3.1927 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2}-\frac{5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^3} \]

[Out]

(5*(c*d^2 - a*e^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])/(8*e^3) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e
^2) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(d + e*x)^2) - (5*(c*
d^2 - a*e^2)^3*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*Sqrt[c]*Sqrt[d]*e^(7/2))

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Rubi [A]  time = 0.424902, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac{5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2}-\frac{5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^3} \]

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)^3,x]

[Out]

(5*(c*d^2 - a*e^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])/(8*e^3) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e
^2) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(d + e*x)^2) - (5*(c*
d^2 - a*e^2)^3*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*Sqrt[c]*Sqrt[d]*e^(7/2))

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Rubi in Sympy [A]  time = 62.9776, size = 238, normalized size = 0.98 \[ - \frac{5 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{2}} - \frac{5 \left (a e^{2} - c d^{2}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{8 e^{3}} + \frac{5 \left (a e^{2} - c d^{2}\right )^{3} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{16 \sqrt{c} \sqrt{d} e^{\frac{7}{2}}} \]

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)**3,x)

[Out]

-5*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*e**2) + 2*(a*d*e + c
*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(e*(d + e*x)**2) - 5*(a*e**2 - c*d**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))/(8*e
**3) + 5*(a*e**2 - c*d**2)**3*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqr
t(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(16*sqrt(c)*sqrt(d
)*e**(7/2))

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Mathematica [A]  time = 0.338842, size = 200, normalized size = 0.82 \[ \frac{\frac{2 (d+e x) (a e+c d x) \left (33 a^2 e^4+2 a c d e^2 (13 e x-20 d)+c^2 d^2 \left (15 d^2-10 d e x+8 e^2 x^2\right )\right )}{3 e^3}-\frac{5 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{c} \sqrt{d} e^{7/2}}}{16 \sqrt{(d+e x) (a e+c d x)}} \]

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)^3,x]

[Out]

((2*(a*e + c*d*x)*(d + e*x)*(33*a^2*e^4 + 2*a*c*d*e^2*(-20*d + 13*e*x) + c^2*d^2
*(15*d^2 - 10*d*e*x + 8*e^2*x^2)))/(3*e^3) - (5*(c*d^2 - a*e^2)^3*Sqrt[a*e + c*d
*x]*Sqrt[d + e*x]*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)])/(Sqrt[c]*Sqrt[d]*e^(7/2)))/(16*Sqrt[(a*e + c*d*x)*(d
+ e*x)])

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Maple [B]  time = 0.018, size = 1531, normalized size = 6.3 \[ \text{result too large to display} \]

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)^3,x)

[Out]

2/e^3/(a*e^2-c*d^2)/(d/e+x)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)-16/3
/e^2*d*c/(a*e^2-c*d^2)^2/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)
+16/3/e*d^2*c^2/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2)+5/
3*e^2*d*c/(a*e^2-c*d^2)^2*a^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)-5/4*
e^4*d*c/(a*e^2-c*d^2)^2*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-15/4
*d^5*c^3/(a*e^2-c*d^2)^2*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+5/16*
e^7/(a*e^2-c*d^2)^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)-25/8*e*d^6*c^3/(a*e^2-
c*d^2)^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)+25/16/e*d^8*c^4/(a*e^2-c*d^2)^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)+10/3*e*d^2*c^2/(a*e^2-c*d^2)^2*a*(c*d*(d/e+x)
^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-5/3/e^2*d^5*c^3/(a*e^2-c*d^2)^2*(c*d*(d/e+x)
^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)+5/8/e^3*d^8*c^4/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2
*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+15/4*e^2*d^3*c^2/(a*e^2-c*d^2)^2*a^2*(c*d*(d/e+x
)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/8*e^5/(a*e^2-c*d^2)^2*a^4*(c*d*(d/e+x)^2*
e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-5/4/e*d^6*c^3/(a*e^2-c*d^2)^2*a*(c*d*(d/e+x)^2*e+
(a*e^2-c*d^2)*(d/e+x))^(1/2)-25/16*e^5*d^2*c/(a*e^2-c*d^2)^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)+25/8*e^3*d^4*c^2/(a*e^2-c*d^2)^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)+5/4/e^2*d^7*c^4/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x)
)^(1/2)*x-5/16/e^3*d^10*c^5/(a*e^2-c*d^2)^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/
4*e^3*d^2*c/(a*e^2-c*d^2)^2*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-10
/3/e*d^4*c^3/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295748, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 40 \, a c d^{2} e^{2} + 33 \, a^{2} e^{4} - 2 \,{\left (5 \, c^{2} d^{3} e - 13 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} + 15 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{96 \, \sqrt{c d e} e^{3}}, \frac{2 \,{\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 40 \, a c d^{2} e^{2} + 33 \, a^{2} e^{4} - 2 \,{\left (5 \, c^{2} d^{3} e - 13 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} - 15 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{48 \, \sqrt{-c d e} e^{3}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[1/96*(4*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 40*a*c*d^2*e^2 + 33*a^2*e^4 - 2*(5*c^
2*d^3*e - 13*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*
e) + 15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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)*e^3), 1/48*(2*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 -
40*a*c*d^2*e^2 + 33*a^2*e^4 - 2*(5*c^2*d^3*e - 13*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 - a^3*e^6)*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)*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)**3,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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError