### 3.2060 $$\int \frac{\sqrt{d+e x}}{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx$$

Optimal. Leaf size=46 $\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}}$

[Out]

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

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Rubi [A]  time = 0.019753, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.026, Rules used = {648} $\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0173811, size = 35, normalized size = 0.76 $\frac{2 \sqrt{(d+e x) (a e+c d x)}}{c d \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 50, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( cdx+ae \right ) \sqrt{ex+d}}{cd\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07004, size = 24, normalized size = 0.52 \begin{align*} \frac{2 \, \sqrt{c d x + a e}}{c d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(c*d*x + a*e)/(c*d)

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Fricas [A]  time = 1.96065, size = 107, normalized size = 2.33 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{c d e x + c d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x}}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out