\(\int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx\) [330]

Optimal result

Integrand size = 19, antiderivative size = 54 \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\sqrt {c d^2+a e^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {739, 212} \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\sqrt {a e^2+c d^2}} \]

[In]

Int[1/((d + e*x)*Sqrt[a + c*x^2]),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(48)=96\).

Time = 0.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.35

method	result	size
default	\(-\frac {\ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(127\)

[In]

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

[Out]

-1/e/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2/e*c*d*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*
c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (49) = 98\).

Time = 0.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.91 \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {\log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt {c d^{2} + a e^{2}}}, -\frac {\sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right )}{c d^{2} + a e^{2}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{\sqrt {a + \frac {c d^{2}}{e^{2}}} e} \]

[In]

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

[Out]

arcsinh(c*d*x/(e*sqrt(a*c/e^2)*abs(e*x + d)) - a/(sqrt(a*c/e^2)*abs(e*x + d)))/(sqrt(a + c*d^2/e^2)*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \, \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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