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

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

[Out]

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

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Rubi [A]  time = 0.126241, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e \sqrt{a e^2+c d^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.9739, size = 75, normalized size = 0.87 \[ \frac{d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e \sqrt{a e^{2} + c d^{2}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.100612, size = 111, normalized size = 1.29 \[ \frac{\frac{d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{d \log (d+e x)}{\sqrt{a e^2+c d^2}}+\frac{\log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 151, normalized size = 1.8 \[{\frac{1}{e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{d}{{e}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.349945, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c} d \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + \sqrt{c d^{2} + a e^{2}} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c} e}, -\frac{2 \, \sqrt{c} d \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) - \sqrt{-c d^{2} - a e^{2}} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{c} e}, \frac{\sqrt{-c} d \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{c d^{2} + a e^{2}} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{-c} e}, -\frac{\sqrt{-c} d \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) - \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{\sqrt{-c d^{2} - a e^{2}} \sqrt{-c} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27681, size = 119, normalized size = 1.38 \[ -\frac{2 \, d \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-1\right )}}{\sqrt{-c d^{2} - a e^{2}}} - \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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