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]

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 1.11, size = 92, normalized size = 2.14 \[ \left [\frac {\sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, \sqrt {c x^{2} + a} e}{2 \, c}, -\frac {\sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - \sqrt {c x^{2} + a} e}{c}\right ] \]

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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giac [A]  time = 0.24, size = 40, normalized size = 0.93 \[ -\frac {d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e}{c} \]

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

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maple [A]  time = 0.04, size = 37, normalized size = 0.86 \[ \frac {d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+\frac {\sqrt {c \,x^{2}+a}\, e}{c} \]

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(c^(1/2)*x+(c*x^2+a)^(1/2))/c^(1/2)

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maxima [A]  time = 1.37, size = 29, normalized size = 0.67 \[ \frac {d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} e}{c} \]

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]

d*arcsinh(c*x/sqrt(a*c))/sqrt(c) + sqrt(c*x^2 + a)*e/c

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mupad [B]  time = 0.62, size = 36, normalized size = 0.84 \[ \frac {e\,\sqrt {c\,x^2+a}}{c}+\frac {d\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2+a}\right )}{\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2)^(1/2),x)

[Out]

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

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sympy [B]  time = 1.43, size = 102, normalized size = 2.37 \[ d \left (\begin {cases} \frac {\sqrt {- \frac {a}{c}} \operatorname {asin}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c < 0 \\\frac {\sqrt {\frac {a}{c}} \operatorname {asinh}{\left (x \sqrt {\frac {c}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge c > 0 \\\frac {\sqrt {- \frac {a}{c}} \operatorname {acosh}{\left (x \sqrt {- \frac {c}{a}} \right )}}{\sqrt {- a}} & \text {for}\: c > 0 \wedge a < 0 \end {cases}\right ) + e \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: c = 0 \\\frac {\sqrt {a + c x^{2}}}{c} & \text {otherwise} \end {cases}\right ) \]

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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