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

Optimal. Leaf size=42 \[ -\frac {d \log (b+c x)}{b^2}+\frac {d \log (x)}{b^2}+\frac {c d-b e}{b c (b+c x)} \]

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Rubi [A]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} -\frac {d \log (b+c x)}{b^2}+\frac {d \log (x)}{b^2}+\frac {c d-b e}{b c (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.90 \begin {gather*} \frac {\frac {b (c d-b e)}{c (b+c x)}-d \log (b+c x)+d \log (x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (d+e x)}{\left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 61, normalized size = 1.45 \begin {gather*} \frac {b c d - b^{2} e - {\left (c^{2} d x + b c d\right )} \log \left (c x + b\right ) + {\left (c^{2} d x + b c d\right )} \log \relax (x)}{b^{2} c^{2} x + b^{3} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 48, normalized size = 1.14 \begin {gather*} -\frac {d \log \left ({\left | c x + b \right |}\right )}{b^{2}} + \frac {d \log \left ({\left | x \right |}\right )}{b^{2}} + \frac {b c d - b^{2} e}{{\left (c x + b\right )} b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 46, normalized size = 1.10 \begin {gather*} \frac {d}{\left (c x +b \right ) b}+\frac {d \ln \relax (x )}{b^{2}}-\frac {d \ln \left (c x +b \right )}{b^{2}}-\frac {e}{\left (c x +b \right ) c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 43, normalized size = 1.02 \begin {gather*} \frac {c d - b e}{b c^{2} x + b^{2} c} - \frac {d \log \left (c x + b\right )}{b^{2}} + \frac {d \log \relax (x)}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.02, size = 40, normalized size = 0.95 \begin {gather*} -\frac {2\,d\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b^2}-\frac {b\,e-c\,d}{b\,c\,\left (b+c\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(d + e*x))/(b*x + c*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.29, size = 32, normalized size = 0.76 \begin {gather*} \frac {- b e + c d}{b^{2} c + b c^{2} x} + \frac {d \left (\log {\relax (x )} - \log {\left (\frac {b}{c} + x \right )}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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