10*b^2*c^4*d^3*e^2 + 5*b^3*c^3*d^2*e^3 + 3*b^4*c^2*d*e^4 - 4*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 32*b^2*c^4*d^4*
e + b^3*c^3*d^3*e^2 + 13*b^4*c^2*d^2*e^3 - 6*b^6*e^5)*x^2 - (4*b^2*c^4*d^5 - 9*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e
^2 + 5*b^5*c*d^2*e^3 - 3*b^6*d*e^4)*x)/((b^4*c^5*d^6*e - 3*b^5*c^4*d^5*e^2 + 3*b^6*c^3*d^4*e^3 - b^7*c^2*d^3*e
^4)*x^5 + (b^4*c^5*d^7 - b^5*c^4*d^6*e - 3*b^6*c^3*d^5*e^2 + 5*b^7*c^2*d^4*e^3 - 2*b^8*c*d^3*e^4)*x^4 + (2*b^5
*c^4*d^7 - 5*b^6*c^3*d^6*e + 3*b^7*c^2*d^5*e^2 + b^8*c*d^4*e^3 - b^9*d^3*e^4)*x^3 + (b^6*c^3*d^7 - 3*b^7*c^2*d
^6*e + 3*b^8*c*d^5*e^2 - b^9*d^4*e^3)*x^2) + 3*(2*c^2*d^2 + 2*b*c*d*e + b^2*e^2)*log(x)/(b^5*d^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (226) = 452\).

Time = 0.29 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.77 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=-\frac {e^{11}}{{\left (c^{3} d^{6} e^{6} - 3 \, b c^{2} d^{5} e^{7} + 3 \, b^{2} c d^{4} e^{8} - b^{3} d^{3} e^{9}\right )} {\left (e x + d\right )}} - \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | -c + \frac {2 \, c d}{e x + d} - \frac {c d^{2}}{{\left (e x + d\right )}^{2}} - \frac {b e}{e x + d} + \frac {b d e}{{\left (e x + d\right )}^{2}} \right |}\right )}{2 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )}} - \frac {3 \, {\left (4 \, c^{6} d^{6} e^{2} - 12 \, b c^{5} d^{5} e^{3} + 10 \, b^{2} c^{4} d^{4} e^{4} - 2 \, b^{5} c d e^{7} + b^{6} e^{8}\right )} \log \left (\frac {{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{e x + d} + b e^{2} - \frac {2 \, b d e^{2}}{e x + d} - e^{2} {\left | b \right |} \right |}}{{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{e x + d} + b e^{2} - \frac {2 \, b d e^{2}}{e x + d} + e^{2} {\left | b \right |} \right |}}\right )}{2 \, {\left (b^{4} c^{4} d^{8} - 4 \, b^{5} c^{3} d^{7} e + 6 \, b^{6} c^{2} d^{6} e^{2} - 4 \, b^{7} c d^{5} e^{3} + b^{8} d^{4} e^{4}\right )} e^{2} {\left | b \right |}} + \frac {12 \, c^{7} d^{5} e - 30 \, b c^{6} d^{4} e^{2} + 16 \, b^{2} c^{5} d^{3} e^{3} + 6 \, b^{3} c^{4} d^{2} e^{4} - 14 \, b^{4} c^{3} d e^{5} + 5 \, b^{5} c^{2} e^{6} - \frac {2 \, {\left (18 \, c^{7} d^{6} e^{2} - 54 \, b c^{6} d^{5} e^{3} + 47 \, b^{2} c^{5} d^{4} e^{4} - 4 \, b^{3} c^{4} d^{3} e^{5} - 29 \, b^{4} c^{3} d^{2} e^{6} + 22 \, b^{5} c^{2} d e^{7} - 5 \, b^{6} c e^{8}\right )}}{{\left (e x + d\right )} e} + \frac {36 \, c^{7} d^{7} e^{3} - 126 \, b c^{6} d^{6} e^{4} + 144 \, b^{2} c^{5} d^{5} e^{5} - 45 \, b^{3} c^{4} d^{4} e^{6} - 70 \, b^{4} c^{3} d^{3} e^{7} + 87 \, b^{5} c^{2} d^{2} e^{8} - 36 \, b^{6} c d e^{9} + 5 \, b^{7} e^{10}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {6 \, {\left (2 \, c^{7} d^{8} e^{4} - 8 \, b c^{6} d^{7} e^{5} + 11 \, b^{2} c^{5} d^{6} e^{6} - 5 \, b^{3} c^{4} d^{5} e^{7} - 5 \, b^{4} c^{3} d^{4} e^{8} + 9 \, b^{5} c^{2} d^{3} e^{9} - 5 \, b^{6} c d^{2} e^{10} + b^{7} d e^{11}\right )}}{{\left (e x + d\right )}^{3} e^{3}}}{2 \, {\left (c d - b e\right )}^{4} b^{4} {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}\right )}^{2} d^{4}} \]

