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

Optimal. Leaf size=54 \[ -\frac{\tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\sqrt{a e^2+c d^2}} \]

[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])

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Rubi [A]  time = 0.057906, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{\tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\sqrt{a e^2+c d^2}} \]

Antiderivative was successfully verified.

[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])

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Rubi in Sympy [A]  time = 8.86406, size = 48, normalized size = 0.89 \[ - \frac{\operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\sqrt{a e^{2} + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0692709, size = 62, normalized size = 1.15 \[ \frac{\log (d+e x)-\log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 127, normalized size = 2.4 \[ -{\frac{1}{e}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238182, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt{c d^{2} + a e^{2}}}, \frac{\arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{\sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + a)*(e*x + d)),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)*sqrt(c
*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a
))/(e^2*x^2 + 2*d*e*x + d^2))/sqrt(c*d^2 + a*e^2), arctan(sqrt(-c*d^2 - a*e^2)*(
c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a)))/sqrt(-c*d^2 - a*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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GIAC/XCAS [A]  time = 0.21616, size = 80, normalized size = 1.48 \[ \frac{2 \, \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{\sqrt{-c d^{2} - a e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + a)*(e*x + d)),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))/sq
rt(-c*d^2 - a*e^2)