3.1949 \(\int \frac{1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8
*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.136367, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8
*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3*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 = 12.0871, size = 116, normalized size = 0.96 \[ \frac{4 c d \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{3 \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{2}{3 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c*d*(2*a*e**2 + 2*c*d**2 + 4*c*d*e*x)/(3*(a*e**2 - c*d**2)**3*sqrt(a*d*e + c*d
*e*x**2 + x*(a*e**2 + c*d**2))) - 2/(3*(d + e*x)*(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 = 0.176988, size = 95, normalized size = 0.79 \[ \frac{2 a^2 e^4-4 a c d e^2 (3 d+2 e x)-2 c^2 d^2 \left (3 d^2+12 d e x+8 e^2 x^2\right )}{3 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*e^4 - 4*a*c*d*e^2*(3*d + 2*e*x) - 2*c^2*d^2*(3*d^2 + 12*d*e*x + 8*e^2*x^2
))/(3*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.013, size = 138, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -8\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-4\,xacd{e}^{3}-12\,x{c}^{2}{d}^{3}e+{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-3\,{c}^{2}{d}^{4} \right ) }{3\,{a}^{3}{e}^{6}-9\,{a}^{2}c{d}^{2}{e}^{4}+9\,{c}^{2}{d}^{4}a{e}^{2}-3\,{c}^{3}{d}^{6}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*d*x+a*e)*(-8*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x-12*c^2*d^3*e*x+a^2*e^4-6*a*c*
d^2*e^2-3*c^2*d^4)/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.523955, size = 413, normalized size = 3.41 \[ -\frac{2 \,{\left (8 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 4 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{3 \,{\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} +{\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} +{\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} +{\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \]

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)),x, algorithm="fricas")

[Out]

-2/3*(8*c^2*d^2*e^2*x^2 + 3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4 + 4*(3*c^2*d^3*e +
a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^3*d^8*e - 3*a^2*
c^2*d^6*e^3 + 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*c^3*d^5*e^4 + 3
*a^2*c^2*d^3*e^6 - a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5*a*c^3*d^6*e^3 + 3*a^2*c^2
*d^4*e^5 + a^3*c*d^2*e^7 - a^4*e^9)*x^2 + (c^4*d^9 - a*c^3*d^7*e^2 - 3*a^2*c^2*d
^5*e^4 + 5*a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, 1]