\(\int \frac {d+e x}{(b x+c x^2)^3} \, dx\) [281]

Optimal result

Integrand size = 17, antiderivative size = 110 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=-\frac {d}{2 b^3 x^2}+\frac {3 c d-b e}{b^4 x}+\frac {c (c d-b e)}{2 b^3 (b+c x)^2}+\frac {c (3 c d-2 b e)}{b^4 (b+c x)}+\frac {3 c (2 c d-b e) \log (x)}{b^5}-\frac {3 c (2 c d-b e) \log (b+c x)}{b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {645} \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=\frac {3 c \log (x) (2 c d-b e)}{b^5}-\frac {3 c (2 c d-b e) \log (b+c x)}{b^5}+\frac {3 c d-b e}{b^4 x}+\frac {c (3 c d-2 b e)}{b^4 (b+c x)}+\frac {c (c d-b e)}{2 b^3 (b+c x)^2}-\frac {d}{2 b^3 x^2} \]

[In]

Int[(d + e*x)/(b*x + c*x^2)^3,x]

[Out]

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

Rule 645

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {b \left (-12 c^3 d x^3+6 b c^2 x^2 (-3 d+e x)+b^3 (d+2 e x)+b^2 c x (-4 d+9 e x)\right )}{x^2 (b+c x)^2}+6 c (2 c d-b e) \log (x)+6 c (-2 c d+b e) \log (b+c x)}{2 b^5} \]

[In]

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

[Out]

(-((b*(-12*c^3*d*x^3 + 6*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 2*e*x) + b^2*c*x*(-4*d + 9*e*x)))/(x^2*(b + c*x)^2)
) + 6*c*(2*c*d - b*e)*Log[x] + 6*c*(-2*c*d + b*e)*Log[b + c*x])/(2*b^5)

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {d}{2 b^{3} x^{2}}-\frac {b e -3 c d}{b^{4} x}-\frac {3 c \left (b e -2 c d \right ) \ln \left (x \right )}{b^{5}}-\frac {c \left (2 b e -3 c d \right )}{b^{4} \left (c x +b \right )}-\frac {\left (b e -c d \right ) c}{2 b^{3} \left (c x +b \right )^{2}}+\frac {3 c \left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{5}}\)	\(106\)
norman	\(\frac {-\frac {d}{2 b}-\frac {\left (b e -2 c d \right ) x}{b^{2}}+\frac {2 c \left (3 b c e -6 c^{2} d \right ) x^{3}}{b^{4}}+\frac {c^{2} \left (9 b c e -18 c^{2} d \right ) x^{4}}{2 b^{5}}}{x^{2} \left (c x +b \right )^{2}}-\frac {3 c \left (b e -2 c d \right ) \ln \left (x \right )}{b^{5}}+\frac {3 c \left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{5}}\)	\(114\)
risch	\(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{3}}{b^{4}}-\frac {9 c \left (b e -2 c d \right ) x^{2}}{2 b^{3}}-\frac {\left (b e -2 c d \right ) x}{b^{2}}-\frac {d}{2 b}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 c \ln \left (-c x -b \right ) e}{b^{4}}-\frac {6 c^{2} \ln \left (-c x -b \right ) d}{b^{5}}-\frac {3 c \ln \left (x \right ) e}{b^{4}}+\frac {6 c^{2} \ln \left (x \right ) d}{b^{5}}\)	\(124\)
parallelrisch	\(-\frac {6 \ln \left (x \right ) x^{4} b \,c^{3} e -12 \ln \left (x \right ) x^{4} c^{4} d -6 \ln \left (c x +b \right ) x^{4} b \,c^{3} e +12 \ln \left (c x +b \right ) x^{4} c^{4} d +12 \ln \left (x \right ) x^{3} b^{2} c^{2} e -24 \ln \left (x \right ) x^{3} b \,c^{3} d -12 \ln \left (c x +b \right ) x^{3} b^{2} c^{2} e +24 \ln \left (c x +b \right ) x^{3} b \,c^{3} d -9 x^{4} b \,c^{3} e +18 x^{4} c^{4} d +6 \ln \left (x \right ) x^{2} b^{3} c e -12 \ln \left (x \right ) x^{2} b^{2} c^{2} d -6 \ln \left (c x +b \right ) x^{2} b^{3} c e +12 \ln \left (c x +b \right ) x^{2} b^{2} c^{2} d -12 x^{3} b^{2} c^{2} e +24 b \,c^{3} d \,x^{3}+2 x \,b^{4} e -4 b^{3} c d x +d \,b^{4}}{2 b^{5} x^{2} \left (c x +b \right )^{2}}\)	\(252\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (106) = 212\).

