3.1962 \(\int \frac{1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=312 \[ \frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))
+ (8*c*d)/(21*(c*d^2 - a*e^2)^2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2)) + (16*c^2*d^2)/(21*(c*d^2 - a*e^2)^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2)) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2
- a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (1024*c^4*d^4*e*(c*d
^2 + a*e^2 + 2*c*d*e*x))/(63*(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 = 0.467573, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))
+ (8*c*d)/(21*(c*d^2 - a*e^2)^2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2)) + (16*c^2*d^2)/(21*(c*d^2 - a*e^2)^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2)) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2
- a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (1024*c^4*d^4*e*(c*d
^2 + a*e^2 + 2*c*d*e*x))/(63*(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 = 83.8096, size = 303, normalized size = 0.97 \[ - \frac{512 c^{4} d^{4} e \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{63 \left (a e^{2} - c d^{2}\right )^{7} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{128 c^{3} d^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right )}{63 \left (a e^{2} - c d^{2}\right )^{5} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{16 c^{2} d^{2}}{21 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} + \frac{8 c d}{21 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{2}{9 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

-512*c**4*d**4*e*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)/(63*(a*e**2 - c*d**2)**7*sqrt
(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 128*c**3*d**3*(a*e**2 + c*d**2 + 2
*c*d*e*x)/(63*(a*e**2 - c*d**2)**5*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(
3/2)) - 16*c**2*d**2/(21*(d + e*x)*(a*e**2 - c*d**2)**3*(a*d*e + c*d*e*x**2 + x*
(a*e**2 + c*d**2))**(3/2)) + 8*c*d/(21*(d + e*x)**2*(a*e**2 - c*d**2)**2*(a*d*e
+ c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 2/(9*(d + e*x)**3*(a*e**2 - c*d**2
)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2))

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Mathematica [A]  time = 1.00603, size = 237, normalized size = 0.76 \[ \frac{2 (d+e x)^3 (a e+c d x)^3 \left (-\frac{357 c^5 d^5 e}{a e+c d x}+\frac{21 c^5 d^5 \left (c d^2-a e^2\right )}{(a e+c d x)^2}+\frac{176 c^3 d^3 e^2 \left (a e^2-c d^2\right )}{(d+e x)^2}-\frac{69 c^2 d^2 e^2 \left (c d^2-a e^2\right )^2}{(d+e x)^3}+\frac{26 c d e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^4}-\frac{7 e^2 \left (c d^2-a e^2\right )^4}{(d+e x)^5}-\frac{667 c^4 d^4 e^2}{d+e x}\right )}{63 \left (a e^2-c d^2\right )^7 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(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*((21*c^5*d^5*(c*d^2 - a*e^2))/(a*e + c*d*x)^2 - (
357*c^5*d^5*e)/(a*e + c*d*x) - (7*e^2*(c*d^2 - a*e^2)^4)/(d + e*x)^5 + (26*c*d*e
^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^4 - (69*c^2*d^2*e^2*(c*d^2 - a*e^2)^2)/(d + e
*x)^3 + (176*c^3*d^3*e^2*(-(c*d^2) + a*e^2))/(d + e*x)^2 - (667*c^4*d^4*e^2)/(d
+ e*x)))/(63*(-(c*d^2) + a*e^2)^7*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [A]  time = 0.025, size = 536, normalized size = 1.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1024\,{c}^{6}{d}^{6}{e}^{6}{x}^{6}+1536\,a{c}^{5}{d}^{5}{e}^{7}{x}^{5}+4608\,{c}^{6}{d}^{7}{e}^{5}{x}^{5}+384\,{a}^{2}{c}^{4}{d}^{4}{e}^{8}{x}^{4}+6912\,a{c}^{5}{d}^{6}{e}^{6}{x}^{4}+8064\,{c}^{6}{d}^{8}{e}^{4}{x}^{4}-64\,{a}^{3}{c}^{3}{d}^{3}{e}^{9}{x}^{3}+1728\,{a}^{2}{c}^{4}{d}^{5}{e}^{7}{x}^{3}+12096\,a{c}^{5}{d}^{7}{e}^{5}{x}^{3}+6720\,{c}^{6}{d}^{9}{e}^{3}{x}^{3}+24\,{a}^{4}{c}^{2}{d}^{2}{e}^{10}{x}^{2}-288\,{a}^{3}{c}^{3}{d}^{4}{e}^{8}{x}^{2}+3024\,{a}^{2}{c}^{4}{d}^{6}{e}^{6}{x}^{2}+10080\,a{c}^{5}{d}^{8}{e}^{4}{x}^{2}+2520\,{c}^{6}{d}^{10}{e}^{2}{x}^{2}-12\,{a}^{5}cd{e}^{11}x+108\,{a}^{4}{c}^{2}{d}^{3}{e}^{9}x-504\,{a}^{3}{c}^{3}{d}^{5}{e}^{7}x+2520\,{a}^{2}{c}^{4}{d}^{7}{e}^{5}x+3780\,a{c}^{5}{d}^{9}{e}^{3}x+252\,{c}^{6}{d}^{11}ex+7\,{a}^{6}{e}^{12}-54\,{a}^{5}c{d}^{2}{e}^{10}+189\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-420\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+945\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}+378\,a{c}^{5}{d}^{10}{e}^{2}-21\,{c}^{6}{d}^{12} \right ) }{63\, \left ({a}^{7}{e}^{14}-7\,{a}^{6}c{d}^{2}{e}^{12}+21\,{a}^{5}{c}^{2}{d}^{4}{e}^{10}-35\,{a}^{4}{c}^{3}{d}^{6}{e}^{8}+35\,{a}^{3}{c}^{4}{d}^{8}{e}^{6}-21\,{a}^{2}{c}^{5}{d}^{10}{e}^{4}+7\,a{c}^{6}{d}^{12}{e}^{2}-{c}^{7}{d}^{14} \right ) \left ( ex+d \right ) ^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/63*(c*d*x+a*e)*(1024*c^6*d^6*e^6*x^6+1536*a*c^5*d^5*e^7*x^5+4608*c^6*d^7*e^5*
x^5+384*a^2*c^4*d^4*e^8*x^4+6912*a*c^5*d^6*e^6*x^4+8064*c^6*d^8*e^4*x^4-64*a^3*c
^3*d^3*e^9*x^3+1728*a^2*c^4*d^5*e^7*x^3+12096*a*c^5*d^7*e^5*x^3+6720*c^6*d^9*e^3
*x^3+24*a^4*c^2*d^2*e^10*x^2-288*a^3*c^3*d^4*e^8*x^2+3024*a^2*c^4*d^6*e^6*x^2+10
080*a*c^5*d^8*e^4*x^2+2520*c^6*d^10*e^2*x^2-12*a^5*c*d*e^11*x+108*a^4*c^2*d^3*e^
9*x-504*a^3*c^3*d^5*e^7*x+2520*a^2*c^4*d^7*e^5*x+3780*a*c^5*d^9*e^3*x+252*c^6*d^
11*e*x+7*a^6*e^12-54*a^5*c*d^2*e^10+189*a^4*c^2*d^4*e^8-420*a^3*c^3*d^6*e^6+945*
a^2*c^4*d^8*e^4+378*a*c^5*d^10*e^2-21*c^6*d^12)/(e*x+d)^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)^(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(1/((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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((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]

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(1/(e*x+d)**3/(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(1/((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]

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