3.1936 \(\int \frac{(d+e x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} \sqrt{e}}+\frac{3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d} \]

[Out]

(3*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^2*d^2) + ((
d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c*d) + (3*(c*d^2 - a*e^
2)^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])])/(8*c^(5/2)*d^(5/2)*Sqrt[e])

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Rubi [A]  time = 0.328844, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} \sqrt{e}}+\frac{3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d} \]

Antiderivative was successfully verified.

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

[Out]

(3*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^2*d^2) + ((
d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c*d) + (3*(c*d^2 - a*e^
2)^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])])/(8*c^(5/2)*d^(5/2)*Sqrt[e])

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Rubi in Sympy [A]  time = 43.2105, size = 184, normalized size = 0.94 \[ \frac{\left (d + e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 c d} - \frac{3 \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 c^{2} d^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 c^{\frac{5}{2}} d^{\frac{5}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*c*d) - 3*(a*e**2 - c
*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*c**2*d**2) + 3*(a*e**2
- c*d**2)**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))))/(8*c**(5/2)*d**(5/2)*sqrt(e))

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Mathematica [A]  time = 0.229891, size = 168, normalized size = 0.86 \[ \frac{\frac{3 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{5/2} d^{5/2} \sqrt{e}}+\frac{2 (d+e x) (a e+c d x) \left (c d (5 d+2 e x)-3 a e^2\right )}{c^2 d^2}}{8 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/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)*(-3*a*e^2 + c*d*(5*d + 2*e*x)))/(c^2*d^2) + (3*(c*d^
2 - a*e^2)^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[
e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(5/2)*d^(5/2)*Sqrt[e])
)/(8*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.015, size = 318, normalized size = 1.6 \[{\frac{3\,{d}^{2}}{8}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{ex}{2\,cd}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,a{e}^{2}}{4\,{c}^{2}{d}^{2}}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{5}{4\,c}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,{a}^{2}{e}^{4}}{8\,{c}^{2}{d}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}-{\frac{3\,a{e}^{2}}{4\,c}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*d^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2))/(d*e*c)^(1/2)+1/2*e*x/d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)-3/4*e^2/d^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+5/4/c*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)+3/8*e^4/d^2/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*
c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^2-3/4*e^2/c*ln
((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2))/(d*e*c)^(1/2)*a

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261253, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + 5 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{c d e} + 3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{16 \, \sqrt{c d e} c^{2} d^{2}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + 5 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt{-c d e} + 3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{8 \, \sqrt{-c d e} c^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + 5*c*d^2 - 3*a*
e^2)*sqrt(c*d*e) + 3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*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)*sqr
t(c*d*e)))/(sqrt(c*d*e)*c^2*d^2), 1/8*(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2
)*x)*(2*c*d*e*x + 5*c*d^2 - 3*a*e^2)*sqrt(-c*d*e) + 3*(c^2*d^4 - 2*a*c*d^2*e^2 +
a^2*e^4)*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)))/(sqrt(-c*d*e)*c^2*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272244, size = 238, normalized size = 1.22 \[ \frac{1}{4} \, \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (\frac{2 \, x e}{c d} + \frac{{\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} e^{\left (-1\right )}}{c^{2} d^{2}}\right )} - \frac{3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{8 \, c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)*(2*x*e/(c*d) + (5*c*d^2*e - 3*a*
e^3)*e^(-1)/(c^2*d^2)) - 3/8*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d)*e^(-1
/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e +
a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^3*d^3)