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.13 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} d - 6 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} - 9 \, {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2} - 2 \, {\left (2 \, b^{3} c d - b^{4} e\right )} x + 6 \, {\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} + {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \, {\left ({\left (2 \, c^{4} d - b c^{3} e\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d - b^{2} c^{2} e\right )} x^{3} + {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \]

[In]

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

[Out]

-1/2*(b^4*d - 6*(2*b*c^3*d - b^2*c^2*e)*x^3 - 9*(2*b^2*c^2*d - b^3*c*e)*x^2 - 2*(2*b^3*c*d - b^4*e)*x + 6*((2*
c^4*d - b*c^3*e)*x^4 + 2*(2*b*c^3*d - b^2*c^2*e)*x^3 + (2*b^2*c^2*d - b^3*c*e)*x^2)*log(c*x + b) - 6*((2*c^4*d
- b*c^3*e)*x^4 + 2*(2*b*c^3*d - b^2*c^2*e)*x^3 + (2*b^2*c^2*d - b^3*c*e)*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. 219 vs. \(2 (104) = 208\).

Time = 0.39 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.99 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} d + x^{3} \left (- 6 b c^{2} e + 12 c^{3} d\right ) + x^{2} \left (- 9 b^{2} c e + 18 b c^{2} d\right ) + x \left (- 2 b^{3} e + 4 b^{2} c d\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac {3 c \left (b e - 2 c d\right ) \log {\left (x + \frac {3 b^{2} c e - 6 b c^{2} d - 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} + \frac {3 c \left (b e - 2 c d\right ) \log {\left (x + \frac {3 b^{2} c e - 6 b c^{2} d + 3 b c \left (b e - 2 c d\right )}{6 b c^{2} e - 12 c^{3} d} \right )}}{b^{5}} \]

[In]

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

[Out]

(-b**3*d + x**3*(-6*b*c**2*e + 12*c**3*d) + x**2*(-9*b**2*c*e + 18*b*c**2*d) + x*(-2*b**3*e + 4*b**2*c*d))/(2*
b**6*x**2 + 4*b**5*c*x**3 + 2*b**4*c**2*x**4) - 3*c*(b*e - 2*c*d)*log(x + (3*b**2*c*e - 6*b*c**2*d - 3*b*c*(b*
e - 2*c*d))/(6*b*c**2*e - 12*c**3*d))/b**5 + 3*c*(b*e - 2*c*d)*log(x + (3*b**2*c*e - 6*b*c**2*d + 3*b*c*(b*e -
2*c*d))/(6*b*c**2*e - 12*c**3*d))/b**5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} d - 6 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 9 \, {\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 2 \, {\left (2 \, b^{2} c d - b^{3} e\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} - \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left (x\right )}{b^{5}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.15 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=\frac {3 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{3} d x^{3} - 6 \, b c^{2} e x^{3} + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c e x^{2} + 4 \, b^{2} c d x - 2 \, b^{3} e x - b^{3} d}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \]

[In]

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

[Out]

3*(2*c^2*d - b*c*e)*log(abs(x))/b^5 - 3*(2*c^3*d - b*c^2*e)*log(abs(c*x + b))/(b^5*c) + 1/2*(12*c^3*d*x^3 - 6*
b*c^2*e*x^3 + 18*b*c^2*d*x^2 - 9*b^2*c*e*x^2 + 4*b^2*c*d*x - 2*b^3*e*x - b^3*d)/((c*x^2 + b*x)^2*b^4)

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.20 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {d}{2\,b}+\frac {x\,\left (b\,e-2\,c\,d\right )}{b^2}+\frac {9\,c\,x^2\,\left (b\,e-2\,c\,d\right )}{2\,b^3}+\frac {3\,c^2\,x^3\,\left (b\,e-2\,c\,d\right )}{b^4}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (b\,e-2\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (6\,c^2\,d-3\,b\,c\,e\right )}\right )\,\left (b\,e-2\,c\,d\right )}{b^5} \]

[In]

int((d + e*x)/(b*x + c*x^2)^3,x)

[Out]

- (d/(2*b) + (x*(b*e - 2*c*d))/b^2 + (9*c*x^2*(b*e - 2*c*d))/(2*b^3) + (3*c^2*x^3*(b*e - 2*c*d))/b^4)/(b^2*x^2
+ c^2*x^4 + 2*b*c*x^3) - (6*c*atanh((3*c*(b*e - 2*c*d)*(b + 2*c*x))/(b*(6*c^2*d - 3*b*c*e)))*(b*e - 2*c*d))/b
^5