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

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

[Out]

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

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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {(c d-b e)^2 \log (b+c x)}{b c^2}+\frac {d^2 \log (x)}{b}+\frac {e^2 x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(b*x + c*x^2),x]

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 1.00 \[ \frac {-(c d-b e)^2 \log (b+c x)+b c e^2 x+c^2 d^2 \log (x)}{b c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.09, size = 53, normalized size = 1.26 \[ \frac {b c e^{2} x + c^{2} d^{2} \log \relax (x) - {\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 54, normalized size = 1.29 \[ \frac {d^{2} \log \left ({\left | x \right |}\right )}{b} + \frac {x e^{2}}{c} - \frac {{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 61, normalized size = 1.45 \[ -\frac {b \,e^{2} \ln \left (c x +b \right )}{c^{2}}+\frac {d^{2} \ln \relax (x )}{b}-\frac {d^{2} \ln \left (c x +b \right )}{b}+\frac {2 d e \ln \left (c x +b \right )}{c}+\frac {e^{2} x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 53, normalized size = 1.26 \[ \frac {e^{2} x}{c} + \frac {d^{2} \log \relax (x)}{b} - \frac {{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \log \left (c x + b\right )}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 49, normalized size = 1.17 \[ \frac {e^2\,x}{c}-\ln \left (b+c\,x\right )\,\left (\frac {d^2}{b}+\frac {b\,e^2}{c^2}-\frac {2\,d\,e}{c}\right )+\frac {d^2\,\ln \relax (x)}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2/(b*x + c*x^2),x)

[Out]

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

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sympy [B]  time = 0.87, size = 73, normalized size = 1.74 \[ \frac {e^{2} x}{c} + \frac {d^{2} \log {\relax (x )}}{b} - \frac {\left (b e - c d\right )^{2} \log {\left (x + \frac {b c d^{2} + \frac {b \left (b e - c d\right )^{2}}{c}}{b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}} \right )}}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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