3.1930 \(\int \frac{(a+b x) (d+e x)}{a^2+2 a b x+b^2 x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0205974, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)/b + ((b*d - a*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 43

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

Mathematica [A]  time = 0.0071223, size = 25, normalized size = 1. \[ \frac{(b d-a e) \log (a+b x)}{b^2}+\frac{e x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 1.3 \begin{align*}{\frac{ex}{b}}-{\frac{\ln \left ( bx+a \right ) ae}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) d}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.957427, size = 34, normalized size = 1.36 \begin{align*} \frac{e x}{b} + \frac{{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/b + (b*d - a*e)*log(b*x + a)/b^2

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Fricas [A]  time = 1.46514, size = 54, normalized size = 2.16 \begin{align*} \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.308245, size = 20, normalized size = 0.8 \begin{align*} \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x/b - (a*e - b*d)*log(a + b*x)/b**2

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Giac [A]  time = 1.09236, size = 38, normalized size = 1.52 \begin{align*} \frac{x e}{b} + \frac{{\left (b d - a e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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