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

Optimal. Leaf size=28 \[ \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]

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Rubi [A]  time = 0.02, antiderivative size = 28, 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 {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \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),x]

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/e

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} \, dx &=c \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{e}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.75 \begin {gather*} \frac {c x (d+e x)}{\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),x]

[Out]

(c*x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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IntegrateAlgebraic [A]  time = 0.03, size = 17, normalized size = 0.61 \begin {gather*} \frac {\sqrt {c (d+e x)^2}}{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),x]

[Out]

Sqrt[c*(d + e*x)^2]/e

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fricas [A]  time = 0.39, size = 31, normalized size = 1.11 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{e x + d} \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),x, algorithm="fricas")

[Out]

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

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 4*1/4/exp(1)*sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))+2*(-c*d^2*exp(1)^2+c*d^2*exp(2))*2/2/exp(1)^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 = 32, normalized size = 1.14 \begin {gather*} \frac {\sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{e x +d} \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),x)

[Out]

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

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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),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 = 0.48, size = 15, normalized size = 0.54 \begin {gather*} \frac {\sqrt {c\,{\left (d+e\,x\right )}^2}}{e} \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),x)

[Out]

(c*(d + e*x)^2)^(1/2)/e

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sympy [A]  time = 2.60, size = 37, normalized size = 1.32 \begin {gather*} \begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {x \sqrt {c d^{2}}}{d} & \text {otherwise} \end {cases} \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),x)

[Out]

Piecewise((sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/e, Ne(e, 0)), (x*sqrt(c*d**2)/d, True))

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