3.1951 \(\int \frac{1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{35 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 c d}{35 (d+e x)^2 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) +
(16*c*d)/(35*(c*d^2 - a*e^2)^2*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2]) + (32*c^2*d^2)/(35*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2]) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(35*(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 = 0.372934, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ -\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{35 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{16 c d}{35 (d+e x)^2 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

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)^(3/2)),x]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) +
(16*c*d)/(35*(c*d^2 - a*e^2)^2*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2]) + (32*c^2*d^2)/(35*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2]) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(35*(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 = 67.7144, size = 231, normalized size = 0.96 \[ \frac{64 c^{3} d^{3} \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{35 \left (a e^{2} - c d^{2}\right )^{5} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{32 c^{2} d^{2}}{35 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{16 c d}{35 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{2}{7 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64*c**3*d**3*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)/(35*(a*e**2 - c*d**2)**5*sqrt(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 32*c**2*d**2/(35*(d + e*x)*(a*e**2 - c
*d**2)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 16*c*d/(35*(d + e*x)
**2*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 2/(7*
(d + e*x)**3*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))

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Mathematica [A]  time = 0.600276, size = 149, normalized size = 0.62 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (\frac{35 c^4 d^4}{a e+c d x}+\frac{29 c^2 d^2 e \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{13 c d e \left (c d^2-a e^2\right )^2}{(d+e x)^3}-\frac{5 e \left (a e^2-c d^2\right )^3}{(d+e x)^4}+\frac{93 c^3 d^3 e}{d+e x}\right )}{35 \left (a e^2-c d^2\right )^5} \]

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)^(3/2)),x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*((35*c^4*d^4)/(a*e + c*d*x) - (5*e*(-(c*d^2) +
a*e^2)^3)/(d + e*x)^4 + (13*c*d*e*(c*d^2 - a*e^2)^2)/(d + e*x)^3 + (29*c^2*d^2*e
*(c*d^2 - a*e^2))/(d + e*x)^2 + (93*c^3*d^3*e)/(d + e*x)))/(35*(-(c*d^2) + a*e^2
)^5)

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Maple [A]  time = 0.018, size = 307, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -128\,{x}^{4}{c}^{4}{d}^{4}{e}^{4}-64\,{x}^{3}a{c}^{3}{d}^{3}{e}^{5}-448\,{x}^{3}{c}^{4}{d}^{5}{e}^{3}+16\,{x}^{2}{a}^{2}{c}^{2}{d}^{2}{e}^{6}-224\,{x}^{2}a{c}^{3}{d}^{4}{e}^{4}-560\,{x}^{2}{c}^{4}{d}^{6}{e}^{2}-8\,x{a}^{3}cd{e}^{7}+56\,x{a}^{2}{c}^{2}{d}^{3}{e}^{5}-280\,xa{c}^{3}{d}^{5}{e}^{3}-280\,{c}^{4}{d}^{7}ex+5\,{a}^{4}{e}^{8}-28\,{a}^{3}c{d}^{2}{e}^{6}+70\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-140\,a{c}^{3}{d}^{6}{e}^{2}-35\,{c}^{4}{d}^{8} \right ) }{35\, \left ({a}^{5}{e}^{10}-5\,{a}^{4}c{d}^{2}{e}^{8}+10\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-10\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+5\,a{c}^{4}{d}^{8}{e}^{2}-{c}^{5}{d}^{10} \right ) \left ( ex+d \right ) ^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-2/35*(c*d*x+a*e)*(-128*c^4*d^4*e^4*x^4-64*a*c^3*d^3*e^5*x^3-448*c^4*d^5*e^3*x^3
+16*a^2*c^2*d^2*e^6*x^2-224*a*c^3*d^4*e^4*x^2-560*c^4*d^6*e^2*x^2-8*a^3*c*d*e^7*
x+56*a^2*c^2*d^3*e^5*x-280*a*c^3*d^5*e^3*x-280*c^4*d^7*e*x+5*a^4*e^8-28*a^3*c*d^
2*e^6+70*a^2*c^2*d^4*e^4-140*a*c^3*d^6*e^2-35*c^4*d^8)/(e*x+d)^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)^(3/2)

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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)^(3/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.0937, size = 990, normalized size = 4.11 \[ -\frac{2 \,{\left (128 \, c^{4} d^{4} e^{4} x^{4} + 35 \, c^{4} d^{8} + 140 \, a c^{3} d^{6} e^{2} - 70 \, a^{2} c^{2} d^{4} e^{4} + 28 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8} + 64 \,{\left (7 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 16 \,{\left (35 \, c^{4} d^{6} e^{2} + 14 \, a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \,{\left (35 \, c^{4} d^{7} e + 35 \, a c^{3} d^{5} e^{3} - 7 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{35 \,{\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} +{\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} +{\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \,{\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \,{\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} +{\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \]

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)^(3/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-2/35*(128*c^4*d^4*e^4*x^4 + 35*c^4*d^8 + 140*a*c^3*d^6*e^2 - 70*a^2*c^2*d^4*e^4
+ 28*a^3*c*d^2*e^6 - 5*a^4*e^8 + 64*(7*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 16*(3
5*c^4*d^6*e^2 + 14*a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 8*(35*c^4*d^7*e + 35*a
*c^3*d^5*e^3 - 7*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d
^2 + a*e^2)*x)/(a*c^5*d^14*e - 5*a^2*c^4*d^12*e^3 + 10*a^3*c^3*d^10*e^5 - 10*a^4
*c^2*d^8*e^7 + 5*a^5*c*d^6*e^9 - a^6*d^4*e^11 + (c^6*d^11*e^4 - 5*a*c^5*d^9*e^6
+ 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^5*e^10 + 5*a^4*c^2*d^3*e^12 - a^5*c*d*e^14)*
x^5 + (4*c^6*d^12*e^3 - 19*a*c^5*d^10*e^5 + 35*a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*
e^9 + 10*a^4*c^2*d^4*e^11 + a^5*c*d^2*e^13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 -
13*a*c^5*d^11*e^4 + 20*a^2*c^4*d^9*e^6 - 10*a^3*c^3*d^7*e^8 - 5*a^4*c^2*d^5*e^1
0 + 7*a^5*c*d^3*e^12 - 2*a^6*d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12*e^3 +
5*a^2*c^4*d^10*e^5 + 10*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a^5*c*d^4*e^11
- 3*a^6*d^2*e^13)*x^2 + (c^6*d^15 - a*c^5*d^13*e^2 - 10*a^2*c^4*d^11*e^4 + 30*a
^3*c^3*d^9*e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*a^6*d^3*e^12)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{3}}\, dx \]

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)**(3/2),x)

[Out]

Integral(1/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)

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

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)^(3/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

[undef, undef, undef, 1]