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

Optimal. Leaf size=195 \[ -\frac{\left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} e^{5/2}}+\frac{2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x) \left (c d^2-a e^2\right )}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d e^2} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(c*d*e^2) + (2*d^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(c*d^2 - a*e^2)*(d + e*x)) - ((3*c*d^2 + a*e^2)
*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^(3/2)*d^(3/2)*e^(5/2))

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Rubi [A]  time = 0.68325, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} e^{5/2}}+\frac{2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x) \left (c d^2-a e^2\right )}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d e^2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(c*d*e^2) + (2*d^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(c*d^2 - a*e^2)*(d + e*x)) - ((3*c*d^2 + a*e^2)
*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^(3/2)*d^(3/2)*e^(5/2))

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Rubi in Sympy [A]  time = 67.3713, size = 262, normalized size = 1.34 \[ - \frac{2 d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{2} \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )} + \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c d e^{2}} - \frac{\sqrt{d} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{\sqrt{c} e^{\frac{5}{2}}} - \frac{\left (a e^{2} + c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 c^{\frac{3}{2}} d^{\frac{3}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(e**2*(d + e*x)*(a*e**2 -
c*d**2)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(c*d*e**2) - sqrt(d)*
atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*
e*x**2 + x*(a*e**2 + c*d**2))))/(sqrt(c)*e**(5/2)) - (a*e**2 + c*d**2)*atanh((a*
e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))))/(2*c**(3/2)*d**(3/2)*e**(5/2))

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Mathematica [A]  time = 0.327766, size = 178, normalized size = 0.91 \[ \frac{\frac{2 (d+e x) (a e+c d x) \left (\frac{2 d^3}{(d+e x) \left (c d^2-a e^2\right )}+\frac{1}{c}\right )}{d e^2}-\frac{\sqrt{d+e x} \left (a e^2+3 c d^2\right ) \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{3/2} d^{3/2} e^{5/2}}}{2 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

((2*(a*e + c*d*x)*(d + e*x)*(c^(-1) + (2*d^3)/((c*d^2 - a*e^2)*(d + e*x))))/(d*e
^2) - ((3*c*d^2 + a*e^2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Log[a*e^2 + 2*Sqrt[c]*S
qrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(3/2)*d^(3
/2)*e^(5/2)))/(2*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.017, size = 241, normalized size = 1.2 \[{\frac{1}{cd{e}^{2}}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{a}{2\,cd}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}-{\frac{3\,d}{2\,{e}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}-2\,{\frac{{d}^{2}}{{e}^{3} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.433189, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt{c d e} +{\left (3 \, c^{2} d^{5} - 2 \, a c d^{3} e^{2} - a^{2} d e^{4} +{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{4 \,{\left (c^{2} d^{4} e^{2} - a c d^{2} e^{4} +{\left (c^{2} d^{3} e^{3} - a c d e^{5}\right )} x\right )} \sqrt{c d e}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt{-c d e} -{\left (3 \, c^{2} d^{5} - 2 \, a c d^{3} e^{2} - a^{2} d e^{4} +{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{2 \,{\left (c^{2} d^{4} e^{2} - a c d^{2} e^{4} +{\left (c^{2} d^{3} e^{3} - a c d e^{5}\right )} x\right )} \sqrt{-c d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d^3 - a*d*e^2 + (c*d^2*
e - a*e^3)*x)*sqrt(c*d*e) + (3*c^2*d^5 - 2*a*c*d^3*e^2 - a^2*d*e^4 + (3*c^2*d^4*
e - 2*a*c*d^2*e^3 - a^2*e^5)*x)*log(-4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*
a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d*e)))/((c^2*d^4*e^2
- a*c*d^2*e^4 + (c^2*d^3*e^3 - a*c*d*e^5)*x)*sqrt(c*d*e)), 1/2*(2*sqrt(c*d*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-c
*d*e) - (3*c^2*d^5 - 2*a*c*d^3*e^2 - a^2*d*e^4 + (3*c^2*d^4*e - 2*a*c*d^2*e^3 -
a^2*e^5)*x)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/((c^2*d^4*e^2 - a*c*d^2*e^4 + (c^2*d^3*e^3
- a*c*d*e^5)*x)*sqrt(-c*d*e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError