3.2028 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c^2*d^2*(d
+ e*x)^(5/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*(d + e
*x)^(3/2))

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Rubi [A]  time = 0.191709, 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 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c^2*d^2*(d
+ e*x)^(5/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*(d + e
*x)^(3/2))

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Rubi in Sympy [A]  time = 33.1893, size = 100, normalized size = 0.92 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 c d \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{35 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(7*c*d*(d + e*x)**(3/2)) - 4
*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(35*c**2*d*
*2*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.102043, size = 55, normalized size = 0.5 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (c d (7 d+5 e x)-2 a e^2\right )}{35 c^2 d^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-2*a*e^2 + c*d*(7*d + 5*e*x)))/(35*c^2*d^2*(
d + e*x)^(5/2))

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Maple [A]  time = 0.008, size = 69, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,cdex+2\,a{e}^{2}-7\,c{d}^{2} \right ) }{35\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*d*x+a*e)*(-5*c*d*e*x+2*a*e^2-7*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(3/2)/c^2/d^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.739787, size = 132, normalized size = 1.21 \[ \frac{2 \,{\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt{c d x + a e}}{35 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*c^3*d^3*e*x^3 + 7*a^2*c*d^2*e^2 - 2*a^3*e^4 + (7*c^3*d^4 + 8*a*c^2*d^2*e
^2)*x^2 + (14*a*c^2*d^3*e + a^2*c*d*e^3)*x)*sqrt(c*d*x + a*e)/(c^2*d^2)

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Fricas [A]  time = 0.214154, size = 296, normalized size = 2.72 \[ \frac{2 \,{\left (5 \, c^{4} d^{4} e^{2} x^{5} + 7 \, a^{3} c d^{3} e^{3} - 2 \, a^{4} d e^{5} +{\left (12 \, c^{4} d^{5} e + 13 \, a c^{3} d^{3} e^{3}\right )} x^{4} +{\left (7 \, c^{4} d^{6} + 34 \, a c^{3} d^{4} e^{2} + 9 \, a^{2} c^{2} d^{2} e^{4}\right )} x^{3} +{\left (21 \, a c^{3} d^{5} e + 30 \, a^{2} c^{2} d^{3} e^{3} - a^{3} c d e^{5}\right )} x^{2} +{\left (21 \, a^{2} c^{2} d^{4} e^{2} + 6 \, a^{3} c d^{2} e^{4} - 2 \, a^{4} e^{6}\right )} x\right )}}{35 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*c^4*d^4*e^2*x^5 + 7*a^3*c*d^3*e^3 - 2*a^4*d*e^5 + (12*c^4*d^5*e + 13*a*c
^3*d^3*e^3)*x^4 + (7*c^4*d^6 + 34*a*c^3*d^4*e^2 + 9*a^2*c^2*d^2*e^4)*x^3 + (21*a
*c^3*d^5*e + 30*a^2*c^2*d^3*e^3 - a^3*c*d*e^5)*x^2 + (21*a^2*c^2*d^4*e^2 + 6*a^3
*c*d^2*e^4 - 2*a^4*e^6)*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out