∫ 1______________√ d + ex√_________ a2 + 2abx + b2x2 dx [1707]

Optimal. Leaf size=72 \[ -\frac {2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}} \]

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

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Rubi [A]
time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {660, 65, 214} \begin {gather*} -\frac {2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}} \end {gather*}

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

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

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

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra

cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,

c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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

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Mathematica [A]
time = 0.04, size = 63, normalized size = 0.88 \begin {gather*} \frac {2 (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} \sqrt {-b d+a e} \sqrt {(a+b x)^2}} \end {gather*}

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

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

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method	result	size

default	\(\frac {2 \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{\sqrt {\left (b x +a \right )^{2}}\, \sqrt {b \left (a e -b d \right )}}\)	\(51\)

int(1/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

integrate(1/(sqrt((b*x + a)^2)*sqrt(x*e + d)), x)

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Fricas [A]
time = 2.12, size = 129, normalized size = 1.79 \begin {gather*} \left [\frac {\log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right )}{\sqrt {b^{2} d - a b e}}, \frac {2 \, \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right )}{b^{2} d - a b e}\right ] \end {gather*}

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

[log((2*b*d + (b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a))/sqrt(b^2*d - a*b*e), 2*sqrt(-b^2*d

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + e x} \sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}

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

Integral(1/(sqrt(d + e*x)*sqrt((a + b*x)**2)), x)

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Giac [A]
time = 0.89, size = 47, normalized size = 0.65 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) \mathrm {sgn}\left (b x + a\right )}{\sqrt {-b^{2} d + a b e}} \end {gather*}

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

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

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

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

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

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3.18.7 \(\int \frac {1}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) [1707]