3.2045 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=366 \[ \frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*e^3*(d + e*x)^(7/2))
+ (c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e^2)*
(d + e*x)^(5/2)) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*
e^3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(8*e^2*(d + e*x)^(11/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2)/(5*e*(d + e*x)^(15/2)) + (3*c^5*d^5*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(128*e^(7/2)*(c*d^2
- a*e^2)^(5/2))

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Rubi [A]  time = 0.834586, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{3 c^5 d^5 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c^4 d^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]

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)^(17/2),x]

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*e^3*(d + e*x)^(7/2))
+ (c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e^2)*
(d + e*x)^(5/2)) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*
e^3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2))/(8*e^2*(d + e*x)^(11/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2)/(5*e*(d + e*x)^(15/2)) + (3*c^5*d^5*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(128*e^(7/2)*(c*d^2
- a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 169.486, size = 343, normalized size = 0.94 \[ - \frac{3 c^{5} d^{5} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{128 e^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{3 c^{4} d^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{128 e^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 e^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{16 e^{3} \left (d + e x\right )^{\frac{7}{2}}} - \frac{c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{8 e^{2} \left (d + e x\right )^{\frac{11}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{15}{2}}} \]

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)**(17/2),x)

[Out]

-3*c**5*d**5*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(
d + e*x)*sqrt(a*e**2 - c*d**2)))/(128*e**(7/2)*(a*e**2 - c*d**2)**(5/2)) + 3*c**
4*d**4*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(128*e**3*(d + e*x)**(3/2)
*(a*e**2 - c*d**2)**2) - c**3*d**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)/(64*e**3*(d + e*x)**(5/2)*(a*e**2 - c*d**2)) - c**2*d**2*sqrt(a*d*e + c*d*e*x*
*2 + x*(a*e**2 + c*d**2))/(16*e**3*(d + e*x)**(7/2)) - c*d*(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))**(3/2)/(8*e**2*(d + e*x)**(11/2)) - (a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))**(5/2)/(5*e*(d + e*x)**(15/2))

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Mathematica [A]  time = 1.14199, size = 243, normalized size = 0.66 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{\frac{15 c^4 d^4 (d+e x)^4}{\left (c d^2-a e^2\right )^2}+\frac{10 c^3 d^3 (d+e x)^3}{c d^2-a e^2}+336 c d (d+e x) \left (c d^2-a e^2\right )-128 \left (c d^2-a e^2\right )^2-248 c^2 d^2 (d+e x)^2}{5 e^3 (d+e x)^5 (a e+c d x)^2}-\frac{3 c^5 d^5 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{e^{7/2} \left (a e^2-c d^2\right )^{5/2} (a e+c d x)^{5/2}}\right )}{128 (d+e x)^{5/2}} \]

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)^(17/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((-128*(c*d^2 - a*e^2)^2 + 336*c*d*(c*d^2 - a*e
^2)*(d + e*x) - 248*c^2*d^2*(d + e*x)^2 + (10*c^3*d^3*(d + e*x)^3)/(c*d^2 - a*e^
2) + (15*c^4*d^4*(d + e*x)^4)/(c*d^2 - a*e^2)^2)/(5*e^3*(a*e + c*d*x)^2*(d + e*x
)^5) - (3*c^5*d^5*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(
e^(7/2)*(-(c*d^2) + a*e^2)^(5/2)*(a*e + c*d*x)^(5/2))))/(128*(d + e*x)^(5/2))

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Maple [B]  time = 0.05, size = 910, normalized size = 2.5 \[ \text{result too large to display} \]

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)^(17/2),x)

[Out]

-1/640*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/(
(a*e^2-c*d^2)*e)^(1/2))*x^5*c^5*d^5*e^5+75*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c
*d^2)*e)^(1/2))*x^4*c^5*d^6*e^4+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e
)^(1/2))*x^3*c^5*d^7*e^3+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)
)*x^2*c^5*d^8*e^2-15*x^4*c^4*d^4*e^4*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+7
5*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*c^5*d^9*e+10*x^3*a*c^3*
d^3*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-70*x^3*c^4*d^5*e^3*(c*d*x+a*e)
^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^
(1/2))*c^5*d^10+248*x^2*a^2*c^2*d^2*e^6*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2
)-466*x^2*a*c^3*d^4*e^4*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*x^2*c^4*d^
6*e^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+336*x*a^3*c*d*e^7*(c*d*x+a*e)^(1
/2)*((a*e^2-c*d^2)*e)^(1/2)-512*x*a^2*c^2*d^3*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^
2)*e)^(1/2)+46*x*a*c^3*d^5*e^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+70*x*c^
4*d^7*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*((a*e^2-c*d^2)*e)^(1/2)*(c
*d*x+a*e)^(1/2)*a^4*e^8-176*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*c*d^2*
e^6+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^2*d^4*e^4+10*((a*e^2-c*d^2
)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^3*d^6*e^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e
)^(1/2)*c^4*d^8)/(e*x+d)^(11/2)/((a*e^2-c*d^2)*e)^(1/2)/e^3/(a*e^2-c*d^2)^2/(c*d
*x+a*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)^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246735, 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)^(17/2),x, algorithm="fricas")

