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

Optimal. Leaf size=259 \[ \frac{8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 \left (x \left (-2 a^3 e^6+a^2 c d^2 e^4+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 x^2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) - (8*(a*d*e*(c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2
*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3
/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(1
5*e*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.670368, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075 \[ \frac{8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 \left (x \left (-2 a^3 e^6+a^2 c d^2 e^4+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2 x^2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2)) - (8*(a*d*e*(c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2
*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3
/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(1
5*e*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 76.0519, size = 252, normalized size = 0.97 \[ - \frac{2 x^{2} \left (a e + c d x\right )}{5 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}} + \frac{8 \left (a d e \left (a e^{2} - c d^{2}\right ) \left (3 a e^{2} + c d^{2}\right ) + x \left (2 a^{3} e^{6} - a^{2} c d^{2} e^{4} - c^{3} d^{6}\right )\right )}{15 e \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{4 \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right ) \left (5 a^{2} e^{4} + 10 a c d^{2} e^{2} + c^{2} d^{4}\right )}{15 e \left (a e^{2} - c d^{2}\right )^{5} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-2*x**2*(a*e + c*d*x)/(5*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d
**2))**(5/2)) + 8*(a*d*e*(a*e**2 - c*d**2)*(3*a*e**2 + c*d**2) + x*(2*a**3*e**6
- a**2*c*d**2*e**4 - c**3*d**6))/(15*e*(a*e**2 - c*d**2)**4*(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))**(3/2)) - 4*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)*(5*a**2*e**
4 + 10*a*c*d**2*e**2 + c**2*d**4)/(15*e*(a*e**2 - c*d**2)**5*sqrt(a*d*e + c*d*e*
x**2 + x*(a*e**2 + c*d**2)))

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Mathematica [A]  time = 1.27308, size = 210, normalized size = 0.81 \[ \frac{2 (d+e x)^3 (a e+c d x)^3 \left (-\frac{15 a^2 e^4+50 a c d^2 e^2+8 c^2 d^4}{d+e x}-\frac{5 a^2 c d e^2 \left (a e^2-c d^2\right )}{(a e+c d x)^2}-\frac{3 \left (c d^3-a d e^2\right )^2}{(d+e x)^3}-\frac{5 a c d e \left (5 a e^2+6 c d^2\right )}{a e+c d x}+\frac{2 d \left (a e^2-c d^2\right ) \left (5 a e^2+2 c d^2\right )}{(d+e x)^2}\right )}{15 \left (a e^2-c d^2\right )^5 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(2*(a*e + c*d*x)^3*(d + e*x)^3*((-5*a^2*c*d*e^2*(-(c*d^2) + a*e^2))/(a*e + c*d*x
)^2 - (5*a*c*d*e*(6*c*d^2 + 5*a*e^2))/(a*e + c*d*x) - (3*(c*d^3 - a*d*e^2)^2)/(d
+ e*x)^3 + (2*d*(-(c*d^2) + a*e^2)*(2*c*d^2 + 5*a*e^2))/(d + e*x)^2 - (8*c^2*d^
4 + 50*a*c*d^2*e^2 + 15*a^2*e^4)/(d + e*x)))/(15*(-(c*d^2) + a*e^2)^5*((a*e + c*
d*x)*(d + e*x))^(5/2))

