3.1928 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=235 \[ -\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}+\frac{15 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac{15 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 e^{7/2}} \]

[Out]

(15*c*d*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) + (
5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*e^2*(d + e*x)) - (2*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(d + e*x)^3) + (15*Sqrt[c]*Sqrt[d]
*(c*d^2 - a*e^2)^2*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])])/(8*e^(7/2))

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Rubi [A]  time = 0.470126, antiderivative size = 235, 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 \[ -\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}+\frac{15 c d \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac{15 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 e^{7/2}} \]

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)^4,x]

[Out]

(15*c*d*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) + (
5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*e^2*(d + e*x)) - (2*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(d + e*x)^3) + (15*Sqrt[c]*Sqrt[d]
*(c*d^2 - a*e^2)^2*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])])/(8*e^(7/2))

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Rubi in Sympy [A]  time = 75.1572, size = 228, normalized size = 0.97 \[ \frac{15 \sqrt{c} \sqrt{d} \left (a e^{2} - c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 e^{\frac{7}{2}}} + \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2 e^{2} \left (d + e x\right )} + \frac{15 c d \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e^{3}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{3}} \]

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)**4,x)

[Out]

15*sqrt(c)*sqrt(d)*(a*e**2 - c*d**2)**2*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*s
qrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(8*e**(7
/2)) + 5*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(2*e**2*(d + e*x)
) + 15*c*d*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*e
**3) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(e*(d + e*x)**3)

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Mathematica [A]  time = 0.605644, size = 202, normalized size = 0.86 \[ \frac{1}{8} ((d+e x) (a e+c d x))^{5/2} \left (\frac{15 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^2 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}-\frac{2 \left (8 a^2 e^4-a c d e^2 (25 d+9 e x)+c^2 d^2 \left (15 d^2+5 d e x-2 e^2 x^2\right )\right )}{e^3 (d+e x)^3 (a e+c d x)^2}\right ) \]

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)^4,x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-2*(8*a^2*e^4 - a*c*d*e^2*(25*d + 9*e*x) + c^
2*d^2*(15*d^2 + 5*d*e*x - 2*e^2*x^2)))/(e^3*(a*e + c*d*x)^2*(d + e*x)^3) + (15*S
qrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^2*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)])/(e^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*x
)^(5/2))))/8

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Maple [B]  time = 0.018, size = 1617, normalized size = 6.9 \[ \text{result too large to display} \]

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)^4,x)

[Out]

-2/e^4/(a*e^2-c*d^2)/(d/e+x)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)+12/
e^3*d*c/(a*e^2-c*d^2)^2/(d/e+x)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)-
32/e^2*d^2*c^2/(a*e^2-c*d^2)^3/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))
^(7/2)+32/e*d^3*c^3/(a*e^2-c*d^2)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2
)-15/2*e^4*d^2*c^2/(a*e^2-c*d^2)^3*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(
1/2)*x-45/2*d^6*c^4/(a*e^2-c*d^2)^3*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1
/2)*x+15/8*e^7*d*c/(a*e^2-c*d^2)^3*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)+20*e*d^
3*c^3/(a*e^2-c*d^2)^3*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+15/2*e^3
*d^3*c^2/(a*e^2-c*d^2)^3*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-20/e*
d^5*c^4/(a*e^2-c*d^2)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-10/e^2*d
^6*c^4/(a*e^2-c*d^2)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)+15/4/e^3*d^
9*c^5/(a*e^2-c*d^2)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+10*e^2*d^2*c
^2/(a*e^2-c*d^2)^3*a^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)+45/2*e^2*d^
4*c^3/(a*e^2-c*d^2)^3*a^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-15/4*e
^5*d*c/(a*e^2-c*d^2)^3*a^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/2/e*
d^7*c^4/(a*e^2-c*d^2)^3*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-75/8*e^5
*d^3*c^2/(a*e^2-c*d^2)^3*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)+75/4*e^3*d^5*c^3/
(a*e^2-c*d^2)^3*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)+15/2/e^2*d^8*c^5/(a*e^2-c*
d^2)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-15/8/e^3*d^11*c^6/(a*e^2-
c*d^2)^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)-75/4*e*d^7*c^4/(a*e^2-c*d^2)^3*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)+75/8/e*d^9*c^5/(a*e^2-c*d^2)^3*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)

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.46461, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} -{\left (5 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{16 \,{\left (e^{4} x + d e^{3}\right )}}, \frac{15 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-\frac{c d}{e}} e}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} -{\left (5 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{8 \,{\left (e^{4} x + d e^{3}\right )}}\right ] \]

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)^4,x, algorithm="fricas")

[Out]

[1/16*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^
2*e^5)*x)*sqrt(c*d/e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4
+ 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*
sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2*c^2*d^2*e^2*x^2 - 15*c^2*d^4 +
25*a*c*d^2*e^2 - 8*a^2*e^4 - (5*c^2*d^3*e - 9*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x))/(e^4*x + d*e^3), 1/8*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^
2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(-c*d/e)*arctan(1/2*(2*c*
d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d/e)
*e)) + 2*(2*c^2*d^2*e^2*x^2 - 15*c^2*d^4 + 25*a*c*d^2*e^2 - 8*a^2*e^4 - (5*c^2*d
^3*e - 9*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^4*x + 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)**4,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)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError