3.5.28 \(\int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=80 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {c d f-a e g}} \]

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Rubi [A]  time = 0.13, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {874, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {c d f-a e g}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]
*Sqrt[c*d*f - a*e*g])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 874

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{\sqrt {g} \sqrt {c d f-a e g}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 93, normalized size = 1.16 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {(d+e x) (a e+c d x)} \sqrt {c d f-a e g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/((f + g*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]*Sqrt[d + e*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[g]*Sqrt[c*d*
f - a*e*g]*Sqrt[(a*e + c*d*x)*(d + e*x)])

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IntegrateAlgebraic [C]  time = 5.27, size = 609, normalized size = 7.61 \begin {gather*} -\frac {2 \left (\sqrt {e} \sqrt {c d f-a e g}-i \sqrt {c} \sqrt {d} \sqrt {d g-e f}\right ) \sqrt {-2 i \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {d g-e f} \sqrt {c d f-a e g}+a e^2 g+c d^2 g-2 c d e f} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {d+e x} \sqrt {-2 i \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {d g-e f} \sqrt {c d f-a e g}+a e^2 g+c d^2 g-2 c d e f}}{e \sqrt {g} \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}-\sqrt {g} \sqrt {c d e} (d+e x)}\right )}{g^{3/2} \left (a e^2-c d^2\right ) \sqrt {c d f-a e g}}-\frac {2 \left (\sqrt {e} \sqrt {c d f-a e g}+i \sqrt {c} \sqrt {d} \sqrt {d g-e f}\right ) \sqrt {2 i \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {d g-e f} \sqrt {c d f-a e g}+a e^2 g+c d^2 g-2 c d e f} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {d+e x} \sqrt {2 i \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {d g-e f} \sqrt {c d f-a e g}+a e^2 g+c d^2 g-2 c d e f}}{e \sqrt {g} \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}-\sqrt {g} \sqrt {c d e} (d+e x)}\right )}{g^{3/2} \left (a e^2-c d^2\right ) \sqrt {c d f-a e g}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((-I)*Sqrt[c]*Sqrt[d]*Sqrt[-(e*f) + d*g] + Sqrt[e]*Sqrt[c*d*f - a*e*g])*Sqrt[-2*c*d*e*f + c*d^2*g + a*e^2*
g - (2*I)*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[-(e*f) + d*g]*Sqrt[c*d*f - a*e*g]]*ArcTanh[(Sqrt[e]*Sqrt[-2*c*d*e*f + c
*d^2*g + a*e^2*g - (2*I)*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[-(e*f) + d*g]*Sqrt[c*d*f - a*e*g]]*Sqrt[d + e*x])/(-(Sqr
t[c*d*e]*Sqrt[g]*(d + e*x)) + e*Sqrt[g]*Sqrt[-((c*d^2*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e])])/
((-(c*d^2) + a*e^2)*g^(3/2)*Sqrt[c*d*f - a*e*g]) - (2*(I*Sqrt[c]*Sqrt[d]*Sqrt[-(e*f) + d*g] + Sqrt[e]*Sqrt[c*d
*f - a*e*g])*Sqrt[-2*c*d*e*f + c*d^2*g + a*e^2*g + (2*I)*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[-(e*f) + d*g]*Sqrt[c*d*f
- a*e*g]]*ArcTanh[(Sqrt[e]*Sqrt[-2*c*d*e*f + c*d^2*g + a*e^2*g + (2*I)*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[-(e*f) +
d*g]*Sqrt[c*d*f - a*e*g]]*Sqrt[d + e*x])/(-(Sqrt[c*d*e]*Sqrt[g]*(d + e*x)) + e*Sqrt[g]*Sqrt[-((c*d^2*(d + e*x)
)/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e])])/((-(c*d^2) + a*e^2)*g^(3/2)*Sqrt[c*d*f - a*e*g])

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fricas [A]  time = 0.42, size = 252, normalized size = 3.15 \begin {gather*} \left [-\frac {\sqrt {-c d f g + a e g^{2}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d f g + a e g^{2}} \sqrt {e x + d}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right )}{c d f g - a e g^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d f g - a e g^{2}} \sqrt {e x + d}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right )}{\sqrt {c d f g - a e g^{2}}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x - 2*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*
x))/(c*d*f*g - a*e*g^2), -2*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*
x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x))/sqrt(c*d*f*g - a*e*g^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)), x)

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maple [A]  time = 0.03, size = 87, normalized size = 1.09 \begin {gather*} -\frac {2 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(e*x+d)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)*arctanh(g*(
c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{\left (f+g\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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