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

Optimal. Leaf size=103 \[ -\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2}-\frac{\sqrt{c} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^2}+\frac{\sqrt{a+c x^2}}{e} \]

[Out]

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

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Rubi [A]  time = 0.235485, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{\sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2}-\frac{\sqrt{c} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^2}+\frac{\sqrt{a+c x^2}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.743, size = 90, normalized size = 0.87 \[ - \frac{\sqrt{c} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{2}} + \frac{\sqrt{a + c x^{2}}}{e} - \frac{\sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.102979, size = 124, normalized size = 1.2 \[ \frac{-\sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+\sqrt{a e^2+c d^2} \log (d+e x)-\sqrt{c} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+e \sqrt{a+c x^2}}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 381, normalized size = 3.7 \[{\frac{1}{e}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}-{\frac{d}{{e}^{2}}\sqrt{c}\ln \left ({1 \left ( -{\frac{cd}{e}}+c \left ({\frac{d}{e}}+x \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) }-{\frac{a}{e}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{c{d}^{2}}{{e}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(1/2)/(e*x+d),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325057, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c} d \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \, \sqrt{c x^{2} + a} e + \sqrt{c d^{2} + a e^{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, e^{2}}, -\frac{2 \, \sqrt{-c} d \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 2 \, \sqrt{c x^{2} + a} e - \sqrt{c d^{2} + a e^{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, e^{2}}, \frac{\sqrt{c} d \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \, \sqrt{c x^{2} + a} e + 2 \, \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right )}{2 \, e^{2}}, -\frac{\sqrt{-c} d \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - \sqrt{c x^{2} + a} e - \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{c d x - a e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}}\right )}{e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223001, size = 147, normalized size = 1.43 \[ \sqrt{c} d e^{\left (-2\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \,{\left (c d^{2} + a e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \sqrt{c x^{2} + a} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*d*e^(-2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 2*(c*d^2 + a*e^2)*arcta
n(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))*e^(-2)/sq
rt(-c*d^2 - a*e^2) + sqrt(c*x^2 + a)*e^(-1)