3.1946 \(\int \frac{(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{\sqrt{e} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2}}-\frac{2 (d+e x)}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.16767, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{\sqrt{e} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2}}-\frac{2 (d+e x)}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.2979, size = 121, normalized size = 0.97 \[ - \frac{2 \left (d + e x\right )}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{\sqrt{e} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}}} \]

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)**(3/2),x)

[Out]

-2*(d + e*x)/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + sqrt(e)*atan
h((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))))/(c**(3/2)*d**(3/2))

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Mathematica [A]  time = 0.181868, size = 128, normalized size = 1.02 \[ \frac{\sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )-2 \sqrt{c} \sqrt{d} (d+e x)}{c^{3/2} d^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.013, size = 717, normalized size = 5.7 \[ 2\,{\frac{{d}^{2} \left ( 2\,cdex+a{e}^{2}+c{d}^{2} \right ) }{ \left ( 4\,ac{d}^{2}{e}^{2}- \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}-{\frac{ex}{cd}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}+{\frac{a{e}^{2}}{2\,{c}^{2}{d}^{2}}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-{\frac{3}{2\,c}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}+{\frac{{e}^{5}x{a}^{2}}{cd \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-2\,{\frac{{e}^{3}dxa}{ \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) \sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}-3\,{\frac{c{d}^{3}ex}{ \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) \sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}+{\frac{{a}^{3}{e}^{6}}{2\,{c}^{2}{d}^{2} \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-{\frac{{a}^{2}{e}^{4}}{2\,c \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-{\frac{5\,a{d}^{2}{e}^{2}}{-2\,{a}^{2}{e}^{4}+4\,ac{d}^{2}{e}^{2}-2\,{c}^{2}{d}^{4}}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-{\frac{3\,c{d}^{4}}{-2\,{a}^{2}{e}^{4}+4\,ac{d}^{2}{e}^{2}-2\,{c}^{2}{d}^{4}}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}+{\frac{e}{cd}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}} \]

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)^(3/2),x)

[Out]

2*d^2*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)-e*x/d/c/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2*e^2/d^
2/c^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-3/2/c/(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)+e^5/d/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)*x*a^2-2*e^3*d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x*a-3*e*d^3*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/2*e^6/d^2/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/
(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-1/2*e^4/c/(-a^2*e^4+2*a*c*d^2*e^2-c^
2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/2*e^2*d^2/(-a^2*e^4+2*a*c*d
^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-3/2*d^4*c/(-a^2*e^4+2*
a*c*d^2*e^2-c^2*d^4)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+e/d/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)

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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/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308338, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c d x + a e\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{2 \,{\left (c^{2} d^{2} x + a c d e\right )}}, \frac{{\left (c d x + a e\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{c^{2} d^{2} x + a c d e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((d + e*x)**2/((d + e*x)*(a*e + c*d*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.259404, size = 309, normalized size = 2.47 \[ -\frac{2 \,{\left (\frac{{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}{c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}} + \frac{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4}}{c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}}\right )}}{\sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{\sqrt{c d} e^{\frac{1}{2}}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")

[Out]

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