3.9.77 \(\int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c e} \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {629} \begin {gather*} \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 629

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

Rubi steps

\begin {align*} \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c e}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.65 \begin {gather*} \frac {x (d+e x)}{\sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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IntegrateAlgebraic [A]  time = 0.03, size = 20, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c (d+e x)^2}}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[c*(d + e*x)^2]/(c*e)

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fricas [A]  time = 0.39, size = 34, normalized size = 1.10 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c e x + c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.38, size = 28, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} e^{\left (-1\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*e^(-1)/c

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maple [A]  time = 0.05, size = 30, normalized size = 0.97 \begin {gather*} \frac {\left (e x +d \right ) x}{\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 29, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.73, size = 109, normalized size = 3.52 \begin {gather*} \frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c\,e}-\frac {c\,d\,e^2\,\ln \left (\sqrt {c\,{\left (d+e\,x\right )}^2}\,\sqrt {c\,e^2}+c\,d\,e+c\,e^2\,x\right )}{{\left (c\,e^2\right )}^{3/2}}+\frac {c\,d\,e^2\,\ln \left (c\,x\,e^2+c\,d\,e\right )\,\mathrm {sign}\left (c\,e\,\left (d+e\,x\right )\right )}{{\left (c\,e^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)/(c*e) - (c*d*e^2*log((c*(d + e*x)^2)^(1/2)*(c*e^2)^(1/2) + c*d*e + c*e^2
*x))/(c*e^2)^(3/2) + (c*d*e^2*log(c*d*e + c*e^2*x)*sign(c*e*(d + e*x)))/(c*e^2)^(3/2)

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sympy [A]  time = 0.89, size = 39, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} & \text {for}\: e \neq 0 \\\frac {d x}{\sqrt {c d^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(c*e), Ne(e, 0)), (d*x/sqrt(c*d**2), True))

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