\(\int \frac {(d+e x)^2}{(b x+c x^2)^3} \, dx\) [280]

Optimal result

Integrand size = 19, antiderivative size = 144 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^2}{2 b^3 x^2}+\frac {d (3 c d-2 b e)}{b^4 x}+\frac {(c d-b e)^2}{2 b^3 (b+c x)^2}+\frac {(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (x)}{b^5}-\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, 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} \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=\frac {d (3 c d-2 b e)}{b^4 x}+\frac {(c d-b e) (3 c d-b e)}{b^4 (b+c x)}+\frac {(c d-b e)^2}{2 b^3 (b+c x)^2}-\frac {d^2}{2 b^3 x^2}+\frac {\log (x) \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{b^5}-\frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5} \]

[In]

Int[(d + e*x)^2/(b*x + c*x^2)^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {b^2 d^2}{x^2}-\frac {2 b d (-3 c d+2 b e)}{x}+\frac {b^2 (c d-b e)^2}{(b+c x)^2}+\frac {2 b \left (3 c^2 d^2-4 b c d e+b^2 e^2\right )}{b+c x}+2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (x)-2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \log (b+c x)}{2 b^5} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {d^{2}}{2 b^{3} x^{2}}+\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {d \left (2 b e -3 c d \right )}{b^{4} x}-\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}+\frac {b^{2} e^{2}-4 b c d e +3 c^{2} d^{2}}{b^{4} \left (c x +b \right )}+\frac {b^{2} e^{2}-2 b c d e +c^{2} d^{2}}{2 b^{3} \left (c x +b \right )^{2}}\)	\(158\)
norman	\(\frac {\frac {\left (b^{2} c^{2} e^{2}-6 b \,c^{3} d e +6 c^{4} d^{2}\right ) x^{3}}{b^{4} c}-\frac {d^{2}}{2 b}-\frac {2 d \left (b e -c d \right ) x}{b^{2}}+\frac {\left (3 b^{2} c^{2} e^{2}-18 b \,c^{3} d e +18 c^{4} d^{2}\right ) x^{2}}{2 c^{2} b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {\left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\)	\(174\)
risch	\(\frac {\frac {c \left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) x^{3}}{b^{4}}+\frac {3 \left (b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}\right ) x^{2}}{2 b^{3}}-\frac {2 d \left (b e -c d \right ) x}{b^{2}}-\frac {d^{2}}{2 b}}{x^{2} \left (c x +b \right )^{2}}+\frac {\ln \left (-x \right ) e^{2}}{b^{3}}-\frac {6 \ln \left (-x \right ) c d e}{b^{4}}+\frac {6 \ln \left (-x \right ) c^{2} d^{2}}{b^{5}}-\frac {\ln \left (c x +b \right ) e^{2}}{b^{3}}+\frac {6 \ln \left (c x +b \right ) c d e}{b^{4}}-\frac {6 \ln \left (c x +b \right ) c^{2} d^{2}}{b^{5}}\)	\(180\)
parallelrisch	\(\frac {4 b^{3} c^{3} d^{2} x -b^{4} c^{2} d^{2}-24 \ln \left (x \right ) x^{3} b^{2} c^{4} d e +24 \ln \left (c x +b \right ) x^{3} b^{2} c^{4} d e -12 \ln \left (x \right ) x^{2} b^{3} c^{3} d e +12 \ln \left (c x +b \right ) x^{2} b^{3} c^{3} d e -12 \ln \left (x \right ) x^{4} b \,c^{5} d e +12 \ln \left (c x +b \right ) x^{4} b \,c^{5} d e +2 x^{3} b^{3} c^{3} e^{2}+12 x^{3} b \,c^{5} d^{2}+3 x^{2} b^{4} c^{2} e^{2}+18 x^{2} b^{2} c^{4} d^{2}+12 \ln \left (x \right ) x^{4} c^{6} d^{2}-12 \ln \left (c x +b \right ) x^{4} c^{6} d^{2}-4 \ln \left (c x +b \right ) x^{3} b^{3} c^{3} e^{2}-24 \ln \left (c x +b \right ) x^{3} b \,c^{5} d^{2}-12 x^{3} b^{2} c^{4} d e -18 x^{2} b^{3} c^{3} d e -4 x \,b^{4} c^{2} d e +2 \ln \left (x \right ) x^{2} b^{4} c^{2} e^{2}+12 \ln \left (x \right ) x^{2} b^{2} c^{4} d^{2}-2 \ln \left (c x +b \right ) x^{2} b^{4} c^{2} e^{2}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{4} d^{2}+2 \ln \left (x \right ) x^{4} b^{2} c^{4} e^{2}-2 \ln \left (c x +b \right ) x^{4} b^{2} c^{4} e^{2}+4 \ln \left (x \right ) x^{3} b^{3} c^{3} e^{2}+24 \ln \left (x \right ) x^{3} b \,c^{5} d^{2}}{2 c^{2} b^{5} x^{2} \left (c x +b \right )^{2}}\)	\(438\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (140) = 280\).

