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

Optimal. Leaf size=86 \[ \frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \sqrt{a e^2+c d^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 0.216069, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \sqrt{a e^2+c d^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.133, size = 73, normalized size = 0.85 \[ \frac{e \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d \sqrt{a e^{2} + c d^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.12827, size = 118, normalized size = 1.37 \[ \frac{\frac{e \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{e \log (d+e x)}{\sqrt{a e^2+c d^2}}-\frac{\log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{\log (x)}{\sqrt{a}}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 158, normalized size = 1.8 \[ -{\frac{1}{d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{1}{d}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313643, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} e \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + \sqrt{c d^{2} + a e^{2}} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{a} d}, -\frac{2 \, \sqrt{a} e \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) - \sqrt{-c d^{2} - a e^{2}} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{a} d}, \frac{\sqrt{-a} e \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt{c d^{2} + a e^{2}} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{-a} d}, -\frac{\sqrt{-a} e \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) + \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{\sqrt{-c d^{2} - a e^{2}} \sqrt{-a} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError