3.2054 \(\int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-16*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (8*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*c^2*d^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2]) + (2*(d + e*x)^(5/2))/(3*c*d*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.331042, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ -\frac{16 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2 (d+e x)^{5/2}}{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-16*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (8*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*c^2*d^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2]) + (2*(d + e*x)^(5/2))/(3*c*d*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 = 53.0508, size = 160, normalized size = 0.94 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{8 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{16 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{3 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

2*(d + e*x)**(5/2)/(3*c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 8*(d
+ e*x)**(3/2)*(a*e**2 - c*d**2)/(3*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**
2 + c*d**2))) - 16*sqrt(d + e*x)*(a*e**2 - c*d**2)**2/(3*c**3*d**3*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2)))

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Mathematica [A]  time = 0.111841, size = 87, normalized size = 0.51 \[ -\frac{2 \sqrt{d+e x} \left (8 a^2 e^4+4 a c d e^2 (e x-3 d)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(8*a^2*e^4 + 4*a*c*d*e^2*(-3*d + e*x) + c^2*d^2*(3*d^2 - 6*d*e
*x - e^2*x^2)))/(3*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.009, size = 110, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -{x}^{2}{c}^{2}{d}^{2}{e}^{2}+4\,xacd{e}^{3}-6\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-12\,ac{d}^{2}{e}^{2}+3\,{c}^{2}{d}^{4} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(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)*(-c^2*d^2*e^2*x^2+4*a*c*d*e^3*x-6*c^2*d^3*e*x+8*a^2*e^4-12*a*c*
d^2*e^2+3*c^2*d^4)*(e*x+d)^(3/2)/c^3/d^3/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)

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Maxima [A]  time = 0.829259, size = 107, normalized size = 0.63 \[ \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \,{\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt{c d x + a e} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.214772, size = 185, normalized size = 1.08 \[ \frac{2 \,{\left (c^{2} d^{2} e^{3} x^{3} - 3 \, c^{2} d^{5} + 12 \, a c d^{3} e^{2} - 8 \, a^{2} d e^{4} +{\left (7 \, c^{2} d^{3} e^{2} - 4 \, a c d e^{4}\right )} x^{2} +{\left (3 \, c^{2} d^{4} e + 8 \, a c d^{2} e^{3} - 8 \, a^{2} e^{5}\right )} x\right )}}{3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="fricas")

[Out]

2/3*(c^2*d^2*e^3*x^3 - 3*c^2*d^5 + 12*a*c*d^3*e^2 - 8*a^2*d*e^4 + (7*c^2*d^3*e^2
- 4*a*c*d*e^4)*x^2 + (3*c^2*d^4*e + 8*a*c*d^2*e^3 - 8*a^2*e^5)*x)/(sqrt(c*d*e*x
^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^3*d^3)

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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((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.624051, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x