3.1084 \(\int \frac {d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, 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} \[ -\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

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

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 )^{5/2}} \, dx &=-\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.88 \[ -\frac {\sqrt {c (d+e x)^2}}{3 c^3 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

-1/3*Sqrt[c*(d + e*x)^2]/(c^3*e*(d + e*x)^4)

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fricas [B]  time = 0.98, size = 83, normalized size = 2.44 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (c^{3} e^{5} x^{4} + 4 \, c^{3} d e^{4} x^{3} + 6 \, c^{3} d^{2} e^{3} x^{2} + 4 \, c^{3} d^{3} e^{2} x + c^{3} d^{4} e\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^5*x^4 + 4*c^3*d*e^4*x^3 + 6*c^3*d^2*e^3*x^2 + 4*c^3*d^3*e^2*x
+ c^3*d^4*e)

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giac [B]  time = 0.44, size = 64, normalized size = 1.88 \[ \frac {6 \, C_{0} d^{3} e^{\left (-3\right )} + 6 \, {\left (3 \, C_{0} d^{2} e^{\left (-2\right )} + {\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x\right )} x - \frac {e^{\left (-1\right )}}{c}}{3 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

1/3*(6*C_0*d^3*e^(-3) + 6*(3*C_0*d^2*e^(-2) + (3*C_0*d*e^(-1) + C_0*x)*x)*x - e^(-1)/c)/(c*x^2*e^2 + 2*c*d*x*e
+ c*d^2)^(3/2)

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maple [A]  time = 0.04, size = 35, normalized size = 1.03 \[ -\frac {\left (e x +d \right )^{2}}{3 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \]

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)^(5/2),x)

[Out]

-1/3*(e*x+d)^2/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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maxima [A]  time = 1.33, size = 30, normalized size = 0.88 \[ -\frac {1}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} \]

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)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.49, size = 37, normalized size = 1.09 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,c^3\,e\,{\left (d+e\,x\right )}^4} \]

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)^(5/2),x)

[Out]

-(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2)/(3*c^3*e*(d + e*x)^4)

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sympy [A]  time = 1.60, size = 124, normalized size = 3.65 \[ \begin {cases} - \frac {1}{3 c^{2} d^{2} e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} + 6 c^{2} d e^{2} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} + 3 c^{2} e^{3} x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

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)**(5/2),x)

[Out]

Piecewise((-1/(3*c**2*d**2*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2) + 6*c**2*d*e**2*x*sqrt(c*d**2 + 2*c*d*e*x
+ c*e**2*x**2) + 3*c**2*e**3*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d**2)**(5/2), Tr
ue))

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