3.2047 \(\int \frac{(d+e x)^{5/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^3 d^3 \sqrt{d+e x}}+\frac{8 \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^2 d^2}+\frac{2 (d+e x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d} \]

[Out]

(16*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^3*d^3*S
qrt[d + e*x]) + (8*(c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(15*c^2*d^2) + (2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2])/(5*c*d)

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Rubi [A]  time = 0.354715, 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 \[ \frac{16 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^3 d^3 \sqrt{d+e x}}+\frac{8 \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^2 d^2}+\frac{2 (d+e x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d} \]

Antiderivative was successfully verified.

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

[Out]

(16*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^3*d^3*S
qrt[d + e*x]) + (8*(c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(15*c^2*d^2) + (2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2])/(5*c*d)

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Rubi in Sympy [A]  time = 53.3725, size = 160, normalized size = 0.94 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 c d} - \frac{8 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 c^{2} d^{2}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 c^{3} d^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(3/2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(5*c*d) - 8*sq
rt(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))/(15
*c**2*d**2) + 16*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d*
*2))/(15*c**3*d**3*sqrt(d + e*x))

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Mathematica [A]  time = 0.118725, size = 87, normalized size = 0.51 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 e^4-4 a c d e^2 (5 d+e x)+c^2 d^2 \left (15 d^2+10 d e x+3 e^2 x^2\right )\right )}{15 c^3 d^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2*(5*d + e*x) + c^2*d^2*
(15*d^2 + 10*d*e*x + 3*e^2*x^2)))/(15*c^3*d^3*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 110, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-4\,xacd{e}^{3}+10\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-20\,ac{d}^{2}{e}^{2}+15\,{c}^{2}{d}^{4} \right ) }{15\,{c}^{3}{d}^{3}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(c*d*x+a*e)*(3*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x+10*c^2*d^3*e*x+8*a^2*e^4-20*a*
c*d^2*e^2+15*c^2*d^4)*(e*x+d)^(1/2)/c^3/d^3/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1
/2)

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Maxima [A]  time = 0.815804, size = 165, normalized size = 0.96 \[ \frac{2 \,{\left (3 \, c^{3} d^{3} e^{2} x^{3} + 15 \, a c^{2} d^{4} e - 20 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} +{\left (10 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (15 \, c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 4 \, a^{2} c d e^{4}\right )} x\right )}}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

2/15*(3*c^3*d^3*e^2*x^3 + 15*a*c^2*d^4*e - 20*a^2*c*d^2*e^3 + 8*a^3*e^5 + (10*c^
3*d^4*e - a*c^2*d^2*e^3)*x^2 + (15*c^3*d^5 - 10*a*c^2*d^3*e^2 + 4*a^2*c*d*e^4)*x
)/(sqrt(c*d*x + a*e)*c^3*d^3)

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Fricas [A]  time = 0.214139, size = 263, normalized size = 1.54 \[ \frac{2 \,{\left (3 \, c^{3} d^{3} e^{3} x^{4} + 15 \, a c^{2} d^{5} e - 20 \, a^{2} c d^{3} e^{3} + 8 \, a^{3} d e^{5} +{\left (13 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{3} +{\left (25 \, c^{3} d^{5} e - 11 \, a c^{2} d^{3} e^{3} + 4 \, a^{2} c d e^{5}\right )} x^{2} +{\left (15 \, c^{3} d^{6} + 5 \, a c^{2} d^{4} e^{2} - 16 \, a^{2} c d^{2} e^{4} + 8 \, a^{3} e^{6}\right )} x\right )}}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

2/15*(3*c^3*d^3*e^3*x^4 + 15*a*c^2*d^5*e - 20*a^2*c*d^3*e^3 + 8*a^3*d*e^5 + (13*
c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^3 + (25*c^3*d^5*e - 11*a*c^2*d^3*e^3 + 4*a^2*c*d*
e^5)*x^2 + (15*c^3*d^6 + 5*a*c^2*d^4*e^2 - 16*a^2*c*d^2*e^4 + 8*a^3*e^6)*x)/(sqr
t(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)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

integrate((e*x + d)^(5/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)