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Maple [A]  time = 0.019, size = 366, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 40\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{4}+80\,a{c}^{3}{d}^{4}{e}^{4}{x}^{4}+8\,{c}^{4}{d}^{6}{e}^{2}{x}^{4}+60\,{a}^{3}cd{e}^{7}{x}^{3}+220\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}{x}^{3}+212\,a{c}^{3}{d}^{5}{e}^{3}{x}^{3}+20\,{c}^{4}{d}^{7}e{x}^{3}+15\,{a}^{4}{e}^{8}{x}^{2}+180\,{a}^{3}c{d}^{2}{e}^{6}{x}^{2}+378\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{x}^{2}+180\,a{c}^{3}{d}^{6}{e}^{2}{x}^{2}+15\,{c}^{4}{d}^{8}{x}^{2}+20\,{a}^{4}d{e}^{7}x+212\,{a}^{3}c{d}^{3}{e}^{5}x+220\,{a}^{2}{c}^{2}{d}^{5}{e}^{3}x+60\,a{c}^{3}{d}^{7}ex+8\,{a}^{4}{d}^{2}{e}^{6}+80\,{a}^{3}c{d}^{4}{e}^{4}+40\,{a}^{2}{c}^{2}{d}^{6}{e}^{2} \right ) }{15\,{a}^{5}{e}^{10}-75\,{a}^{4}c{d}^{2}{e}^{8}+150\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-150\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+75\,a{c}^{4}{d}^{8}{e}^{2}-15\,{c}^{5}{d}^{10}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/15*(c*d*x+a*e)*(40*a^2*c^2*d^2*e^6*x^4+80*a*c^3*d^4*e^4*x^4+8*c^4*d^6*e^2*x^4
+60*a^3*c*d*e^7*x^3+220*a^2*c^2*d^3*e^5*x^3+212*a*c^3*d^5*e^3*x^3+20*c^4*d^7*e*x
^3+15*a^4*e^8*x^2+180*a^3*c*d^2*e^6*x^2+378*a^2*c^2*d^4*e^4*x^2+180*a*c^3*d^6*e^
2*x^2+15*c^4*d^8*x^2+20*a^4*d*e^7*x+212*a^3*c*d^3*e^5*x+220*a^2*c^2*d^5*e^3*x+60
*a*c^3*d^7*e*x+8*a^4*d^2*e^6+80*a^3*c*d^4*e^4+40*a^2*c^2*d^6*e^2)/(a^5*e^10-5*a^
4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(c*d
*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)

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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(x^2/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.85, size = 1107, normalized size = 4.27 \[ \frac{2 \,{\left (40 \, a^{2} c^{2} d^{6} e^{2} + 80 \, a^{3} c d^{4} e^{4} + 8 \, a^{4} d^{2} e^{6} + 8 \,{\left (c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{4} + 4 \,{\left (5 \, c^{4} d^{7} e + 53 \, a c^{3} d^{5} e^{3} + 55 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7}\right )} x^{3} + 3 \,{\left (5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 126 \, a^{2} c^{2} d^{4} e^{4} + 60 \, a^{3} c d^{2} e^{6} + 5 \, a^{4} e^{8}\right )} x^{2} + 4 \,{\left (15 \, a c^{3} d^{7} e + 55 \, a^{2} c^{2} d^{5} e^{3} + 53 \, a^{3} c d^{3} e^{5} + 5 \, a^{4} d e^{7}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{15 \,{\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} +{\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} +{\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} +{\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} +{\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} +{\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

2/15*(40*a^2*c^2*d^6*e^2 + 80*a^3*c*d^4*e^4 + 8*a^4*d^2*e^6 + 8*(c^4*d^6*e^2 + 1
0*a*c^3*d^4*e^4 + 5*a^2*c^2*d^2*e^6)*x^4 + 4*(5*c^4*d^7*e + 53*a*c^3*d^5*e^3 + 5
5*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7)*x^3 + 3*(5*c^4*d^8 + 60*a*c^3*d^6*e^2 + 126*
a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 + 5*a^4*e^8)*x^2 + 4*(15*a*c^3*d^7*e + 55*a^2
*c^2*d^5*e^3 + 53*a^3*c*d^3*e^5 + 5*a^4*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a
^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6*d^10*
e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^5*c^2*d^2
*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5*d^9*e^6 - 10*a^3*c
^4*d^7*e^8 - 5*a^4*c^3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2*a^6*c*d*e^14)*x^4 + (3*
c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 25*a^3*c^4*d^8*e^7 - 35*a^4*c
^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 +
a*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e^8 -
a^5*c^2*d^5*e^10 + 9*a^6*c*d^3*e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^14*e - 7*a
^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 20*a^5*c^2*d^6*e^9 +
13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)

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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(x**2/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]