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

Optimal. Leaf size=77 \[ \frac {2 \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b} \]

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Rubi [A]  time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {699, 1130, 208} \begin {gather*} \frac {2 \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(b*x + c*x^2),x]

[Out]

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

Rule 208

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

Rule 699

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

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{b x+c x^2} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=\frac {(2 c d) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b}+\left (e \left (1+\frac {-2 c d+b e}{b e}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )\\ &=-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}+\frac {2 \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 75, normalized size = 0.97 \begin {gather*} \frac {2 \left (\frac {\sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{\sqrt {c}}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.11, size = 87, normalized size = 1.13 \begin {gather*} -\frac {2 \sqrt {b e-c d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b \sqrt {c}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-(c*d) + b*e]*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b*Sqrt[c]) - (2*Sqrt[d
]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b

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fricas [A]  time = 0.44, size = 352, normalized size = 4.57 \begin {gather*} \left [\frac {\sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right ) + \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b}, \frac {2 \, \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b}, \frac {2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + \sqrt {\frac {c d - b e}{c}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, \sqrt {e x + d} c \sqrt {\frac {c d - b e}{c}}}{c x + b}\right )}{b}, \frac {2 \, {\left (\sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {e x + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right )\right )}}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + sqrt(d)*l
og((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, (2*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b
*e)/c)/(c*d - b*e)) + sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/b, (2*sqrt(-d)*arctan(sqrt(e*x + d
)*sqrt(-d)/d) + sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b
)))/b, 2*(sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + sqrt(-d)*arctan(sqr
t(e*x + d)*sqrt(-d)/d))/b]

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giac [A]  time = 0.18, size = 80, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (c d - b e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b} + \frac {2 \, d \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 100, normalized size = 1.30 \begin {gather*} -\frac {2 c d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}}-\frac {2 \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)-2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/
2)/((b*e-c*d)*c)^(1/2)*c)*c*d-2*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 0.16, size = 100, normalized size = 1.30 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {16\,b\,c^2\,d\,e^3\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{16\,b\,c^3\,d^2\,e^3-16\,b^2\,c^2\,d\,e^4}\right )\,\sqrt {c^2\,d-b\,c\,e}}{b\,c}-\frac {2\,\sqrt {d}\,\mathrm {atanh}\left (\frac {\sqrt {d+e\,x}}{\sqrt {d}}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(b*x + c*x^2),x)

[Out]

(2*atanh((16*b*c^2*d*e^3*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(16*b*c^3*d^2*e^3 - 16*b^2*c^2*d*e^4))*(c^2*d
- b*c*e)^(1/2))/(b*c) - (2*d^(1/2)*atanh((d + e*x)^(1/2)/d^(1/2)))/b

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sympy [A]  time = 6.78, size = 78, normalized size = 1.01 \begin {gather*} \frac {2 \left (\frac {d e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b \sqrt {- d}} + \frac {e \left (b e - c d\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b c \sqrt {\frac {b e - c d}{c}}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*e*atan(sqrt(d + e*x)/sqrt(-d))/(b*sqrt(-d)) + e*(b*e - c*d)*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*c*
sqrt((b*e - c*d)/c)))/e

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