3.1035 \(\int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac {c}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 30, 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} \[ -\frac {c}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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]

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 {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^3} \, dx &=c^2 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {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 = 19, normalized size = 0.63 \[ -\frac {c}{e \sqrt {c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(e*Sqrt[c*(d + e*x)^2]))

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fricas [A]  time = 0.69, size = 46, normalized size = 1.53 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-(-c*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2
*exp(2))-sqrt(c*exp(2))*x)^3*exp(1)^3+2*c*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*
exp(1)-c*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d*exp(1)^2-2*c*exp(2)*sqr
t(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d+c^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^
2*exp(2))-sqrt(c*exp(2))*x)*d^2*exp(1)^3+2*c^2*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*
x)*d^2*exp(1)-c^2*sqrt(c*exp(2))*d^3*exp(1)^2)/2/exp(1)^2/(-(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*ex
p(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^
2+c/2/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*sqrt(c*exp(2))+(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(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 = 1.17 \[ -\frac {\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}}{\left (e x +d \right )^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(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 [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.42, size = 34, normalized size = 1.13 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{e\,{\left (d+e\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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