3.2019 \(\int \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.190825, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.9349, size = 100, normalized size = 0.92 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{5 c d \sqrt{d + e x}} - \frac{4 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{15 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0702423, size = 55, normalized size = 0.5 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (c d (5 d+3 e x)-2 a e^2\right )}{15 c^2 d^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-2*a*e^2 + c*d*(5*d + 3*e*x)))/(15*c^2*d^2*(
d + e*x)^(3/2))

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Maple [A]  time = 0.006, size = 69, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -3\,cdex+2\,a{e}^{2}-5\,c{d}^{2} \right ) }{15\,{c}^{2}{d}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]  time = 0.747597, size = 112, normalized size = 1.03 \[ \frac{2 \,{\left (3 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (5 \, c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{15 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 2*a^2*e^3 + (5*c^2*d^3 + a*c*d*e^2)*x)*sqr
t(c*d*x + a*e)*(e*x + d)/(c^2*d^2*e*x + c^2*d^3)

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Fricas [A]  time = 0.219021, size = 234, normalized size = 2.15 \[ \frac{2 \,{\left (3 \, c^{3} d^{3} e^{2} x^{4} + 5 \, a^{2} c d^{3} e^{2} - 2 \, a^{3} d e^{4} + 4 \,{\left (2 \, c^{3} d^{4} e + a c^{2} d^{2} e^{3}\right )} x^{3} +{\left (5 \, c^{3} d^{5} + 14 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x^{2} + 2 \,{\left (5 \, a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x\right )}}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*c^3*d^3*e^2*x^4 + 5*a^2*c*d^3*e^2 - 2*a^3*d*e^4 + 4*(2*c^3*d^4*e + a*c^2
*d^2*e^3)*x^3 + (5*c^3*d^5 + 14*a*c^2*d^3*e^2 - a^2*c*d*e^4)*x^2 + 2*(5*a*c^2*d^
4*e + 2*a^2*c*d^2*e^3 - a^3*e^5)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*sqrt(e*x + d)*c^2*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))*sqrt(d + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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