3.1911 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=232 \[ \frac{3 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{5/2} d^{5/2} e^{5/2}}-\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c^2 d^2 e^2}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e} \]

[Out]

(-3*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(64*c^2*d^2*e^2) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(3/2))/(8*c*d*e) + (3*(c*d^2 - a*e^2)^4*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])])/(128*c^(5/2)*d^(5/2)*e^(5/2))

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Rubi [A]  time = 0.233575, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{3 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{5/2} d^{5/2} e^{5/2}}-\frac{3 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c^2 d^2 e^2}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(64*c^2*d^2*e^2) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(3/2))/(8*c*d*e) + (3*(c*d^2 - a*e^2)^4*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])])/(128*c^(5/2)*d^(5/2)*e^(5/2))

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Rubi in Sympy [A]  time = 31.7866, size = 223, normalized size = 0.96 \[ \frac{\left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{8 c d e} - \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 c^{2} d^{2} e^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{4} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{128 c^{\frac{5}{2}} d^{\frac{5}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/
(8*c*d*e) - 3*(a*e**2 - c*d**2)**2*(a*e**2 + c*d**2 + 2*c*d*e*x)*sqrt(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))/(64*c**2*d**2*e**2) + 3*(a*e**2 - c*d**2)**4*ata
nh((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))))/(128*c**(5/2)*d**(5/2)*e**(5/2))

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Mathematica [A]  time = 0.543209, size = 253, normalized size = 1.09 \[ \frac{1}{128} ((d+e x) (a e+c d x))^{3/2} \left (\frac{-6 a^3 e^6+2 a^2 c d e^4 (11 d+2 e x)+2 a c^2 d^2 e^2 \left (11 d^2+44 d e x+24 e^2 x^2\right )+c^3 \left (-6 d^6+4 d^5 e x+48 d^4 e^2 x^2+32 d^3 e^3 x^3\right )}{c^2 d^2 e^2 (d+e x) (a e+c d x)}+\frac{3 \left (c d^2-a e^2\right )^4 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{5/2} d^{5/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((-6*a^3*e^6 + 2*a^2*c*d*e^4*(11*d + 2*e*x) + 2
*a*c^2*d^2*e^2*(11*d^2 + 44*d*e*x + 24*e^2*x^2) + c^3*(-6*d^6 + 4*d^5*e*x + 48*d
^4*e^2*x^2 + 32*d^3*e^3*x^3))/(c^2*d^2*e^2*(a*e + c*d*x)*(d + e*x)) + (3*(c*d^2
- a*e^2)^4*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)*e^(5/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2
))))/128

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Maple [B]  time = 0.008, size = 671, normalized size = 2.9 \[{\frac{2\,cdex+a{e}^{2}+c{d}^{2}}{8\,dec} \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{3}x{a}^{2}}{32\,cd}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,adex}{16}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,c{d}^{3}x}{32\,e}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,{e}^{4}{a}^{3}}{64\,{c}^{2}{d}^{2}}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,{a}^{2}{e}^{2}}{64\,c}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,a{d}^{2}}{64}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,c{d}^{4}}{64\,{e}^{2}}\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,{a}^{4}{e}^{6}}{128\,{c}^{2}{d}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}-{\frac{3\,{e}^{4}{a}^{3}}{32\,c}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{9\,{d}^{2}{e}^{2}{a}^{2}}{64}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}-{\frac{3\,c{d}^{4}a}{32}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{3\,{d}^{6}{c}^{2}}{128\,{e}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-3/32/d
*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+3/16*d*e*(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x*a-3/32*d^3/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x
-3/64/d^2*e^4/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+3/64*e^2/c*(a*e*d+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+3/64*d^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*a-3/64*d^4/e^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/128/d^2*e^6/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^4-3/32*e^4/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^3+9/64*d^2*e^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^2-3/32*d^4*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+3/128*d^6/e^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)

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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((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 = 0.247085, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (16 \, c^{3} d^{3} e^{3} x^{3} - 3 \, c^{3} d^{6} + 11 \, a c^{2} d^{4} e^{2} + 11 \, a^{2} c d^{2} e^{4} - 3 \, a^{3} e^{6} + 24 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} + 3 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{256 \, \sqrt{c d e} c^{2} d^{2} e^{2}}, \frac{2 \,{\left (16 \, c^{3} d^{3} e^{3} x^{3} - 3 \, c^{3} d^{6} + 11 \, a c^{2} d^{4} e^{2} + 11 \, a^{2} c d^{2} e^{4} - 3 \, a^{3} e^{6} + 24 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} + 3 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{128 \, \sqrt{-c d e} c^{2} d^{2} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(4*(16*c^3*d^3*e^3*x^3 - 3*c^3*d^6 + 11*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4
- 3*a^3*e^6 + 24*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 2*(c^3*d^5*e + 22*a*c^2*d^3
*e^3 + a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) +
3*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*l
og(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)))/(sqrt(c*d*e)*c^2*d^2*e^2), 1/128*(2*(16*c^3*d^3
*e^3*x^3 - 3*c^3*d^6 + 11*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 3*a^3*e^6 + 24*(c^3
*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 2*(c^3*d^5*e + 22*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) + 3*(c^4*d^8 - 4*a*c^
3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*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*e^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.253765, size = 414, normalized size = 1.78 \[ \frac{1}{64} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \, c d x e + \frac{3 \,{\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x + \frac{{\left (c^{4} d^{6} e^{2} + 22 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x - \frac{{\left (3 \, c^{4} d^{7} e - 11 \, a c^{3} d^{5} e^{3} - 11 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} - \frac{3 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{128 \, c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*c*d*x*e + 3*(c^4*d^5*e
^3 + a*c^3*d^3*e^5)*e^(-3)/(c^3*d^3))*x + (c^4*d^6*e^2 + 22*a*c^3*d^4*e^4 + a^2*
c^2*d^2*e^6)*e^(-3)/(c^3*d^3))*x - (3*c^4*d^7*e - 11*a*c^3*d^5*e^3 - 11*a^2*c^2*
d^3*e^5 + 3*a^3*c*d*e^7)*e^(-3)/(c^3*d^3)) - 3/128*(c^4*d^8 - 4*a*c^3*d^6*e^2 +
6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(c*d)*e^(-5/2)*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 + c*d^2*x + a*x*e^2
+ a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^3*d^3)