3.1952 \(\int \frac{1}{(d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=301 \[ -\frac{256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) +
(20*c*d)/(63*(c*d^2 - a*e^2)^2*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(6
3*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.516342, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ -\frac{256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) +
(20*c*d)/(63*(c*d^2 - a*e^2)^2*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(6
3*(c*d^2 - a*e^2)^6*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 = 96.3153, size = 289, normalized size = 0.96 \[ - \frac{128 c^{4} d^{4} \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{63 \left (a e^{2} - c d^{2}\right )^{6} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{64 c^{3} d^{3}}{63 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{32 c^{2} d^{2}}{63 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{20 c d}{63 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{2}{9 \left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

-128*c**4*d**4*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)/(63*(a*e**2 - c*d**2)**6*sqrt(a
*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) + 64*c**3*d**3/(63*(d + e*x)*(a*e**2 -
c*d**2)**4*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 32*c**2*d**2/(63*(
d + e*x)**2*(a*e**2 - c*d**2)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))
+ 20*c*d/(63*(d + e*x)**3*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e
**2 + c*d**2))) - 2/(9*(d + e*x)**4*(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 = 0.671941, size = 179, normalized size = 0.59 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-\frac{63 c^5 d^5}{a e+c d x}+\frac{65 c^3 d^3 e \left (a e^2-c d^2\right )}{(d+e x)^2}-\frac{33 c^2 d^2 e \left (c d^2-a e^2\right )^2}{(d+e x)^3}+\frac{17 c d e \left (a e^2-c d^2\right )^3}{(d+e x)^4}-\frac{7 e \left (c d^2-a e^2\right )^4}{(d+e x)^5}-\frac{193 c^4 d^4 e}{d+e x}\right )}{63 \left (c d^2-a e^2\right )^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(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)]*((-63*c^5*d^5)/(a*e + c*d*x) - (7*e*(c*d^2 - a*
e^2)^4)/(d + e*x)^5 + (17*c*d*e*(-(c*d^2) + a*e^2)^3)/(d + e*x)^4 - (33*c^2*d^2*
e*(c*d^2 - a*e^2)^2)/(d + e*x)^3 + (65*c^3*d^3*e*(-(c*d^2) + a*e^2))/(d + e*x)^2
- (193*c^4*d^4*e)/(d + e*x)))/(63*(c*d^2 - a*e^2)^6)

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Maple [A]  time = 0.021, size = 412, normalized size = 1.4 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}+128\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}+1152\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-32\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}+576\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}+2016\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-144\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}+1008\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}+1680\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-10\,{a}^{4}cd{e}^{9}x+72\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-252\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x+840\,a{c}^{4}{d}^{7}{e}^{3}x+630\,{c}^{5}{d}^{9}ex+7\,{a}^{5}{e}^{10}-45\,{a}^{4}c{d}^{2}{e}^{8}+126\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+315\,a{c}^{4}{d}^{8}{e}^{2}+63\,{c}^{5}{d}^{10} \right ) }{63\, \left ({a}^{6}{e}^{12}-6\,{a}^{5}c{d}^{2}{e}^{10}+15\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-20\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+15\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}-6\,a{c}^{5}{d}^{10}{e}^{2}+{c}^{6}{d}^{12} \right ) \left ( ex+d \right ) ^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-2/63*(c*d*x+a*e)*(256*c^5*d^5*e^5*x^5+128*a*c^4*d^4*e^6*x^4+1152*c^5*d^6*e^4*x^
4-32*a^2*c^3*d^3*e^7*x^3+576*a*c^4*d^5*e^5*x^3+2016*c^5*d^7*e^3*x^3+16*a^3*c^2*d
^2*e^8*x^2-144*a^2*c^3*d^4*e^6*x^2+1008*a*c^4*d^6*e^4*x^2+1680*c^5*d^8*e^2*x^2-1
0*a^4*c*d*e^9*x+72*a^3*c^2*d^3*e^7*x-252*a^2*c^3*d^5*e^5*x+840*a*c^4*d^7*e^3*x+6
30*c^5*d^9*e*x+7*a^5*e^10-45*a^4*c*d^2*e^8+126*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e
^4+315*a*c^4*d^8*e^2+63*c^5*d^10)/(e*x+d)^3/(a^6*e^12-6*a^5*c*d^2*e^10+15*a^4*c^
2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d^12)/(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)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 15.9069, size = 1355, normalized size = 4.5 \[ \text{result too large to display} \]

