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

Optimal. Leaf size=112 \[ \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 50, 63, 208} \begin {gather*} \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)*Sqrt[d + e*x])/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh[(Sqrt[b]*
Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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,
b, c, d, m, n, x]

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 646

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]

Rubi steps

\begin {align*} \int \frac {\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 {\sqrt {d+e x}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \operatorname {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 )}{b^2 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 81, normalized size = 0.72 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x}-\sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{b^{3/2} \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x] - Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]))/(b^(3
/2)*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 9.98, size = 104, normalized size = 0.93 \begin {gather*} \frac {(-a e-b e x) \left (-\frac {2 \sqrt {a e-b d} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{3/2}}-\frac {2 \sqrt {d+e x}}{b}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[d + e*x]/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((-(a*e) - b*e*x)*((-2*Sqrt[d + e*x])/b - (2*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*
x])/(b*d - a*e)])/b^(3/2)))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [A]  time = 0.42, size = 143, normalized size = 1.28 \begin {gather*} \left [\frac {\sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt {e x + d}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - \sqrt {e x + d}\right )}}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.23, size = 85, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b} + \frac {2 \, \sqrt {x e + d} \mathrm {sgn}\left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 104, normalized size = 0.93 \begin {gather*} \frac {2 \left (b x +a \right ) \left (-a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+b d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\right )}{\sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x+a)*(-arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*e+arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*b*d+(e
*x+d)^(1/2)*((a*e-b*d)*b)^(1/2))/((b*x+a)^2)^(1/2)/b/((a*e-b*d)*b)^(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 {{\left (b x + a\right )}^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.56, size = 95, normalized size = 0.85 \begin {gather*} \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} \log {\left (- b \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} + \sqrt {d + e x} \right )} - \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} \log {\left (b \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} + \sqrt {d + e x} \right )} + \frac {2 \sqrt {d + e x}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a*e/b**3 + d/b**2)*log(-b*sqrt(-a*e/b**3 + d/b**2) + sqrt(d + e*x)) - sqrt(-a*e/b**3 + d/b**2)*log(b*sqr
t(-a*e/b**3 + d/b**2) + sqrt(d + e*x)) + 2*sqrt(d + e*x)/b

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