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

Optimal. Leaf size=345 \[ -\frac{\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac{1}{24} x^2 \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac{x^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e} \]

[Out]

((a/(c*d) - (7*d)/e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/24 + (x^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) - ((105*c^3*d^6 - 25*a*c^2*
d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^4 - 6*a*c*d^2*e^2 -
5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*c^3*d^3*e^4) + (
(c*d^2 - a*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*Ar
cTanh[(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^(7/2)*d^(7/2)*e^(9/2))

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Rubi [A]  time = 1.35247, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac{1}{24} x^2 \left (\frac{a}{c d}-\frac{7 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac{x^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

((a/(c*d) - (7*d)/e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/24 + (x^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) - ((105*c^3*d^6 - 25*a*c^2*
d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^4 - 6*a*c*d^2*e^2 -
5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*c^3*d^3*e^4) + (
(c*d^2 - a*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*Ar
cTanh[(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^(7/2)*d^(7/2)*e^(9/2))

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Rubi in Sympy [A]  time = 144.696, size = 348, normalized size = 1.01 \[ - x^{2} \left (- \frac{a}{24 c d} + \frac{7 d}{24 e^{2}}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} + \frac{x^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e} + \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (\frac{15 a^{3} e^{6}}{8} + \frac{17 a^{2} c d^{2} e^{4}}{8} + \frac{25 a c^{2} d^{4} e^{2}}{8} - \frac{105 c^{3} d^{6}}{8} - \frac{c d e x \left (5 a^{2} e^{4} + 6 a c d^{2} e^{2} - 35 c^{2} d^{4}\right )}{4}\right )}{24 c^{3} d^{3} e^{4}} - \frac{\left (a e^{2} - c d^{2}\right ) \left (5 a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} + 15 a c^{2} d^{4} e^{2} + 35 c^{3} d^{6}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{128 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)

[Out]

-x**2*(-a/(24*c*d) + 7*d/(24*e**2))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2
)) + x**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*e) + sqrt(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))*(15*a**3*e**6/8 + 17*a**2*c*d**2*e**4/8 + 25*a*c
**2*d**4*e**2/8 - 105*c**3*d**6/8 - c*d*e*x*(5*a**2*e**4 + 6*a*c*d**2*e**2 - 35*
c**2*d**4)/4)/(24*c**3*d**3*e**4) - (a*e**2 - c*d**2)*(5*a**3*e**6 + 9*a**2*c*d*
*2*e**4 + 15*a*c**2*d**4*e**2 + 35*c**3*d**6)*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))))/(1
28*c**(7/2)*d**(7/2)*e**(9/2))

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Mathematica [A]  time = 0.497908, size = 276, normalized size = 0.8 \[ \frac{1}{384} \sqrt{(d+e x) (a e+c d x)} \left (\frac{30 a^3 e^2}{c^3 d^3}+\frac{4 x \left (-\frac{5 a^2 e^4}{c^2 d^2}-\frac{6 a e^2}{c}+35 d^2\right )}{e^3}+\frac{34 a^2}{c^2 d}+\frac{3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{9/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{16 x^2 \left (\frac{a}{c}-\frac{7 d^2}{e^2}\right )}{d}+\frac{50 a d}{c e^2}-\frac{210 d^3}{e^4}+\frac{96 x^3}{e}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((34*a^2)/(c^2*d) - (210*d^3)/e^4 + (50*a*d)/(c*e
^2) + (30*a^3*e^2)/(c^3*d^3) + (4*(35*d^2 - (6*a*e^2)/c - (5*a^2*e^4)/(c^2*d^2))
*x)/e^3 + (16*(a/c - (7*d^2)/e^2)*x^2)/d + (96*x^3)/e + (3*(c*d^2 - a*e^2)*(35*c
^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*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^(7/2)*d^(7
/2)*e^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/384

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Maple [B]  time = 0.031, size = 946, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)

[Out]

19/64/c^2/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+29/32*d^2/e^3*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/32*e/c^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)*x*a^2-5/128*e^4/c^3/d^3*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^4-1/32*e^2/c^2*a^3/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)+1/4/e^2*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-5/24/e/c
^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+7/16/e/c*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x*a+5/64*e^2/c^3/d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
^3-13/24/e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+29/64*d^3/e^4*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/64*d/c*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^2+11/32*d^3/e^2*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-29/128*d^5/e^4*c*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)+43/64*d/e^2/c*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-d^3/e^4*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d
/e))^(1/2)-1/2*d^3/e^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d
*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)*a+1/2*d^5/e^4*ln((1/2*a
*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/
e))^(1/2))/(c*d*e)^(1/2)*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.375192, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} - 105 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} + 17 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} - 8 \,{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} - 3 \,{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{768 \, \sqrt{c d e} c^{3} d^{3} e^{4}}, \frac{2 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} - 105 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} + 17 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} - 8 \,{\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} + 3 \,{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{384 \, \sqrt{-c d e} c^{3} d^{3} e^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*c^3*d^3*e^3*x^3 - 105*c^3*d^6 + 25*a*c^2*d^4*e^2 + 17*a^2*c*d^2*e^
4 + 15*a^3*e^6 - 8*(7*c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(35*c^3*d^5*e - 6*a*c
^2*d^3*e^3 - 5*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*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6
- 5*a^4*e^8)*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^4), 1/384
*(2*(48*c^3*d^3*e^3*x^3 - 105*c^3*d^6 + 25*a*c^2*d^4*e^2 + 17*a^2*c*d^2*e^4 + 15
*a^3*e^6 - 8*(7*c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 2*(35*c^3*d^5*e - 6*a*c^2*d^3
*e^3 - 5*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*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*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^3*d^3*e^4)]

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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(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)

[Out]

Timed out

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

[Out]

Exception raised: TypeError