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

Optimal. Leaf size=111 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+c x^2}}{a d x} \]

[Out]

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

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Rubi [A]  time = 0.260002, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+c x^2}}{a d x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.3921, size = 95, normalized size = 0.86 \[ - \frac{e^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{2} \sqrt{a e^{2} + c d^{2}}} - \frac{\sqrt{a + c x^{2}}}{a d x} + \frac{e \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.170995, size = 144, normalized size = 1.3 \[ \frac{-\frac{e^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{e^2 \log (d+e x)}{\sqrt{a e^2+c d^2}}-\frac{d \sqrt{a+c x^2}}{a x}+\frac{e \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}-\frac{e \log (x)}{\sqrt{a}}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 180, normalized size = 1.6 \[ -{\frac{1}{adx}\sqrt{c{x}^{2}+a}}-{\frac{e}{{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.336558, size = 1, normalized size = 0.01 \[ \left [\frac{a^{\frac{3}{2}} e^{2} x \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + \sqrt{c d^{2} + a e^{2}} a e x \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} \sqrt{a} d}{2 \, \sqrt{c d^{2} + a e^{2}} a^{\frac{3}{2}} d^{2} x}, \frac{2 \, a^{\frac{3}{2}} e^{2} x \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) + \sqrt{-c d^{2} - a e^{2}} a e x \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} \sqrt{a} d}{2 \, \sqrt{-c d^{2} - a e^{2}} a^{\frac{3}{2}} d^{2} x}, \frac{\sqrt{-a} a e^{2} x \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d^{2} + a e^{2}} a e x \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - 2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} \sqrt{-a} d}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{-a} a d^{2} x}, \frac{\sqrt{-a} a e^{2} x \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) + \sqrt{-c d^{2} - a e^{2}} a e x \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} \sqrt{-a} d}{\sqrt{-c d^{2} - a e^{2}} \sqrt{-a} a d^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(a^(3/2)*e^2*x*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)*a*e*x*log(-((
c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2) - 2*sqrt(c*d^2 + a*e^2)*sqrt(c*
x^2 + a)*sqrt(a)*d)/(sqrt(c*d^2 + a*e^2)*a^(3/2)*d^2*x), 1/2*(2*a^(3/2)*e^2*x*ar
ctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) + sqr
t(-c*d^2 - a*e^2)*a*e*x*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2)
- 2*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(a)*d)/(sqrt(-c*d^2 - a*e^2)*a^(3/2
)*d^2*x), 1/2*(sqrt(-a)*a*e^2*x*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)
*a*e*x*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - 2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*
sqrt(-a)*d)/(sqrt(c*d^2 + a*e^2)*sqrt(-a)*a*d^2*x), (sqrt(-a)*a*e^2*x*arctan(sqr
t(-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)*a*e*x*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - sqrt(-c*d^2 - a*e^2)*sqrt(c*x
^2 + a)*sqrt(-a)*d)/(sqrt(-c*d^2 - a*e^2)*sqrt(-a)*a*d^2*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279711, size = 192, normalized size = 1.73 \[ 2 \, c{\left (\frac{\arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{2}}{\sqrt{-c d^{2} - a e^{2}} c d^{2}} - \frac{\arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a} c d^{2}} + \frac{1}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )} \sqrt{c} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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