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

Optimal. Leaf size=40 \[ \frac {c (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {642, 608, 31} \begin {gather*} \frac {c (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[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^2,x]

[Out]

(c*(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 608

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]

Rule 642

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/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/2]

Rubi steps

\begin {align*} \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx &=c \int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac {\left (c \left (c d e+c e^2 x\right )\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 {c (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 = 0.01, size = 29, normalized size = 0.72 \begin {gather*} \frac {c (d+e x) \log (d+e x)}{e \sqrt {c (d+e x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 0.28, size = 133, normalized size = 3.32 \begin {gather*} -\frac {\sqrt {c e^2} \log \left (x \left (c d e+c e^2 x\right )-x \sqrt {c e^2} \sqrt {c d^2+2 c d e x+c e^2 x^2}\right )}{2 e^2}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {x \sqrt {c e^2}}{\sqrt {c} d}-\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{\sqrt {c} d}\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 40, normalized size = 1.00 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{e^{2} x + d e} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.57, size = 8, normalized size = 0.20 \begin {gather*} 2 \, C_{0} \sqrt {c} e^{3} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

2*C_0*sqrt(c)*e^3

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maple [A]  time = 0.05, size = 40, normalized size = 1.00 \begin {gather*} \frac {\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, \ln \left (e x +d \right )}{\left (e x +d \right ) e} \end {gather*}

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)^2,x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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)^2,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 [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}

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)^2,x)

[Out]

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

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

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)**2,x)

[Out]

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

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