3.9.86 \(\int \frac {d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 32, 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 {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

-(1/(c*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]

Rubi steps

\begin {align*} \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}{c 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 = 21, normalized size = 0.66 \begin {gather*} -\frac {1}{c e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

-(1/(c*e*Sqrt[c*(d + e*x)^2]))

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IntegrateAlgebraic [A]  time = 0.03, size = 28, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {c (d+e x)^2}}{c^2 e (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.39, size = 55, normalized size = 1.72 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

-sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^2*e^3*x^2 + 2*c^2*d*e^2*x + c^2*d^2*e)

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giac [A]  time = 0.40, size = 41, normalized size = 1.28 \begin {gather*} \frac {2 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac {e^{\left (-1\right )}}{c}}{\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \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)^(3/2),x, algorithm="giac")

[Out]

(2*C_0*d*e^(-1) + 2*C_0*x - e^(-1)/c)/sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)

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maple [A]  time = 0.04, size = 35, normalized size = 1.09 \begin {gather*} -\frac {\left (e x +d \right )^{2}}{\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \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)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.33, size = 30, normalized size = 0.94 \begin {gather*} -\frac {1}{\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)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.46, size = 37, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^2\,e\,{\left (d+e\,x\right )}^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)^(3/2),x)

[Out]

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

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sympy [A]  time = 1.23, size = 42, normalized size = 1.31 \begin {gather*} \begin {cases} - \frac {1}{c e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {3}{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)**(3/2),x)

[Out]

Piecewise((-1/(c*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d**2)**(3/2), True))

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