3.9.79 \(\int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac {1}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {643, 629} \begin {gather*} -\frac {1}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]

[Out]

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

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]

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.62 \begin {gather*} -\frac {1}{e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]

[Out]

-(1/(e*Sqrt[c*(d + e*x)^2]))

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IntegrateAlgebraic [A]  time = 0.04, size = 28, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c (d+e x)^2}}{c e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((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*(d + e*x)^2))

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fricas [A]  time = 0.40, size = 49, normalized size = 1.69 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)/(c*e^3*x^2 + 2*c*d*e^2*x + c*d^2*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*s
qrt(c*exp(2))+(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)*exp(1))/d/sqrt(c*exp(1)^2-c*exp(2)))

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maple [A]  time = 0.05, size = 28, normalized size = 0.97 \begin {gather*} -\frac {1}{\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, size = 19, normalized size = 0.66 \begin {gather*} -\frac {1}{\sqrt {c} e^{2} x + \sqrt {c} d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(c)*e^2*x + sqrt(c)*d*e)

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mupad [B]  time = 0.43, size = 27, normalized size = 0.93 \begin {gather*} -\frac {1}{e\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.83, size = 41, normalized size = 1.41 \begin {gather*} \begin {cases} - \frac {1}{e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x}{d \sqrt {c d^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (x/(d*sqrt(c*d**2)), True))

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