3.11.69 \(\int \frac {1}{(d+e x)^3 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1069]

Optimal. Leaf size=32 \[ -\frac {c}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[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 = 0.02, antiderivative size = 32, 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 = {657, 643} \begin {gather*} -\frac {c}{3 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[1/((d + e*x)^3*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]

[Out]

-1/3*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]

Rule 657

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)^3 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c^2 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c}{3 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 = 21, normalized size = 0.66 \begin {gather*} -\frac {c}{3 e \left (c (d+e x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*c/(e*(c*(d + e*x)^2)^(3/2))

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Maple [A]
time = 0.59, size = 35, normalized size = 1.09

method	result	size
risch	\(-\frac {1}{3 \left (e x +d \right )^{2} \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(24\)
gosper	\(-\frac {1}{3 \left (e x +d \right )^{2} e \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\)	\(35\)
default	\(-\frac {1}{3 \left (e x +d \right )^{2} e \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\)	\(35\)
trager	\(\frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 d^{3} c \left (e x +d \right )^{4}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 45, normalized size = 1.41 \begin {gather*} -\frac {1}{3 \, {\left (\sqrt {c} x^{3} e^{4} + 3 \, \sqrt {c} d x^{2} e^{3} + 3 \, \sqrt {c} d^{2} x e^{2} + \sqrt {c} d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
time = 2.61, size = 70, normalized size = 2.19 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{3 \, {\left (c x^{4} e^{5} + 4 \, c d x^{3} e^{4} + 6 \, c d^{2} x^{2} e^{3} + 4 \, c d^{3} x e^{2} + c d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(c*(d + e*x)**2)*(d + e*x)**3), x)

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Giac [A]
time = 2.88, size = 24, normalized size = 0.75 \begin {gather*} -\frac {e^{\left (-1\right )}}{3 \, {\left (x e + d\right )}^{3} \sqrt {c} \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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

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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}}{3\,c\,e\,{\left (d+e\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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