3.11.66 \(\int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\) [1066]

Optimal. Leaf size=39 \[ \frac {(d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {622, 31} \begin {gather*} \frac {(d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

((d + e*x)*Log[d + e*x])/(e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 622

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac {\left (c d e+c e^2 x\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{\sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac {(d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 28, normalized size = 0.72 \begin {gather*} \frac {(d+e x) \log (d+e x)}{e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])

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Maple [A]
time = 0.54, size = 38, normalized size = 0.97

method	result	size
risch	\(\frac {\left (e x +d \right ) \ln \left (e x +d \right )}{\sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(27\)
default	\(\frac {\left (e x +d \right ) \ln \left (e x +d \right )}{e \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.33 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (d e^{\left (-1\right )} + x\right )}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log(d*e^(-1) + x)/sqrt(c)

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Fricas [A]
time = 2.75, size = 43, normalized size = 1.10 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \log \left (x e + d\right )}{c x e^{2} + c d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.60, size = 36, normalized size = 0.92 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |} \sqrt {{\left | c \right |}} {\left | \mathrm {sgn}\left (x e + d\right ) \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.47, size = 29, normalized size = 0.74 \begin {gather*} \frac {\ln \left (c\,x\,e^2+c\,d\,e\right )\,\mathrm {sign}\left (c\,e\,\left (d+e\,x\right )\right )}{\sqrt {c\,e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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