3.4.43 \(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)} \, dx\)

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

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Rubi [A]  time = 0.06, 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 = {707, 1093, 208} \begin {gather*} \frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b \sqrt {c d-b e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(b*x + c*x^2)),x]

[Out]

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

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 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1093

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 75, normalized size = 0.97 \begin {gather*} \frac {\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{\sqrt {c d-b e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 87, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b \sqrt {b e-c d}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[d + e*x]*(b*x + c*x^2)),x]

[Out]

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

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fricas [A]  time = 0.44, size = 396, normalized size = 5.14 \begin {gather*} \left [\frac {d \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b d}, \frac {2 \, d \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right )}{b d}, \frac {d \sqrt {\frac {c}{c d - b e}} \log \left (\frac {c e x + 2 \, c d - b e + 2 \, {\left (c d - b e\right )} \sqrt {e x + d} \sqrt {\frac {c}{c d - b e}}}{c x + b}\right ) + 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right )}{b d}, \frac {2 \, {\left (d \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {e x + d} \sqrt {-\frac {c}{c d - b e}}}{c e x + c d}\right ) + \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right )\right )}}{b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 71, normalized size = 0.92 \begin {gather*} -\frac {2 \, c \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 \, \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(1/(e*x+d)^(1/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 62, normalized size = 0.81 \begin {gather*} -\frac {2 c \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 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/(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 = 0.34, size = 625, normalized size = 8.12 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d+e\,x}}{\sqrt {d}}\right )}{b\,\sqrt {d}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (16\,c^3\,e^2\,\sqrt {d+e\,x}+\frac {\sqrt {c^2\,d-b\,c\,e}\,\left (8\,b^2\,c^2\,e^3+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{b^2\,e-b\,c\,d}\right )}{b^2\,e-b\,c\,d}\right )\,\sqrt {c^2\,d-b\,c\,e}\,1{}\mathrm {i}}{b^2\,e-b\,c\,d}+\frac {\left (16\,c^3\,e^2\,\sqrt {d+e\,x}-\frac {\sqrt {c^2\,d-b\,c\,e}\,\left (8\,b^2\,c^2\,e^3-\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{b^2\,e-b\,c\,d}\right )}{b^2\,e-b\,c\,d}\right )\,\sqrt {c^2\,d-b\,c\,e}\,1{}\mathrm {i}}{b^2\,e-b\,c\,d}}{\frac {\left (16\,c^3\,e^2\,\sqrt {d+e\,x}+\frac {\sqrt {c^2\,d-b\,c\,e}\,\left (8\,b^2\,c^2\,e^3+\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{b^2\,e-b\,c\,d}\right )}{b^2\,e-b\,c\,d}\right )\,\sqrt {c^2\,d-b\,c\,e}}{b^2\,e-b\,c\,d}-\frac {\left (16\,c^3\,e^2\,\sqrt {d+e\,x}-\frac {\sqrt {c^2\,d-b\,c\,e}\,\left (8\,b^2\,c^2\,e^3-\frac {\left (8\,b^3\,c^2\,e^3-16\,b^2\,c^3\,d\,e^2\right )\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{b^2\,e-b\,c\,d}\right )}{b^2\,e-b\,c\,d}\right )\,\sqrt {c^2\,d-b\,c\,e}}{b^2\,e-b\,c\,d}}\right )\,\sqrt {c^2\,d-b\,c\,e}\,2{}\mathrm {i}}{b^2\,e-b\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((16*c^3*e^2*(d + e*x)^(1/2) + ((c^2*d - b*c*e)^(1/2)*(8*b^2*c^2*e^3 + ((8*b^3*c^2*e^3 - 16*b^2*c^3*d*e
^2)*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(b^2*e - b*c*d)))/(b^2*e - b*c*d))*(c^2*d - b*c*e)^(1/2)*1i)/(b^2*e
- b*c*d) + ((16*c^3*e^2*(d + e*x)^(1/2) - ((c^2*d - b*c*e)^(1/2)*(8*b^2*c^2*e^3 - ((8*b^3*c^2*e^3 - 16*b^2*c^
3*d*e^2)*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(b^2*e - b*c*d)))/(b^2*e - b*c*d))*(c^2*d - b*c*e)^(1/2)*1i)/(
b^2*e - b*c*d))/(((16*c^3*e^2*(d + e*x)^(1/2) + ((c^2*d - b*c*e)^(1/2)*(8*b^2*c^2*e^3 + ((8*b^3*c^2*e^3 - 16*b
^2*c^3*d*e^2)*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(b^2*e - b*c*d)))/(b^2*e - b*c*d))*(c^2*d - b*c*e)^(1/2))
/(b^2*e - b*c*d) - ((16*c^3*e^2*(d + e*x)^(1/2) - ((c^2*d - b*c*e)^(1/2)*(8*b^2*c^2*e^3 - ((8*b^3*c^2*e^3 - 16
*b^2*c^3*d*e^2)*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(b^2*e - b*c*d)))/(b^2*e - b*c*d))*(c^2*d - b*c*e)^(1/2
))/(b^2*e - b*c*d)))*(c^2*d - b*c*e)^(1/2)*2i)/(b^2*e - b*c*d) - (2*atanh((d + e*x)^(1/2)/d^(1/2)))/(b*d^(1/2)
)

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sympy [A]  time = 27.30, size = 80, normalized size = 1.04 \begin {gather*} \frac {2 c \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {c}{b e - c d}} \sqrt {d + e x}} \right )}}{b \sqrt {\frac {c}{b e - c d}} \left (b e - c d\right )} + \frac {2 \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{b d \sqrt {- \frac {1}{d}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*atan(1/(sqrt(c/(b*e - c*d))*sqrt(d + e*x)))/(b*sqrt(c/(b*e - c*d))*(b*e - c*d)) + 2*atan(1/(sqrt(-1/d)*sqr
t(d + e*x)))/(b*d*sqrt(-1/d))

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