3.1910 \(\int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=283 \[ \frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}}-\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 c d} \]

[Out]

(-3*(c*d^2 - a*e^2)^3*(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])/(128*c^3*d^3*e^2) + ((c*d^2 - a*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))/(16*c^2*d^2*e) + (a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(5/2)/(5*c*d) + (3*(c*d^2 - a*e^2)^5*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])])/(256*c^(7/2)*d^(7/2)*e^(5/2))

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Rubi [A]  time = 0.397723, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}}-\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(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)^3*(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])/(128*c^3*d^3*e^2) + ((c*d^2 - a*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))/(16*c^2*d^2*e) + (a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(5/2)/(5*c*d) + (3*(c*d^2 - a*e^2)^5*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])])/(256*c^(7/2)*d^(7/2)*e^(5/2))

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Rubi in Sympy [A]  time = 54.2571, size = 269, normalized size = 0.95 \[ \left (- \frac{a e}{16 c^{2} d^{2}} + \frac{1}{16 c e}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} + \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 c d} + \frac{3 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{128 c^{3} d^{3} e^{2}} - \frac{3 \left (a e^{2} - c d^{2}\right )^{5} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{256 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a*e/(16*c**2*d**2) + 1/(16*c*e))*(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) + (a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*
*(5/2)/(5*c*d) + 3*(a*e**2 - c*d**2)**3*(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))/(128*c**3*d**3*e**2) - 3*(a*e**2 - c*d**2)*
*5*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))))/(256*c**(7/2)*d**(7/2)*e**(5/2))

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Mathematica [A]  time = 0.734058, size = 309, normalized size = 1.09 \[ \frac{1}{256} ((d+e x) (a e+c d x))^{3/2} \left (\frac{30 a^4 e^8-20 a^3 c d e^6 (7 d+e x)+4 a^2 c^2 d^2 e^4 \left (64 d^2+23 d e x+4 e^2 x^2\right )+4 a c^3 d^3 e^2 \left (35 d^3+233 d^2 e x+256 d e^2 x^2+88 e^3 x^3\right )+2 c^4 d^4 \left (-15 d^4+10 d^3 e x+248 d^2 e^2 x^2+336 d e^3 x^3+128 e^4 x^4\right )}{5 c^3 d^3 e^2 (d+e x) (a e+c d x)}+\frac{3 \left (c d^2-a e^2\right )^5 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(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)*((30*a^4*e^8 - 20*a^3*c*d*e^6*(7*d + e*x) + 4*a
^2*c^2*d^2*e^4*(64*d^2 + 23*d*e*x + 4*e^2*x^2) + 4*a*c^3*d^3*e^2*(35*d^3 + 233*d
^2*e*x + 256*d*e^2*x^2 + 88*e^3*x^3) + 2*c^4*d^4*(-15*d^4 + 10*d^3*e*x + 248*d^2
*e^2*x^2 + 336*d*e^3*x^3 + 128*e^4*x^4))/(5*c^3*d^3*e^2*(a*e + c*d*x)*(d + e*x))
+ (3*(c*d^2 - a*e^2)^5*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^(7/2)*d^(7/2)*e^(5/2)*(a*e + c*d*x)^(3/2)*(
d + e*x)^(3/2))))/256

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Maple [B]  time = 0.015, size = 917, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d-1/16/d^2*e^3/c^2*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^2+3/256*d^7/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)-3/128*d^5/e
^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/128/d^3*e^6/c^3*(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*a^4+15/256/d*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^4-15/128*d*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+1/16*d^2/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2
)+3/64*d^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1/8*d*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(3/2)*x-9/64*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+9/
64*d^2*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-3/64*d^4/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^5/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)*x*a^3-3/256/d^3*e^8/c^3*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^5-1/8/d*e^2/c*(a*e*d+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-3/64/d*e^4/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*a^3+15/128*d^3*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^2-15/256*d^5*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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26146, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)*(e*x + d),x, algorithm="fricas")

[Out]

[1/2560*(4*(128*c^4*d^4*e^4*x^4 - 15*c^4*d^8 + 70*a*c^3*d^6*e^2 + 128*a^2*c^2*d^
4*e^4 - 70*a^3*c*d^2*e^6 + 15*a^4*e^8 + 16*(21*c^4*d^5*e^3 + 11*a*c^3*d^3*e^5)*x
^3 + 8*(31*c^4*d^6*e^2 + 64*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(5*c^4*d^7*
e + 233*a*c^3*d^5*e^3 + 23*a^2*c^2*d^3*e^5 - 5*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2
*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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)))/(sqrt(c*d*e)*c^3*d^3*e^2), 1/1280*(2*(128*c^4*d^4*e^4*x^4 - 15
*c^4*d^8 + 70*a*c^3*d^6*e^2 + 128*a^2*c^2*d^4*e^4 - 70*a^3*c*d^2*e^6 + 15*a^4*e^
8 + 16*(21*c^4*d^5*e^3 + 11*a*c^3*d^3*e^5)*x^3 + 8*(31*c^4*d^6*e^2 + 64*a*c^3*d^
4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(5*c^4*d^7*e + 233*a*c^3*d^5*e^3 + 23*a^2*c^2*d
^3*e^5 - 5*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d
*e) + 15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 +
5*a^4*c*d^2*e^8 - a^5*e^10)*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^3*d^3*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((e*x+d)*(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 = 0.254326, size = 536, normalized size = 1.89 \[ \frac{1}{640} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c d x e^{2} + \frac{{\left (21 \, c^{5} d^{6} e^{5} + 11 \, a c^{4} d^{4} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (31 \, c^{5} d^{7} e^{4} + 64 \, a c^{4} d^{5} e^{6} + a^{2} c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (5 \, c^{5} d^{8} e^{3} + 233 \, a c^{4} d^{6} e^{5} + 23 \, a^{2} c^{3} d^{4} e^{7} - 5 \, a^{3} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac{{\left (15 \, c^{5} d^{9} e^{2} - 70 \, a c^{4} d^{7} e^{4} - 128 \, a^{2} c^{3} d^{5} e^{6} + 70 \, a^{3} c^{2} d^{3} e^{8} - 15 \, a^{4} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} - \frac{3 \,{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{256 \, c^{4} d^{4}} \]

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)*(e*x + d),x, algorithm="giac")

[Out]

1/640*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*c*d*x*e^2 + (21*c^
5*d^6*e^5 + 11*a*c^4*d^4*e^7)*e^(-4)/(c^4*d^4))*x + (31*c^5*d^7*e^4 + 64*a*c^4*d
^5*e^6 + a^2*c^3*d^3*e^8)*e^(-4)/(c^4*d^4))*x + (5*c^5*d^8*e^3 + 233*a*c^4*d^6*e
^5 + 23*a^2*c^3*d^4*e^7 - 5*a^3*c^2*d^2*e^9)*e^(-4)/(c^4*d^4))*x - (15*c^5*d^9*e
^2 - 70*a*c^4*d^7*e^4 - 128*a^2*c^3*d^5*e^6 + 70*a^3*c^2*d^3*e^8 - 15*a^4*c*d*e^
10)*e^(-4)/(c^4*d^4)) - 3/256*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 -
10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(c*d)*e^(-5/2)*ln(abs(-sqr
t(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^4*d^4)