### 3.1590 $$\int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx$$

Optimal. Leaf size=69 $\frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}$

[Out]

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

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Rubi [A]  time = 0.0233859, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {640, 608, 31} $\frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{\left (2 b^2 d-2 a b e\right ) \int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{\left (\left (2 b^2 d-2 a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{a b+b^2 x} \, dx}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2}+\frac{(b d-a e) (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0172236, size = 40, normalized size = 0.58 $\frac{(a+b x) ((b d-a e) \log (a+b x)+b e x)}{b^2 \sqrt{(a+b x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.164, size = 45, normalized size = 0.7 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) ae-\ln \left ( bx+a \right ) bd-bxe \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/((b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.01973, size = 80, normalized size = 1.16 \begin{align*} \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{a \sqrt{\frac{1}{b^{2}}} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e}{b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48752, size = 54, normalized size = 0.78 \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((e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.369018, size = 20, normalized size = 0.29 \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((e*x+d)/((b*x+a)**2)**(1/2),x)

[Out]

e*x/b - (a*e - b*d)*log(a + b*x)/b**2

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Giac [A]  time = 1.13261, size = 62, normalized size = 0.9 \begin{align*} \frac{x e \mathrm{sgn}\left (b x + a\right )}{b} + \frac{{\left (b d \mathrm{sgn}\left (b x + a\right ) - a e \mathrm{sgn}\left (b x + a\right )\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((e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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