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)^4),x, algorithm="fricas")

[Out]

-2/63*(256*c^5*d^5*e^5*x^5 + 63*c^5*d^10 + 315*a*c^4*d^8*e^2 - 210*a^2*c^3*d^6*e
^4 + 126*a^3*c^2*d^4*e^6 - 45*a^4*c*d^2*e^8 + 7*a^5*e^10 + 128*(9*c^5*d^6*e^4 +
a*c^4*d^4*e^6)*x^4 + 32*(63*c^5*d^7*e^3 + 18*a*c^4*d^5*e^5 - a^2*c^3*d^3*e^7)*x^
3 + 16*(105*c^5*d^8*e^2 + 63*a*c^4*d^6*e^4 - 9*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8
)*x^2 + 2*(315*c^5*d^9*e + 420*a*c^4*d^7*e^3 - 126*a^2*c^3*d^5*e^5 + 36*a^3*c^2*
d^3*e^7 - 5*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^6*d
^17*e - 6*a^2*c^5*d^15*e^3 + 15*a^3*c^4*d^13*e^5 - 20*a^4*c^3*d^11*e^7 + 15*a^5*
c^2*d^9*e^9 - 6*a^6*c*d^7*e^11 + a^7*d^5*e^13 + (c^7*d^13*e^5 - 6*a*c^6*d^11*e^7
+ 15*a^2*c^5*d^9*e^9 - 20*a^3*c^4*d^7*e^11 + 15*a^4*c^3*d^5*e^13 - 6*a^5*c^2*d^
3*e^15 + a^6*c*d*e^17)*x^6 + (5*c^7*d^14*e^4 - 29*a*c^6*d^12*e^6 + 69*a^2*c^5*d^
10*e^8 - 85*a^3*c^4*d^8*e^10 + 55*a^4*c^3*d^6*e^12 - 15*a^5*c^2*d^4*e^14 - a^6*c
*d^2*e^16 + a^7*e^18)*x^5 + 5*(2*c^7*d^15*e^3 - 11*a*c^6*d^13*e^5 + 24*a^2*c^5*d
^11*e^7 - 25*a^3*c^4*d^9*e^9 + 10*a^4*c^3*d^7*e^11 + 3*a^5*c^2*d^5*e^13 - 4*a^6*
c*d^3*e^15 + a^7*d*e^17)*x^4 + 10*(c^7*d^16*e^2 - 5*a*c^6*d^14*e^4 + 9*a^2*c^5*d
^12*e^6 - 5*a^3*c^4*d^10*e^8 - 5*a^4*c^3*d^8*e^10 + 9*a^5*c^2*d^6*e^12 - 5*a^6*c
*d^4*e^14 + a^7*d^2*e^16)*x^3 + 5*(c^7*d^17*e - 4*a*c^6*d^15*e^3 + 3*a^2*c^5*d^1
3*e^5 + 10*a^3*c^4*d^11*e^7 - 25*a^4*c^3*d^9*e^9 + 24*a^5*c^2*d^7*e^11 - 11*a^6*
c*d^5*e^13 + 2*a^7*d^3*e^15)*x^2 + (c^7*d^18 - a*c^6*d^16*e^2 - 15*a^2*c^5*d^14*
e^4 + 55*a^3*c^4*d^12*e^6 - 85*a^4*c^3*d^10*e^8 + 69*a^5*c^2*d^8*e^10 - 29*a^6*c
*d^6*e^12 + 5*a^7*d^4*e^14)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(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)**4), 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)^4),x, algorithm="giac")

[Out]

[undef, undef, undef, 1]