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

Optimal. Leaf size=32 \[ \frac {e \log (b+c x)}{c^2}-\frac {c d-b e}{c^2 (b+c x)} \]

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Rubi [A]  time = 0.03, antiderivative size = 32, 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} \begin {gather*} \frac {e \log (b+c x)}{c^2}-\frac {c d-b e}{c^2 (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.97 \begin {gather*} \frac {b e-c d}{c^2 (b+c x)}+\frac {e \log (b+c x)}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 39, normalized size = 1.22 \begin {gather*} -\frac {c d - b e - {\left (c e x + b e\right )} \log \left (c x + b\right )}{c^{3} x + b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 35, normalized size = 1.09 \begin {gather*} \frac {e \log \left ({\left | c x + b \right |}\right )}{c^{2}} - \frac {c d - b e}{{\left (c x + b\right )} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 39, normalized size = 1.22 \begin {gather*} \frac {b e}{\left (c x +b \right ) c^{2}}-\frac {d}{\left (c x +b \right ) c}+\frac {e \ln \left (c x +b \right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 35, normalized size = 1.09 \begin {gather*} -\frac {c d - b e}{c^{3} x + b c^{2}} + \frac {e \log \left (c x + b\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 31, normalized size = 0.97 \begin {gather*} \frac {b\,e-c\,d}{c^2\,\left (b+c\,x\right )}+\frac {e\,\ln \left (b+c\,x\right )}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.20, size = 27, normalized size = 0.84 \begin {gather*} \frac {b e - c d}{b c^{2} + c^{3} x} + \frac {e \log {\left (b + c x \right )}}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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