3.9.81 \(\int \frac {1}{(d+e x)^3 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\)

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

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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 = {643, 629} \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]

-c/(3*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]

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)^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 = 0.02, 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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IntegrateAlgebraic [A]  time = 0.04, 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]

IntegrateAlgebraic[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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fricas [B]  time = 0.39, size = 73, normalized size = 2.28 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (c e^{5} x^{4} + 4 \, c d e^{4} x^{3} + 6 \, c d^{2} e^{3} x^{2} + 4 \, c d^{3} e^{2} x + 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*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c*e^5*x^4 + 4*c*d*e^4*x^3 + 6*c*d^2*e^3*x^2 + 4*c*d^3*e^2*x + c*d^4*
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)^3/(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*((-3*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2
+c*d^2)-sqrt(c*exp(2))*x)^3*exp(1)^3+2*exp(2)*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^3*exp
(1)+9*sqrt(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^2*d*exp(1)^2-6*exp(2)*sqrt(c*e
xp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)^2*d-5*c*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*
d^2)-sqrt(c*exp(2))*x)*d^2*exp(1)^3+2*c*exp(2)*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)*d^2*
exp(1)+c*sqrt(c*exp(2))*d^3*exp(1)^2)/(-2*d^2*exp(1)^2+2*exp(2)*d^2)/(-(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2
)-sqrt(c*exp(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sqrt(c*exp(2)*x^2+c*d*exp(1)*x*2+c*d^2)-sqrt(c*exp(2))*x)*d-c*d
^2*exp(1))^2+(3*exp(1)^2-2*exp(2))/2/(d^2*exp(1)^2-exp(2)*d^2)/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*sqrt(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 = 35, normalized size = 1.09 \begin {gather*} -\frac {1}{3 \left (e x +d \right )^{2} \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)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[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 = 1.35, size = 47, normalized size = 1.47 \begin {gather*} -\frac {1}{3 \, {\left (\sqrt {c} e^{4} x^{3} + 3 \, \sqrt {c} d e^{3} x^{2} + 3 \, \sqrt {c} d^{2} e^{2} x + \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)*e^4*x^3 + 3*sqrt(c)*d*e^3*x^2 + 3*sqrt(c)*d^2*e^2*x + sqrt(c)*d^3*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}}{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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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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