3.1961 \(\int \frac{1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=252 \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))
+ (4*c*d)/(7*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^2 - a*e^2)^4*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*
e*x))/(21*(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 = 0.336548, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))
+ (4*c*d)/(7*(c*d^2 - a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^2 - a*e^2)^4*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*
e*x))/(21*(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 = 56.3197, size = 245, normalized size = 0.97 \[ \frac{128 c^{3} d^{3} e \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{21 \left (a e^{2} - c d^{2}\right )^{6} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{32 c^{2} d^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right )}{21 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} + \frac{4 c d}{7 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{2}{7 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

128*c**3*d**3*e*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)/(21*(a*e**2 - c*d**2)**6*sqrt(
a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 32*c**2*d**2*(a*e**2 + c*d**2 + 2*c
*d*e*x)/(21*(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*c*d/(7*(d + e*x)*(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/(7*(d + e*x)**2*(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 = 0.90154, size = 203, normalized size = 0.81 \[ \frac{2 (d+e x)^3 (a e+c d x)^3 \left (\frac{98 c^4 d^4 e}{a e+c d x}+\frac{7 c^4 d^4 \left (a e^2-c d^2\right )}{(a e+c d x)^2}+\frac{37 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{12 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^3}-\frac{3 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^4}+\frac{158 c^3 d^3 e^2}{d+e x}\right )}{21 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(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*((7*c^4*d^4*(-(c*d^2) + a*e^2))/(a*e + c*d*x)^2 +
(98*c^4*d^4*e)/(a*e + c*d*x) - (3*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^4 + (12*c
*d*(c*d^2*e - a*e^3)^2)/(d + e*x)^3 + (37*c^2*d^2*e^2*(c*d^2 - a*e^2))/(d + e*x)
^2 + (158*c^3*d^3*e^2)/(d + e*x)))/(21*(c*d^2 - a*e^2)^6*((a*e + c*d*x)*(d + e*x
))^(5/2))

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Maple [A]  time = 0.022, size = 412, normalized size = 1.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}-384\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}-896\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-96\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}-1344\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}-1120\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-336\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}-1680\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}-560\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-6\,{a}^{4}cd{e}^{9}x+56\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-420\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x-840\,a{c}^{4}{d}^{7}{e}^{3}x-70\,{c}^{5}{d}^{9}ex+3\,{a}^{5}{e}^{10}-21\,{a}^{4}c{d}^{2}{e}^{8}+70\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}-105\,a{c}^{4}{d}^{8}{e}^{2}+7\,{c}^{5}{d}^{10} \right ) }{ \left ( 21\,ex+21\,d \right ) \left ({a}^{6}{e}^{12}-6\,{a}^{5}c{d}^{2}{e}^{10}+15\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-20\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+15\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}-6\,a{c}^{5}{d}^{10}{e}^{2}+{c}^{6}{d}^{12} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(c*d*x+a*e)*(-256*c^5*d^5*e^5*x^5-384*a*c^4*d^4*e^6*x^4-896*c^5*d^6*e^4*x^
4-96*a^2*c^3*d^3*e^7*x^3-1344*a*c^4*d^5*e^5*x^3-1120*c^5*d^7*e^3*x^3+16*a^3*c^2*
d^2*e^8*x^2-336*a^2*c^3*d^4*e^6*x^2-1680*a*c^4*d^6*e^4*x^2-560*c^5*d^8*e^2*x^2-6
*a^4*c*d*e^9*x+56*a^3*c^2*d^3*e^7*x-420*a^2*c^3*d^5*e^5*x-840*a*c^4*d^7*e^3*x-70
*c^5*d^9*e*x+3*a^5*e^10-21*a^4*c*d^2*e^8+70*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e^4-
105*a*c^4*d^8*e^2+7*c^5*d^10)/(e*x+d)/(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)^(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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.1878, size = 1428, normalized size = 5.67 \[ \text{result too large to display} \]

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)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

2/21*(256*c^5*d^5*e^5*x^5 - 7*c^5*d^10 + 105*a*c^4*d^8*e^2 + 210*a^2*c^3*d^6*e^4
- 70*a^3*c^2*d^4*e^6 + 21*a^4*c*d^2*e^8 - 3*a^5*e^10 + 128*(7*c^5*d^6*e^4 + 3*a
*c^4*d^4*e^6)*x^4 + 32*(35*c^5*d^7*e^3 + 42*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x
^3 + 16*(35*c^5*d^8*e^2 + 105*a*c^4*d^6*e^4 + 21*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e
^8)*x^2 + 2*(35*c^5*d^9*e + 420*a*c^4*d^7*e^3 + 210*a^2*c^3*d^5*e^5 - 28*a^3*c^2
*d^3*e^7 + 3*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^
6*d^16*e^2 - 6*a^3*c^5*d^14*e^4 + 15*a^4*c^4*d^12*e^6 - 20*a^5*c^3*d^10*e^8 + 15
*a^6*c^2*d^8*e^10 - 6*a^7*c*d^6*e^12 + a^8*d^4*e^14 + (c^8*d^14*e^4 - 6*a*c^7*d^
12*e^6 + 15*a^2*c^6*d^10*e^8 - 20*a^3*c^5*d^8*e^10 + 15*a^4*c^4*d^6*e^12 - 6*a^5
*c^3*d^4*e^14 + a^6*c^2*d^2*e^16)*x^6 + 2*(2*c^8*d^15*e^3 - 11*a*c^7*d^13*e^5 +
24*a^2*c^6*d^11*e^7 - 25*a^3*c^5*d^9*e^9 + 10*a^4*c^4*d^7*e^11 + 3*a^5*c^3*d^5*e
^13 - 4*a^6*c^2*d^3*e^15 + a^7*c*d*e^17)*x^5 + (6*c^8*d^16*e^2 - 28*a*c^7*d^14*e
^4 + 43*a^2*c^6*d^12*e^6 - 6*a^3*c^5*d^10*e^8 - 55*a^4*c^4*d^8*e^10 + 64*a^5*c^3
*d^6*e^12 - 27*a^6*c^2*d^4*e^14 + 2*a^7*c*d^2*e^16 + a^8*e^18)*x^4 + 4*(c^8*d^17
*e - 3*a*c^7*d^15*e^3 - 2*a^2*c^6*d^13*e^5 + 19*a^3*c^5*d^11*e^7 - 30*a^4*c^4*d^
9*e^9 + 19*a^5*c^3*d^7*e^11 - 2*a^6*c^2*d^5*e^13 - 3*a^7*c*d^3*e^15 + a^8*d*e^17
)*x^3 + (c^8*d^18 + 2*a*c^7*d^16*e^2 - 27*a^2*c^6*d^14*e^4 + 64*a^3*c^5*d^12*e^6
- 55*a^4*c^4*d^10*e^8 - 6*a^5*c^3*d^8*e^10 + 43*a^6*c^2*d^6*e^12 - 28*a^7*c*d^4
*e^14 + 6*a^8*d^2*e^16)*x^2 + 2*(a*c^7*d^17*e - 4*a^2*c^6*d^15*e^3 + 3*a^3*c^5*d
^13*e^5 + 10*a^4*c^4*d^11*e^7 - 25*a^5*c^3*d^9*e^9 + 24*a^6*c^2*d^7*e^11 - 11*a^
7*c*d^5*e^13 + 2*a^8*d^3*e^15)*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(1/(e*x+d)**2/(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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError