3.11.84 \(\int \frac {d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\) [1084]

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 = {643} \begin {gather*} -\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 643

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 \begin {gather*} -\frac {\sqrt {c (d+e x)^2}}{3 c^3 e (d+e x)^4} \end {gather*}

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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Maple [A]
time = 0.54, size = 35, normalized size = 1.03

method	result	size
risch	\(-\frac {1}{3 c^{2} \left (e x +d \right )^{2} \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(27\)
gosper	\(-\frac {\left (e x +d \right )^{2}}{3 e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
default	\(-\frac {\left (e x +d \right )^{2}}{3 e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
trager	\(\frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 c^{3} d^{3} \left (e x +d \right )^{4}}\)	\(57\)

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,method=_RETURNVERBOSE)

[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 = 0.27, size = 29, normalized size = 0.85 \begin {gather*} -\frac {e^{\left (-1\right )}}{3 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}} 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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (29) = 58\).
time = 3.16, size = 80, normalized size = 2.35 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{3 \, {\left (c^{3} x^{4} e^{5} + 4 \, c^{3} d x^{3} e^{4} + 6 \, c^{3} d^{2} x^{2} e^{3} + 4 \, c^{3} d^{3} x e^{2} + c^{3} d^{4} e\right )}} \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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (32) = 64\).
time = 0.57, size = 124, normalized size = 3.65 \begin {gather*} \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} \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)**(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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Giac [A]
time = 1.01, size = 24, normalized size = 0.71 \begin {gather*} -\frac {e^{\left (-1\right )}}{3 \, {\left (x e + d\right )}^{3} c^{\frac {5}{2}} \mathrm {sgn}\left (x e + d\right )} \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)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.49, size = 37, normalized size = 1.09 \begin {gather*} -\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} \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)^(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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