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-e^11/((c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*c*d^4*e^8 - b^3*d^3*e^9)*(e*x + d)) - 3/2*(2*c*d*e^5 - b*e^6)*lo
g(abs(-c + 2*c*d/(e*x + d) - c*d^2/(e*x + d)^2 - b*e/(e*x + d) + b*d*e/(e*x + d)^2))/(c^4*d^8 - 4*b*c^3*d^7*e
+ 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4) - 3/2*(4*c^6*d^6*e^2 - 12*b*c^5*d^5*e^3 + 10*b^2*c^4*d^4*
e^4 - 2*b^5*c*d*e^7 + b^6*e^8)*log(abs(-2*c*d*e + 2*c*d^2*e/(e*x + d) + b*e^2 - 2*b*d*e^2/(e*x + d) - e^2*abs(
b))/abs(-2*c*d*e + 2*c*d^2*e/(e*x + d) + b*e^2 - 2*b*d*e^2/(e*x + d) + e^2*abs(b)))/((b^4*c^4*d^8 - 4*b^5*c^3*
d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3 + b^8*d^4*e^4)*e^2*abs(b)) + 1/2*(12*c^7*d^5*e - 30*b*c^6*d^4*e^2
+ 16*b^2*c^5*d^3*e^3 + 6*b^3*c^4*d^2*e^4 - 14*b^4*c^3*d*e^5 + 5*b^5*c^2*e^6 - 2*(18*c^7*d^6*e^2 - 54*b*c^6*d^5
*e^3 + 47*b^2*c^5*d^4*e^4 - 4*b^3*c^4*d^3*e^5 - 29*b^4*c^3*d^2*e^6 + 22*b^5*c^2*d*e^7 - 5*b^6*c*e^8)/((e*x + d
)*e) + (36*c^7*d^7*e^3 - 126*b*c^6*d^6*e^4 + 144*b^2*c^5*d^5*e^5 - 45*b^3*c^4*d^4*e^6 - 70*b^4*c^3*d^3*e^7 + 8
7*b^5*c^2*d^2*e^8 - 36*b^6*c*d*e^9 + 5*b^7*e^10)/((e*x + d)^2*e^2) - 6*(2*c^7*d^8*e^4 - 8*b*c^6*d^7*e^5 + 11*b
^2*c^5*d^6*e^6 - 5*b^3*c^4*d^5*e^7 - 5*b^4*c^3*d^4*e^8 + 9*b^5*c^2*d^3*e^9 - 5*b^6*c*d^2*e^10 + b^7*d*e^11)/((
e*x + d)^3*e^3))/((c*d - b*e)^4*b^4*(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)
^2)^2*d^4)

Mupad [B] (verification not implemented)

Time = 10.42 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.62 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx=\frac {\ln \left (x\right )\,\left (3\,b^2\,e^2+6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (3\,b\,e^6-6\,c\,d\,e^5\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\ln \left (b+c\,x\right )\,\left (15\,b^2\,c^4\,e^2-18\,b\,c^5\,d\,e+6\,c^6\,d^2\right )}{b^9\,e^4-4\,b^8\,c\,d\,e^3+6\,b^7\,c^2\,d^2\,e^2-4\,b^6\,c^3\,d^3\,e+b^5\,c^4\,d^4}-\frac {\frac {1}{2\,b\,d}-\frac {x\,\left (3\,b\,e+4\,c\,d\right )}{2\,b^2\,d^2}+\frac {x^2\,\left (-6\,b^5\,e^5+13\,b^3\,c^2\,d^2\,e^3+b^2\,c^3\,d^3\,e^2-32\,b\,c^4\,d^4\,e+18\,c^5\,d^5\right )}{2\,b^3\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {3\,x^3\,\left (-4\,b^5\,c\,e^5+3\,b^4\,c^2\,d\,e^4+5\,b^3\,c^3\,d^2\,e^3-10\,b^2\,c^4\,d^3\,e^2-2\,b\,c^5\,d^4\,e+4\,c^6\,d^5\right )}{2\,b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {3\,c^2\,e\,x^4\,\left (-b^4\,e^4+b^3\,c\,d\,e^3+b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^3\,\left (e\,b^2+2\,c\,d\,b\right )+x^4\,\left (d\,c^2+2\,b\,e\,c\right )+b^2\,d\,x^2+c^2\,e\,x^5} \]

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)^2),x)

[Out]

(log(x)*(3*b^2*e^2 + 6*c^2*d^2 + 6*b*c*d*e))/(b^5*d^4) - (log(d + e*x)*(3*b*e^6 - 6*c*d*e^5))/(c^4*d^8 + b^4*d
^4*e^4 - 4*b^3*c*d^5*e^3 + 6*b^2*c^2*d^6*e^2 - 4*b*c^3*d^7*e) - (log(b + c*x)*(6*c^6*d^2 + 15*b^2*c^4*e^2 - 18
*b*c^5*d*e))/(b^9*e^4 + b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3) - (1/(2*b*d) - (x*(
3*b*e + 4*c*d))/(2*b^2*d^2) + (x^2*(18*c^5*d^5 - 6*b^5*e^5 + b^2*c^3*d^3*e^2 + 13*b^3*c^2*d^2*e^3 - 32*b*c^4*d
^4*e))/(2*b^3*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)) + (3*x^3*(4*c^6*d^5 - 4*b^5*c*e^5 + 3*b
^4*c^2*d*e^4 - 10*b^2*c^4*d^3*e^2 + 5*b^3*c^3*d^2*e^3 - 2*b*c^5*d^4*e))/(2*b^4*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^
2*d^2*e - 3*b^2*c*d*e^2)) + (3*c^2*e*x^4*(2*c^4*d^4 - b^4*e^4 + b^2*c^2*d^2*e^2 - 4*b*c^3*d^3*e + b^3*c*d*e^3)
)/(b^4*d^3*(b^3*e^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2)))/(x^3*(b^2*e + 2*b*c*d) + x^4*(c^2*d + 2*b*c*e
) + b^2*d*x^2 + c^2*e*x^5)