### 3.564 $$\int \frac{d+e x}{\sqrt{a+c x^2}} \, dx$$

Optimal. Leaf size=43 $\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}$

[Out]

(e*Sqrt[a + c*x^2])/c + (d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c]

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Rubi [A]  time = 0.0129593, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {641, 217, 206} $\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/Sqrt[a + c*x^2],x]

[Out]

(e*Sqrt[a + c*x^2])/c + (d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/Sqrt[c]

Rule 641

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0212389, size = 46, normalized size = 1.07 $\frac{d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 37, normalized size = 0.9 \begin{align*}{\frac{e}{c}\sqrt{c{x}^{2}+a}}+{d\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8446, size = 223, normalized size = 5.19 \begin{align*} \left [\frac{\sqrt{c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \, \sqrt{c x^{2} + a} e}{2 \, c}, -\frac{\sqrt{-c} d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - \sqrt{c x^{2} + a} e}{c}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(c)*d*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*sqrt(c*x^2 + a)*e)/c, -(sqrt(-c)*d*arctan(
sqrt(-c)*x/sqrt(c*x^2 + a)) - sqrt(c*x^2 + a)*e)/c]

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Sympy [B]  time = 1.12576, size = 102, normalized size = 2.37 \begin{align*} d \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

d*Piecewise((sqrt(-a/c)*asin(x*sqrt(-c/a))/sqrt(a), (a > 0) & (c < 0)), (sqrt(a/c)*asinh(x*sqrt(c/a))/sqrt(a),
(a > 0) & (c > 0)), (sqrt(-a/c)*acosh(x*sqrt(-c/a))/sqrt(-a), (c > 0) & (a < 0))) + e*Piecewise((x**2/(2*sqrt
(a)), Eq(c, 0)), (sqrt(a + c*x**2)/c, True))

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Giac [A]  time = 1.52074, size = 54, normalized size = 1.26 \begin{align*} -\frac{d \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2} + a} e}{c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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