3.1926 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^3} \]

[Out]

(5*(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*d*e^3) + (5*(a - (c*d^2)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2))/24 + ((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2))/(4*e) - (5*(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^(3/2)*d^(3
/2)*e^(7/2))

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Rubi [A]  time = 0.559384, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{7/2}}+\frac{5}{24} \left (a-\frac{c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^3} \]

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)^2,x]

[Out]

(5*(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*d*e^3) + (5*(a - (c*d^2)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2))/24 + ((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2))/(4*e) - (5*(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^(3/2)*d^(3
/2)*e^(7/2))

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Rubi in Sympy [A]  time = 91.1189, size = 252, normalized size = 0.97 \[ \frac{\left (a e + c d x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 e} + \frac{5 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{24 e^{2}} + \frac{5 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 c d e^{3}} - \frac{5 \left (a e^{2} - c d^{2}\right )^{4} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{128 c^{\frac{3}{2}} d^{\frac{3}{2}} e^{\frac{7}{2}}} \]

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)**2,x)

[Out]

(a*e + c*d*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(4*e) + 5*(a*e**
2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(24*e**2) + 5*(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*d*e**3) - 5*(a*e**2 - c*d**2)**4*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
))))/(128*c**(3/2)*d**(3/2)*e**(7/2))

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Mathematica [A]  time = 0.44232, size = 234, normalized size = 0.9 \[ \frac{1}{384} \sqrt{(d+e x) (a e+c d x)} \left (\frac{30 a^3 e^3}{c d}+4 x \left (59 a^2 e^2+18 a c d^2-\frac{5 c^2 d^4}{e^2}\right )+146 a^2 d e-\frac{15 \left (c d^2-a e^2\right )^4 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{3/2} d^{3/2} e^{7/2} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{110 a c d^3}{e}+\frac{16 c d x^2 \left (17 a e^2+c d^2\right )}{e}+\frac{30 c^2 d^5}{e^3}+96 c^2 d^2 x^3\right ) \]

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)^2,x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((30*c^2*d^5)/e^3 - (110*a*c*d^3)/e + 146*a^2*d*e
+ (30*a^3*e^3)/(c*d) + 4*(18*a*c*d^2 - (5*c^2*d^4)/e^2 + 59*a^2*e^2)*x + (16*c*
d*(c*d^2 + 17*a*e^2)*x^2)/e + 96*c^2*d^2*x^3 - (15*(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^(3/2)*d^(3/2)*e^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/384

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Maple [B]  time = 0.017, size = 1455, normalized size = 5.6 \[ \text{result too large to display} \]

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

[Out]

2/3/e^2/(a*e^2-c*d^2)/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)-2/
3/e*d*c/(a*e^2-c*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2)+5/32/e*d^5*c
^2/(a*e^2-c*d^2)*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+25/128*e^5*d/(a
*e^2-c*d^2)*a^4*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x
)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+5/32*e^4/(a*e^2-c*d^2)*a^3*(c*
d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+15/32*d^4*c^2/(a*e^2-c*d^2)*a*(c*d*
(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/128*e^7/d/c/(a*e^2-c*d^2)*a^5*ln((1
/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(
d/e+x))^(1/2))/(d*e*c)^(1/2)+25/64*e*d^5*c^2/(a*e^2-c*d^2)*a^2*ln((1/2*a*e^2-1/2
*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2
))/(d*e*c)^(1/2)-25/128/e*d^7*c^3/(a*e^2-c*d^2)*a*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*
(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1
/2)-5/64/e^3*d^7*c^3/(a*e^2-c*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)
+5/128/e^3*d^9*c^4/(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^
(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-5/12*e*d*c/(a
*e^2-c*d^2)*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-5/32*e^3*d/(a*e^2-
c*d^2)*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+5/12/e*d^3*c^2/(a*e^2-c
*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+5/24/e^2*d^4*c^2/(a*e^2-c*
d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)-25/64*e^3*d^3*c/(a*e^2-c*d^2)
*a^3*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^
2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-5/32/e^2*d^6*c^3/(a*e^2-c*d^2)*(c*d*(d/e+
x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/24*e^2/(a*e^2-c*d^2)*a^2*(c*d*(d/e+x)^2*
e+(a*e^2-c*d^2)*(d/e+x))^(3/2)-15/32*e^2*d^2*c/(a*e^2-c*d^2)*a^2*(c*d*(d/e+x)^2*
e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+5/64*e^5/d/c/(a*e^2-c*d^2)*a^4*(c*d*(d/e+x)^2*e
+(a*e^2-c*d^2)*(d/e+x))^(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260049, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{6} - 55 \, a c^{2} d^{4} e^{2} + 73 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 8 \,{\left (c^{3} d^{4} e^{2} + 17 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (5 \, c^{3} d^{5} e - 18 \, a c^{2} d^{3} e^{3} - 59 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} + 15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{768 \, \sqrt{c d e} c d e^{3}}, \frac{2 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{6} - 55 \, a c^{2} d^{4} e^{2} + 73 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 8 \,{\left (c^{3} d^{4} e^{2} + 17 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (5 \, c^{3} d^{5} e - 18 \, a c^{2} d^{3} e^{3} - 59 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} - 15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{384 \, \sqrt{-c d e} c d e^{3}}\right ] \]

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

[Out]

[1/768*(4*(48*c^3*d^3*e^3*x^3 + 15*c^3*d^6 - 55*a*c^2*d^4*e^2 + 73*a^2*c*d^2*e^4
+ 15*a^3*e^6 + 8*(c^3*d^4*e^2 + 17*a*c^2*d^2*e^4)*x^2 - 2*(5*c^3*d^5*e - 18*a*c
^2*d^3*e^3 - 59*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) + 15*(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)*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*d*e^3), 1/384*(2*(48*
c^3*d^3*e^3*x^3 + 15*c^3*d^6 - 55*a*c^2*d^4*e^2 + 73*a^2*c*d^2*e^4 + 15*a^3*e^6
+ 8*(c^3*d^4*e^2 + 17*a*c^2*d^2*e^4)*x^2 - 2*(5*c^3*d^5*e - 18*a*c^2*d^3*e^3 - 5
9*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) - 15*
(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)*arct
an(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*d*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)**2,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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError