3.488 \(\int \frac{x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=341 \[ -\frac{128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac{2 x^2}{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \]

[Out]

(2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/
2)) - (8*(2*a*d*e*(c*d^2 + 2*a*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/
(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) + (16*(3*
c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2
- a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4
+ 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)
^7*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.766494, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac{8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac{2 x^2}{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \]

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)^(7/2)),x]

[Out]

(2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/
2)) - (8*(2*a*d*e*(c*d^2 + 2*a*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/
(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) + (16*(3*
c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2
- a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4
+ 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)
^7*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 = 94.5367, size = 340, normalized size = 1. \[ \frac{64 c d \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right ) \left (7 a^{2} e^{4} + 14 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{105 \left (a e^{2} - c d^{2}\right )^{7} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{2 x^{2} \left (a e + c d x\right )}{7 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}} + \frac{4 \left (4 a d e \left (2 a e^{2} + c d^{2}\right ) + x \left (6 a^{2} e^{4} + 2 a c d^{2} e^{2} + 4 c^{2} d^{4}\right )\right )}{35 e \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}} - \frac{16 \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (7 a^{2} e^{4} + 14 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{105 e \left (a e^{2} - c d^{2}\right )^{5} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

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)**(7/2),x)

[Out]

64*c*d*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)*(7*a**2*e**4 + 14*a*c*d**2*e**2 + 3*c**
2*d**4)/(105*(a*e**2 - c*d**2)**7*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
) - 2*x**2*(a*e + c*d*x)/(7*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))**(7/2)) + 4*(4*a*d*e*(2*a*e**2 + c*d**2) + x*(6*a**2*e**4 + 2*a*c*d**2*
e**2 + 4*c**2*d**4))/(35*e*(a*e**2 - c*d**2)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))**(5/2)) - 16*(a*e**2 + c*d**2 + 2*c*d*e*x)*(7*a**2*e**4 + 14*a*c*d**2
*e**2 + 3*c**2*d**4)/(105*e*(a*e**2 - c*d**2)**5*(a*d*e + c*d*e*x**2 + x*(a*e**2
+ c*d**2))**(3/2))

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Mathematica [A]  time = 2.43307, size = 317, normalized size = 0.93 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (\frac{21 a^2 c^2 d^2 e^2 \left (c d^2-a e^2\right )^2}{(a e+c d x)^3}+\frac{7 c^2 d^2 \left (73 a^2 e^4+110 a c d^2 e^2+15 c^2 d^4\right )}{a e+c d x}+\frac{c d e \left (385 a^2 e^4+1022 a c d^2 e^2+279 c^2 d^4\right )}{d+e x}-\frac{e \left (a e^2-c d^2\right ) \left (35 a^2 e^4+196 a c d^2 e^2+87 c^2 d^4\right )}{(d+e x)^2}+\frac{14 a c^2 d^2 e \left (a e^2-c d^2\right ) \left (7 a e^2+5 c d^2\right )}{(a e+c d x)^2}+\frac{15 d^2 e \left (c d^2-a e^2\right )^3}{(d+e x)^4}+\frac{3 d e \left (14 a e^2+13 c d^2\right ) \left (c d^2-a e^2\right )^2}{(d+e x)^3}\right )}{105 \left (a e^2-c d^2\right )^7} \]

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)^(7/2)),x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*((21*a^2*c^2*d^2*e^2*(c*d^2 - a*e^2)^2)/(a*e +
c*d*x)^3 + (14*a*c^2*d^2*e*(-(c*d^2) + a*e^2)*(5*c*d^2 + 7*a*e^2))/(a*e + c*d*x)
^2 + (7*c^2*d^2*(15*c^2*d^4 + 110*a*c*d^2*e^2 + 73*a^2*e^4))/(a*e + c*d*x) + (15
*d^2*e*(c*d^2 - a*e^2)^3)/(d + e*x)^4 + (3*d*e*(c*d^2 - a*e^2)^2*(13*c*d^2 + 14*
a*e^2))/(d + e*x)^3 - (e*(-(c*d^2) + a*e^2)*(87*c^2*d^4 + 196*a*c*d^2*e^2 + 35*a
^2*e^4))/(d + e*x)^2 + (c*d*e*(279*c^2*d^4 + 1022*a*c*d^2*e^2 + 385*a^2*e^4))/(d
+ e*x)))/(105*(-(c*d^2) + a*e^2)^7)

