3.460 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=274 \[ -\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]

[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^2*d^2*e^3) + ((a/(c*d) - d/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 + (a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(5/2)/(5*e) - (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])])/(2
56*c^(5/2)*d^(5/2)*e^(7/2))

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Rubi [A]  time = 0.377339, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 59.8112, size = 262, normalized size = 0.96 \[ - \left (- \frac{a}{16 c d} + \frac{d}{16 e^{2}}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} + \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 e} - \frac{3 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{128 c^{2} d^{2} e^{3}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{5} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{256 c^{\frac{5}{2}} d^{\frac{5}{2}} e^{\frac{7}{2}}} \]

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)**(5/2)/(e*x+d),x)

[Out]

-(-a/(16*c*d) + d/(16*e**2))*(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*e) - 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**2*d**2*e**3) + 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**(5/2)*d**(5/2)*e**(7/2))

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Mathematica [A]  time = 0.704413, size = 309, normalized size = 1.13 \[ \frac{1}{256} ((d+e x) (a e+c d x))^{3/2} \left (\frac{-30 a^4 e^8+20 a^3 c d e^6 (7 d+e x)+4 a^2 c^2 d^2 e^4 \left (64 d^2+233 d e x+124 e^2 x^2\right )+4 a c^3 d^3 e^2 \left (-35 d^3+23 d^2 e x+256 d e^2 x^2+168 e^3 x^3\right )+2 c^4 d^4 \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )}{5 c^2 d^2 e^3 (d+e x) (a e+c d x)}-\frac{3 \left (c d^2-a e^2\right )^5 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{5/2} d^{5/2} e^{7/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x),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 + 233*d*e*x + 124*e^2*x^2) + 4*a*c^3*d^3*e^2*(-35*d^3 +
23*d^2*e*x + 256*d*e^2*x^2 + 168*e^3*x^3) + 2*c^4*d^4*(15*d^4 - 10*d^3*e*x + 8*d
^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4))/(5*c^2*d^2*e^3*(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^(5/2)*d^(5/2)*e^(7/2)*(a*e + c*d*x)^(3/2)
*(d + e*x)^(3/2))))/256

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Maple [B]  time = 0.011, size = 1123, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),x)

[Out]

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

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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)^(5/2)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.321004, 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)^(5/2)/(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*(11*c^4*d^5*e^3 + 21*a*c^3*d^3*e^5)*x
^3 + 8*(c^4*d^6*e^2 + 64*a*c^3*d^4*e^4 + 31*a^2*c^2*d^2*e^6)*x^2 - 2*(5*c^4*d^7*
e - 23*a*c^3*d^5*e^3 - 233*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^2*d^2*e^3), 1/1280*(2*(128*c^4*d^4*e^4*x^4 + 1
5*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*(11*c^4*d^5*e^3 + 21*a*c^3*d^3*e^5)*x^3 + 8*(c^4*d^6*e^2 + 64*a*c^3*d^4*
e^4 + 31*a^2*c^2*d^2*e^6)*x^2 - 2*(5*c^4*d^7*e - 23*a*c^3*d^5*e^3 - 233*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^2*d^2*e^
3)]

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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)**(5/2)/(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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError