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

Optimal. Leaf size=57 \[ -\frac {d \log (b+c x)}{b^3}+\frac {d \log (x)}{b^3}+\frac {d}{b^2 (b+c x)}+\frac {c d-b e}{2 b c (b+c x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ \frac {d}{b^2 (b+c x)}-\frac {d \log (b+c x)}{b^3}+\frac {d \log (x)}{b^3}+\frac {c d-b e}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 53, normalized size = 0.93 \[ \frac {\frac {b \left (b^2 (-e)+3 b c d+2 c^2 d x\right )}{c (b+c x)^2}-2 d \log (b+c x)+2 d \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 109, normalized size = 1.91 \[ \frac {2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e - 2 \, {\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (c x + b\right ) + 2 \, {\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \relax (x)}{2 \, {\left (b^{3} c^{3} x^{2} + 2 \, b^{4} c^{2} x + b^{5} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 60, normalized size = 1.05 \[ -\frac {d \log \left ({\left | c x + b \right |}\right )}{b^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e}{2 \, {\left (c x + b\right )}^{2} b^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 59, normalized size = 1.04 \[ \frac {d}{2 \left (c x +b \right )^{2} b}-\frac {e}{2 \left (c x +b \right )^{2} c}+\frac {d}{\left (c x +b \right ) b^{2}}+\frac {d \ln \relax (x )}{b^{3}}-\frac {d \ln \left (c x +b \right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.92, size = 68, normalized size = 1.19 \[ \frac {2 \, c^{2} d x + 3 \, b c d - b^{2} e}{2 \, {\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )}} - \frac {d \log \left (c x + b\right )}{b^{3}} + \frac {d \log \relax (x)}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 62, normalized size = 1.09 \[ -\frac {\frac {b\,e-3\,c\,d}{2\,b\,c}-\frac {c\,d\,x}{b^2}}{b^2+2\,b\,c\,x+c^2\,x^2}-\frac {2\,d\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 63, normalized size = 1.11 \[ \frac {- b^{2} e + 3 b c d + 2 c^{2} d x}{2 b^{4} c + 4 b^{3} c^{2} x + 2 b^{2} c^{3} x^{2}} + \frac {d \left (\log {\relax (x )} - \log {\left (\frac {b}{c} + x \right )}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b**2*e + 3*b*c*d + 2*c**2*d*x)/(2*b**4*c + 4*b**3*c**2*x + 2*b**2*c**3*x**2) + d*(log(x) - log(b/c + x))/b**
3

________________________________________________________________________________________