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Maple [B]  time = 0.025, size = 663, normalized size = 1.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -896\,{a}^{2}{c}^{4}{d}^{4}{e}^{8}{x}^{6}-1792\,a{c}^{5}{d}^{6}{e}^{6}{x}^{6}-384\,{c}^{6}{d}^{8}{e}^{4}{x}^{6}-2240\,{a}^{3}{c}^{3}{d}^{3}{e}^{9}{x}^{5}-7616\,{a}^{2}{c}^{4}{d}^{5}{e}^{7}{x}^{5}-7232\,a{c}^{5}{d}^{7}{e}^{5}{x}^{5}-1344\,{c}^{6}{d}^{9}{e}^{3}{x}^{5}-1680\,{a}^{4}{c}^{2}{d}^{2}{e}^{10}{x}^{4}-11200\,{a}^{3}{c}^{3}{d}^{4}{e}^{8}{x}^{4}-20320\,{a}^{2}{c}^{4}{d}^{6}{e}^{6}{x}^{4}-11200\,a{c}^{5}{d}^{8}{e}^{4}{x}^{4}-1680\,{c}^{6}{d}^{10}{e}^{2}{x}^{4}-280\,{a}^{5}cd{e}^{11}{x}^{3}-6440\,{a}^{4}{c}^{2}{d}^{3}{e}^{9}{x}^{3}-21680\,{a}^{3}{c}^{3}{d}^{5}{e}^{7}{x}^{3}-24080\,{a}^{2}{c}^{4}{d}^{7}{e}^{5}{x}^{3}-8120\,a{c}^{5}{d}^{9}{e}^{3}{x}^{3}-840\,{c}^{6}{d}^{11}e{x}^{3}+35\,{a}^{6}{e}^{12}{x}^{2}-910\,{a}^{5}c{d}^{2}{e}^{10}{x}^{2}-9295\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}{x}^{2}-20020\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}{x}^{2}-13195\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}{x}^{2}-2590\,a{c}^{5}{d}^{10}{e}^{2}{x}^{2}-105\,{c}^{6}{d}^{12}{x}^{2}+28\,{a}^{6}d{e}^{11}x-764\,{a}^{5}c{d}^{3}{e}^{9}x-6440\,{a}^{4}{c}^{2}{d}^{5}{e}^{7}x-8120\,{a}^{3}{c}^{3}{d}^{7}{e}^{5}x-2996\,{a}^{2}{c}^{4}{d}^{9}{e}^{3}x-140\,a{c}^{5}{d}^{11}ex+8\,{a}^{6}{d}^{2}{e}^{10}-224\,{a}^{5}c{d}^{4}{e}^{8}-1680\,{a}^{4}{c}^{2}{d}^{6}{e}^{6}-1120\,{a}^{3}{c}^{3}{d}^{8}{e}^{4}-56\,{a}^{2}{c}^{4}{d}^{10}{e}^{2} \right ) }{105\,{a}^{7}{e}^{14}-735\,{a}^{6}c{d}^{2}{e}^{12}+2205\,{a}^{5}{c}^{2}{d}^{4}{e}^{10}-3675\,{a}^{4}{c}^{3}{d}^{6}{e}^{8}+3675\,{a}^{3}{c}^{4}{d}^{8}{e}^{6}-2205\,{a}^{2}{c}^{5}{d}^{10}{e}^{4}+735\,a{c}^{6}{d}^{12}{e}^{2}-105\,{c}^{7}{d}^{14}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{7}{2}}}} \]

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)^(7/2),x)

[Out]

-2/105*(c*d*x+a*e)*(-896*a^2*c^4*d^4*e^8*x^6-1792*a*c^5*d^6*e^6*x^6-384*c^6*d^8*
e^4*x^6-2240*a^3*c^3*d^3*e^9*x^5-7616*a^2*c^4*d^5*e^7*x^5-7232*a*c^5*d^7*e^5*x^5
-1344*c^6*d^9*e^3*x^5-1680*a^4*c^2*d^2*e^10*x^4-11200*a^3*c^3*d^4*e^8*x^4-20320*
a^2*c^4*d^6*e^6*x^4-11200*a*c^5*d^8*e^4*x^4-1680*c^6*d^10*e^2*x^4-280*a^5*c*d*e^
11*x^3-6440*a^4*c^2*d^3*e^9*x^3-21680*a^3*c^3*d^5*e^7*x^3-24080*a^2*c^4*d^7*e^5*
x^3-8120*a*c^5*d^9*e^3*x^3-840*c^6*d^11*e*x^3+35*a^6*e^12*x^2-910*a^5*c*d^2*e^10
*x^2-9295*a^4*c^2*d^4*e^8*x^2-20020*a^3*c^3*d^6*e^6*x^2-13195*a^2*c^4*d^8*e^4*x^
2-2590*a*c^5*d^10*e^2*x^2-105*c^6*d^12*x^2+28*a^6*d*e^11*x-764*a^5*c*d^3*e^9*x-6
440*a^4*c^2*d^5*e^7*x-8120*a^3*c^3*d^7*e^5*x-2996*a^2*c^4*d^9*e^3*x-140*a*c^5*d^
11*e*x+8*a^6*d^2*e^10-224*a^5*c*d^4*e^8-1680*a^4*c^2*d^6*e^6-1120*a^3*c^3*d^8*e^
4-56*a^2*c^4*d^10*e^2)/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*e^10-35*a^4*c^3
*d^6*e^8+35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(c*d*
e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(7/2)

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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)^(7/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(7/2)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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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)**(7/2),x)

[Out]

Timed out

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

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)^(7/2)*(e*x + d)),x, algorithm="giac")

[Out]

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