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

Optimal. Leaf size=109 \[ -\frac{d^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \sqrt{a e^2+c d^2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^2}+\frac{\sqrt{a+c x^2}}{c e} \]

[Out]

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

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Rubi [A]  time = 0.269357, antiderivative size = 109, 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 \[ -\frac{d^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \sqrt{a e^2+c d^2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^2}+\frac{\sqrt{a+c x^2}}{c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.915, size = 95, normalized size = 0.87 \[ - \frac{d^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2} \sqrt{a e^{2} + c d^{2}}} + \frac{\sqrt{a + c x^{2}}}{c e} - \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.158137, size = 133, normalized size = 1.22 \[ \frac{-\frac{d^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{d^2 \log (d+e x)}{\sqrt{a e^2+c d^2}}-\frac{d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}+\frac{e \sqrt{a+c x^2}}{c}}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 172, normalized size = 1.6 \[{\frac{1}{ce}\sqrt{c{x}^{2}+a}}-{\frac{d}{{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{d}^{2}}{{e}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2+a)^(1/2)/c/e-1/e^2*d*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-d^2/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*(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^2/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.402016, size = 1, normalized size = 0.01 \[ \left [\frac{c^{\frac{3}{2}} d^{2} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + \sqrt{c d^{2} + a e^{2}} c d \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} \sqrt{c} e}{2 \, \sqrt{c d^{2} + a e^{2}} c^{\frac{3}{2}} e^{2}}, \frac{2 \, c^{\frac{3}{2}} d^{2} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) + \sqrt{-c d^{2} - a e^{2}} c d \log \left (2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \, \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} \sqrt{c} e}{2 \, \sqrt{-c d^{2} - a e^{2}} c^{\frac{3}{2}} e^{2}}, \frac{\sqrt{-c} c d^{2} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt{c d^{2} + a e^{2}} c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + 2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} \sqrt{-c} e}{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{-c} c e^{2}}, \frac{\sqrt{-c} c d^{2} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right ) - \sqrt{-c d^{2} - a e^{2}} c d \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} \sqrt{-c} e}{\sqrt{-c d^{2} - a e^{2}} \sqrt{-c} c e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275943, size = 142, normalized size = 1.3 \[ \frac{2 \, d^{2} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{d e^{\left (-2\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2} + a} e^{\left (-1\right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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