3.17.49 \(\int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx\) [1649]

Optimal. Leaf size=70 \[ -\frac {\sqrt {d+e x}}{b (a+b x)}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}} \]

[Out]

-e*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(1/2)-(e*x+d)^(1/2)/b/(b*x+a)

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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43, 65, 214} \begin {gather*} -\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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

Rule 65

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 214

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

Rubi steps

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

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Mathematica [A]
time = 0.17, size = 69, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {e \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{3/2} \sqrt {-b d+a e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.65, size = 73, normalized size = 1.04

method	result	size
derivativedivides	\(2 e \left (-\frac {\sqrt {e x +d}}{2 b \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 b \sqrt {b \left (a e -b d \right )}}\right )\)	\(73\)
default	\(2 e \left (-\frac {\sqrt {e x +d}}{2 b \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 b \sqrt {b \left (a e -b d \right )}}\right )\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*(-1/2/b*(e*x+d)^(1/2)/((e*x+d)*b+a*e-b*d)+1/2/b/(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(-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)/(b^2*x^2+2*a*b*x+a^2),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*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.66, size = 243, normalized size = 3.47 \begin {gather*} \left [\frac {\sqrt {b^{2} d - a b e} {\left (b x + a\right )} e \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 ) - 2 \, {\left (b^{2} d - a b e\right )} \sqrt {x e + d}}{2 \, {\left (b^{4} d x + a b^{3} d - {\left (a b^{3} x + a^{2} b^{2}\right )} e\right )}}, \frac {\sqrt {-b^{2} d + a b e} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e - {\left (b^{2} d - a b e\right )} \sqrt {x e + d}}{b^{4} d x + a b^{3} d - {\left (a b^{3} x + a^{2} b^{2}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(b^2*d - a*b*e)*(b*x + a)*e*log((2*b*d + (b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a
)) - 2*(b^2*d - a*b*e)*sqrt(x*e + d))/(b^4*d*x + a*b^3*d - (a*b^3*x + a^2*b^2)*e), (sqrt(-b^2*d + a*b*e)*(b*x
+ a)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(x*e + d)/(b*x*e + b*d))*e - (b^2*d - a*b*e)*sqrt(x*e + d))/(b^4*d*x + a*
b^3*d - (a*b^3*x + a^2*b^2)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (58) = 116\).
time = 10.31, size = 573, normalized size = 8.19 \begin {gather*} - \frac {2 a e^{2} \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac {a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {2 d e \sqrt {d + e x}}{2 a^{2} e^{2} - 2 a b d e + 2 a b e^{2} x - 2 b^{2} d e x} + \frac {2 e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{2} \sqrt {\frac {a e}{b} - d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x) + a*e**2*sqrt(-1/(b*(a
*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*
sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - a*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/
(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d +
e*x))/(2*b) - d*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/
(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + d*e*sqrt(-1/(b*(a*e - b*d)**3
))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a
*e - b*d)**3)) + sqrt(d + e*x))/2 + 2*d*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x - 2*b**2*d*e*x
) + 2*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**2*sqrt(a*e/b - d))

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Giac [A]
time = 0.73, size = 80, normalized size = 1.14 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e}{\sqrt {-b^{2} d + a b e} b} - \frac {\sqrt {x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 61, normalized size = 0.87 \begin {gather*} \frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{b^{3/2}\,\sqrt {a\,e-b\,d}}-\frac {e\,\sqrt {d+e\,x}}{e\,x\,b^2+a\,e\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(b^(3/2)*(a*e - b*d)^(1/2)) - (e*(d + e*x)^(1/2))/(a*b*e
+ b^2*e*x)

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