Time = 0.27 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.27 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} d^{2} - 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} - 3 \, {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2} - 4 \, {\left (b^{3} c d^{2} - b^{4} d e\right )} x + 2 \, {\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (6 \, c^{4} d^{2} - 6 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{3} d^{2} - 6 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{2} d^{2} - 6 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (136) = 272\).

Time = 0.56 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.40 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} d^{2} + x^{3} \cdot \left (2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}\right ) + x^{2} \cdot \left (3 b^{3} e^{2} - 18 b^{2} c d e + 18 b c^{2} d^{2}\right ) + x \left (- 4 b^{3} d e + 4 b^{2} c d^{2}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac {\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} - b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} - \frac {\left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {b^{3} e^{2} - 6 b^{2} c d e + 6 b c^{2} d^{2} + b \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{2 b^{2} c e^{2} - 12 b c^{2} d e + 12 c^{3} d^{2}} \right )}}{b^{5}} \]

[In]

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

[Out]

(-b**3*d**2 + x**3*(2*b**2*c*e**2 - 12*b*c**2*d*e + 12*c**3*d**2) + x**2*(3*b**3*e**2 - 18*b**2*c*d*e + 18*b*c
**2*d**2) + x*(-4*b**3*d*e + 4*b**2*c*d**2))/(2*b**6*x**2 + 4*b**5*c*x**3 + 2*b**4*c**2*x**4) + (b**2*e**2 - 6
*b*c*d*e + 6*c**2*d**2)*log(x + (b**3*e**2 - 6*b**2*c*d*e + 6*b*c**2*d**2 - b*(b**2*e**2 - 6*b*c*d*e + 6*c**2*
d**2))/(2*b**2*c*e**2 - 12*b*c**2*d*e + 12*c**3*d**2))/b**5 - (b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*log(x + (b
**3*e**2 - 6*b**2*c*d*e + 6*b*c**2*d**2 + b*(b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2))/(2*b**2*c*e**2 - 12*b*c**2*
d*e + 12*c**3*d**2))/b**5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} d^{2} - 2 \, {\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \, {\left (6 \, b c^{2} d^{2} - 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 4 \, {\left (b^{2} c d^{2} - b^{3} d e\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=\frac {{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {{\left (6 \, c^{3} d^{2} - 6 \, b c^{2} d e + b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d e x^{3} + 2 \, b^{2} c e^{2} x^{3} + 18 \, b c^{2} d^{2} x^{2} - 18 \, b^{2} c d e x^{2} + 3 \, b^{3} e^{2} x^{2} + 4 \, b^{2} c d^{2} x - 4 \, b^{3} d e x - b^{3} d^{2}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \]

[In]

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

[Out]

(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*log(abs(x))/b^5 - (6*c^3*d^2 - 6*b*c^2*d*e + b^2*c*e^2)*log(abs(c*x + b))/(b
^5*c) + 1/2*(12*c^3*d^2*x^3 - 12*b*c^2*d*e*x^3 + 2*b^2*c*e^2*x^3 + 18*b*c^2*d^2*x^2 - 18*b^2*c*d*e*x^2 + 3*b^3
*e^2*x^2 + 4*b^2*c*d^2*x - 4*b^3*d*e*x - b^3*d^2)/((c*x^2 + b*x)^2*b^4)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {d^2}{2\,b}-\frac {3\,x^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{2\,b^3}-\frac {c\,x^3\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^4}+\frac {2\,d\,x\,\left (b\,e-c\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5} \]

[In]

int((d + e*x)^2/(b*x + c*x^2)^3,x)

[Out]

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