[Out]

[1/1280*(2*(15*c^4*d^4*e^4*x^4 - 15*c^4*d^8 - 10*a*c^3*d^6*e^2 - 8*a^2*c^2*d^4*e
^4 + 176*a^3*c*d^2*e^6 - 128*a^4*e^8 + 10*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 -
2*(64*c^4*d^6*e^2 - 233*a*c^3*d^4*e^4 + 124*a^2*c^2*d^2*e^6)*x^2 - 2*(35*c^4*d^7
*e + 23*a*c^3*d^5*e^3 - 256*a^2*c^2*d^3*e^5 + 168*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^2*e + a*e^3)*sqrt(e*x + d) + 15*(c^5*d^5
*e^6*x^6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*d^7*e^4*x^4 + 20*c^5*d^8*e^3*x^3 + 15*c^5*
d^9*e^2*x^2 + 6*c^5*d^10*e*x + c^5*d^11)*log(-(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(c*d^2*e - a*e^3)*sqrt(e*x + d) + (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3
+ 2*a*d*e^2)*sqrt(-c*d^2*e + a*e^3))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^10*e^3
- 2*a*c*d^8*e^5 + a^2*d^6*e^7 + (c^2*d^4*e^9 - 2*a*c*d^2*e^11 + a^2*e^13)*x^6 +
6*(c^2*d^5*e^8 - 2*a*c*d^3*e^10 + a^2*d*e^12)*x^5 + 15*(c^2*d^6*e^7 - 2*a*c*d^4*
e^9 + a^2*d^2*e^11)*x^4 + 20*(c^2*d^7*e^6 - 2*a*c*d^5*e^8 + a^2*d^3*e^10)*x^3 +
15*(c^2*d^8*e^5 - 2*a*c*d^6*e^7 + a^2*d^4*e^9)*x^2 + 6*(c^2*d^9*e^4 - 2*a*c*d^7*
e^6 + a^2*d^5*e^8)*x)*sqrt(-c*d^2*e + a*e^3)), 1/640*((15*c^4*d^4*e^4*x^4 - 15*c
^4*d^8 - 10*a*c^3*d^6*e^2 - 8*a^2*c^2*d^4*e^4 + 176*a^3*c*d^2*e^6 - 128*a^4*e^8
+ 10*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 - 2*(64*c^4*d^6*e^2 - 233*a*c^3*d^4*e^4
+ 124*a^2*c^2*d^2*e^6)*x^2 - 2*(35*c^4*d^7*e + 23*a*c^3*d^5*e^3 - 256*a^2*c^2*d
^3*e^5 + 168*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^2*e - a*e^3)*sqrt(e*x + d) - 15*(c^5*d^5*e^6*x^6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*
d^7*e^4*x^4 + 20*c^5*d^8*e^3*x^3 + 15*c^5*d^9*e^2*x^2 + 6*c^5*d^10*e*x + c^5*d^1
1)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt
(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)))/((c^2*d^10*e^3 - 2*a*c
*d^8*e^5 + a^2*d^6*e^7 + (c^2*d^4*e^9 - 2*a*c*d^2*e^11 + a^2*e^13)*x^6 + 6*(c^2*
d^5*e^8 - 2*a*c*d^3*e^10 + a^2*d*e^12)*x^5 + 15*(c^2*d^6*e^7 - 2*a*c*d^4*e^9 + a
^2*d^2*e^11)*x^4 + 20*(c^2*d^7*e^6 - 2*a*c*d^5*e^8 + a^2*d^3*e^10)*x^3 + 15*(c^2
*d^8*e^5 - 2*a*c*d^6*e^7 + a^2*d^4*e^9)*x^2 + 6*(c^2*d^9*e^4 - 2*a*c*d^7*e^6 + a
^2*d^5*e^8)*x)*sqrt(c*d^2*e - a*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)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^(17/2),x, algorithm="giac")

[Out]

Timed out