3.323 \(\int \frac{\sqrt{a+c x^2}}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=160 \[ -\frac{\sqrt{a} e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{e \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 0.504806, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{\sqrt{a} e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{e \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^2]/(x^3*(d + e*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 57.7648, size = 141, normalized size = 0.88 \[ - \frac{\sqrt{a} e^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{d^{3}} - \frac{\sqrt{a + c x^{2}}}{2 d x^{2}} + \frac{e \sqrt{a + c x^{2}}}{d^{2} x} + \frac{e \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{3}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 \sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(1/2)/x**3/(e*x+d),x)

[Out]

-sqrt(a)*e**2*atanh(sqrt(a + c*x**2)/sqrt(a))/d**3 - sqrt(a + c*x**2)/(2*d*x**2)
+ e*sqrt(a + c*x**2)/(d**2*x) + e*sqrt(a*e**2 + c*d**2)*atanh((a*e - c*d*x)/(sq
rt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/d**3 - c*atanh(sqrt(a + c*x**2)/sqrt(a))/
(2*sqrt(a)*d)

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Mathematica [A]  time = 0.263801, size = 170, normalized size = 1.06 \[ \frac{-\frac{\left (2 a e^2+c d^2\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+2 e \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+\frac{\log (x) \left (2 a e^2+c d^2\right )}{\sqrt{a}}-2 e \sqrt{a e^2+c d^2} \log (d+e x)+\frac{d \sqrt{a+c x^2} (2 e x-d)}{x^2}}{2 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^2]/(x^3*(d + e*x)),x]

[Out]

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

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Maple [B]  time = 0.017, size = 567, normalized size = 3.5 \[ -{\frac{1}{2\,ad{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{c}{2\,ad}\sqrt{c{x}^{2}+a}}-{\frac{{e}^{2}}{{d}^{3}}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{{e}^{2}}{{d}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{{e}^{2}}{{d}^{3}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}+{\frac{e}{{d}^{2}}\sqrt{c}\ln \left ({1 \left ( -{\frac{cd}{e}}+c \left ( x+{\frac{d}{e}} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) }+{\frac{a{e}^{2}}{{d}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{c}{d}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{{d}^{2}ax} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{cex}{{d}^{2}a}\sqrt{c{x}^{2}+a}}-{\frac{e}{{d}^{2}}\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(1/2)/x^3/(e*x+d),x)

[Out]

-1/2/d/a/x^2*(c*x^2+a)^(3/2)-1/2/d*c/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/
x)+1/2/d*c/a*(c*x^2+a)^(1/2)-1/d^3*e^2*a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2)
)/x)+1/d^3*e^2*(c*x^2+a)^(1/2)-1/d^3*e^2*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d
^2)/e^2)^(1/2)+1/d^2*e*c^(1/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c-2*c*d/
e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))+1/d^3*e^2/((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))*a+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))*c+e/d^2/a/x*(c*x^2+a)^(3/2)-e/d^
2*c/a*x*(c*x^2+a)^(1/2)-e/d^2*c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}}{{\left (e x + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/((e*x + d)*x^3),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)/((e*x + d)*x^3), x)

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Fricas [A]  time = 0.342078, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{a} e x^{2} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) +{\left (c d^{2} + 2 \, a e^{2}\right )} x^{2} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (2 \, d e x - d^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{4 \, \sqrt{a} d^{3} x^{2}}, -\frac{4 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{a} e x^{2} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) -{\left (c d^{2} + 2 \, a e^{2}\right )} x^{2} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (2 \, d e x - d^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{4 \, \sqrt{a} d^{3} x^{2}}, \frac{\sqrt{c d^{2} + a e^{2}} \sqrt{-a} e x^{2} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) -{\left (c d^{2} + 2 \, a e^{2}\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, d e x - d^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} d^{3} x^{2}}, -\frac{2 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{-a} e x^{2} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right ) +{\left (c d^{2} + 2 \, a e^{2}\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (2 \, d e x - d^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} d^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/((e*x + d)*x^3),x, algorithm="fricas")

[Out]

[1/4*(2*sqrt(c*d^2 + a*e^2)*sqrt(a)*e*x^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)) + (c*d^2 + 2*a*e^2)*x^2*log(-((c*x^2 + 2*a)*sqrt(
a) - 2*sqrt(c*x^2 + a)*a)/x^2) + 2*(2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(a))/(sqr
t(a)*d^3*x^2), -1/4*(4*sqrt(-c*d^2 - a*e^2)*sqrt(a)*e*x^2*arctan((c*d*x - a*e)/(
sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) - (c*d^2 + 2*a*e^2)*x^2*log(-((c*x^2 + 2*
a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*(2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(
a))/(sqrt(a)*d^3*x^2), 1/2*(sqrt(c*d^2 + a*e^2)*sqrt(-a)*e*x^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)) - (c*d^2 + 2*a*e^2)*x^2*arct
an(sqrt(-a)/sqrt(c*x^2 + a)) + (2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-
a)*d^3*x^2), -1/2*(2*sqrt(-c*d^2 - a*e^2)*sqrt(-a)*e*x^2*arctan((c*d*x - a*e)/(s
qrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + (c*d^2 + 2*a*e^2)*x^2*arctan(sqrt(-a)/sq
rt(c*x^2 + a)) - (2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*d^3*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}}}{x^{3} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + c*x**2)/(x**3*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.280075, size = 311, normalized size = 1.94 \[ -\frac{2 \,{\left (c d^{2} e + a e^{3}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{\sqrt{-c d^{2} - a e^{2}} d^{3}} + \frac{{\left (c d^{2} + 2 \, a e^{2}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} d^{3}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c d - 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a \sqrt{c} e +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt{c} e}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/((e*x + d)*x^3),x, algorithm="giac")

[Out]

-2*(c*d^2*e + a*e^3)*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)*d^3) + (c*d^2 + 2*a*e^2)*arctan(-(sqrt(c)
*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*d^3) + ((sqrt(c)*x - sqrt(c*x^2 + a))^
3*c*d - 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*sqrt(c)*e + (sqrt(c)*x - sqrt(c*x^2
+ a))*a*c*d + 2*a^2*sqrt(c)*e)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^2*d^2)