3.16.9 \(\int \frac {d+e x}{a^2+2 a b x+b^2 x^2} \, dx\) [1509]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \begin {gather*} \frac {e \log (a+b x)}{b^2}-\frac {b d-a e}{b^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.59, size = 33, normalized size = 1.03

method	result	size
norman	\(\frac {a e -b d}{b^{2} \left (b x +a \right )}+\frac {e \ln \left (b x +a \right )}{b^{2}}\)	\(32\)
default	\(-\frac {-a e +b d}{b^{2} \left (b x +a \right )}+\frac {e \ln \left (b x +a \right )}{b^{2}}\)	\(33\)
risch	\(\frac {a e}{b^{2} \left (b x +a \right )}-\frac {d}{b \left (b x +a \right )}+\frac {e \ln \left (b x +a \right )}{b^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 37, normalized size = 1.16 \begin {gather*} -\frac {b d - a e}{b^{3} x + a b^{2}} + \frac {e \log \left (b x + a\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.40, size = 37, normalized size = 1.16 \begin {gather*} \frac {{\left (b x + a\right )} e \log \left (b x + a\right ) - b d + a e}{b^{3} x + a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.08, size = 27, normalized size = 0.84 \begin {gather*} \frac {a e - b d}{a b^{2} + b^{3} x} + \frac {e \log {\left (a + b x \right )}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.16, size = 35, normalized size = 1.09 \begin {gather*} \frac {e \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac {b d - a e}{{\left (b x + a\right )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.50, size = 31, normalized size = 0.97 \begin {gather*} \frac {a\,e-b\,d}{b^2\,\left (a+b\,x\right )}+\frac {e\,\ln \left (a+b\,x\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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