3.1929 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=226 \[ -\frac{5 c^{3/2} d^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 e^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 e (d+e x)^4}-\frac{10 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^2} \]

[Out]

(5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/e^3 - (10*c*d*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e^2*(d + e*x)^2) - (2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(3*e*(d + e*x)^4) - (5*c^(3/2)*d^(3/2)*(c*d^2 - a*e
^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])])/(2*e^(7/2))

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Rubi [A]  time = 0.430171, antiderivative size = 226, 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 \[ -\frac{5 c^{3/2} d^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 e^{7/2}}+\frac{5 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 e (d+e x)^4}-\frac{10 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2 (d+e x)^2} \]

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)^5,x]

[Out]

(5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/e^3 - (10*c*d*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e^2*(d + e*x)^2) - (2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(3*e*(d + e*x)^4) - (5*c^(3/2)*d^(3/2)*(c*d^2 - a*e
^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])])/(2*e^(7/2))

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Rubi in Sympy [A]  time = 71.93, size = 221, normalized size = 0.98 \[ \frac{5 c^{\frac{3}{2}} d^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 e^{\frac{7}{2}}} + \frac{5 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{3}} - \frac{10 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{4}} \]

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)**5,x)

[Out]

5*c**(3/2)*d**(3/2)*(a*e**2 - c*d**2)*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqr
t(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(2*e**(7/2
)) + 5*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/e**3 - 10*c*d*(a
*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*e**2*(d + e*x)**2) - 2*(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(3*e*(d + e*x)**4)

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Mathematica [A]  time = 0.708885, size = 203, normalized size = 0.9 \[ \frac{1}{2} ((d+e x) (a e+c d x))^{5/2} \left (\frac{-4 a^2 e^4-4 a c d e^2 (5 d+7 e x)+2 c^2 d^2 \left (15 d^2+20 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^4 (a e+c d x)^2}-\frac{5 c^{3/2} d^{3/2} \left (c d^2-a e^2\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right ) \]

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)^5,x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-4*a^2*e^4 - 4*a*c*d*e^2*(5*d + 7*e*x) + 2*c^
2*d^2*(15*d^2 + 20*d*e*x + 3*e^2*x^2))/(3*e^3*(a*e + c*d*x)^2*(d + e*x)^4) - (5*
c^(3/2)*d^(3/2)*(c*d^2 - a*e^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))))/2

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Maple [B]  time = 0.017, size = 1695, normalized size = 7.5 \[ \text{result too large to display} \]

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)^5,x)

[Out]

-2/3/e^5/(a*e^2-c*d^2)/(d/e+x)^5*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/2)-8
/3/e^4*d*c/(a*e^2-c*d^2)^2/(d/e+x)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(7/
2)+16/e^3*d^2*c^2/(a*e^2-c*d^2)^3/(d/e+x)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+
x))^(7/2)-128/3/e^2*d^3*c^3/(a*e^2-c*d^2)^4/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*
d^2)*(d/e+x))^(7/2)+128/3/e*d^4*c^4/(a*e^2-c*d^2)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^
2)*(d/e+x))^(5/2)+40/3*e^2*d^3*c^3/(a*e^2-c*d^2)^4*a^2*(c*d*(d/e+x)^2*e+(a*e^2-c
*d^2)*(d/e+x))^(3/2)+30*e^2*d^5*c^4/(a*e^2-c*d^2)^4*a^2*(c*d*(d/e+x)^2*e+(a*e^2-
c*d^2)*(d/e+x))^(1/2)*x-30*d^7*c^5/(a*e^2-c*d^2)^4*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d
^2)*(d/e+x))^(1/2)*x+5/2*e^7*d^2*c^2/(a*e^2-c*d^2)^4*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*e*d^8*c^5/(a*e^2-c*d^2)^4*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
/2/e*d^10*c^6/(a*e^2-c*d^2)^4*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*e^4*d^3*c^3
/(a*e^2-c*d^2)^4*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+80/3*e*d^4*
c^4/(a*e^2-c*d^2)^4*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+10*e^3*d^4
*c^3/(a*e^2-c*d^2)^4*a^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-80/3/e*d^
6*c^5/(a*e^2-c*d^2)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-40/3/e^2*d
^7*c^5/(a*e^2-c*d^2)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)+5/e^3*d^10*
c^6/(a*e^2-c*d^2)^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-5*e^5*d^2*c^2/
(a*e^2-c*d^2)^4*a^4*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-10/e*d^8*c^5/(
a*e^2-c*d^2)^4*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-25/2*e^5*d^4*c^3/
(a*e^2-c*d^2)^4*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*e^3*d^6*c^4/(a*e^2-c*d^
2)^4*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)+10/e^2*d^9*c^6/(a*e^2-c*d^2)^4*(c*d*(
d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/2/e^3*d^12*c^7/(a*e^2-c*d^2)^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)

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.558059, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (c^{2} d^{5} - a c d^{3} e^{2} +{\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{4} e - a c d^{2} e^{3}\right )} x\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} + 2 \,{\left (10 \, c^{2} d^{3} e - 7 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{12 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}, -\frac{15 \,{\left (c^{2} d^{5} - a c d^{3} e^{2} +{\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{4} e - a c d^{2} e^{3}\right )} x\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-\frac{c d}{e}} e}\right ) - 2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} + 2 \,{\left (10 \, c^{2} d^{3} e - 7 \, a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{6 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}\right ] \]

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)^5,x, algorithm="fricas")

[Out]

[1/12*(15*(c^2*d^5 - a*c*d^3*e^2 + (c^2*d^3*e^2 - a*c*d*e^4)*x^2 + 2*(c^2*d^4*e
- a*c*d^2*e^3)*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*(3*c^2*d^2*e^2*x^2 + 15*c
^2*d^4 - 10*a*c*d^2*e^2 - 2*a^2*e^4 + 2*(10*c^2*d^3*e - 7*a*c*d*e^3)*x)*sqrt(c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^5*x^2 + 2*d*e^4*x + d^2*e^3), -1/6*(15*(
c^2*d^5 - a*c*d^3*e^2 + (c^2*d^3*e^2 - a*c*d*e^4)*x^2 + 2*(c^2*d^4*e - a*c*d^2*e
^3)*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*(3*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 1
0*a*c*d^2*e^2 - 2*a^2*e^4 + 2*(10*c^2*d^3*e - 7*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x))/(e^5*x^2 + 2*d*e^4*x + 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)**5,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)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError