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

Optimal. Leaf size=25 \[ \frac {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, 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 {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(b*x + c*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {(c d-b e) \log (b+c x)}{c^2}+\frac {e x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)/c + ((c*d - b*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 (d+e x)}{b x+c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 24, normalized size = 0.96 \begin {gather*} \frac {c e x + {\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 28, normalized size = 1.12 \begin {gather*} \frac {x e}{c} + \frac {{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 32, normalized size = 1.28 \begin {gather*} -\frac {b e \ln \left (c x +b \right )}{c^{2}}+\frac {d \ln \left (c x +b \right )}{c}+\frac {e x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.94, size = 25, normalized size = 1.00 \begin {gather*} \frac {e x}{c} + \frac {{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/c + (c*d - b*e)*log(c*x + b)/c^2

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mupad [B]  time = 0.05, size = 26, normalized size = 1.04 \begin {gather*} \frac {e\,x}{c}-\frac {\ln \left (b+c\,x\right )\,\left (b\,e-c\,d\right )}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(d + e*x))/(b*x + c*x^2),x)

[Out]

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

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sympy [A]  time = 0.16, size = 20, normalized size = 0.80 \begin {gather*} \frac {e x}{c} - \frac {\left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/c - (b*e - c*d)*log(b + c*x)/c**2

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