3.2027 \(\int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt{d+e x}} \]

[Out]

(16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*d^
3*(d + e*x)^(5/2)) + (8*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2))/(63*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)
^(5/2))/(9*c*d*Sqrt[d + e*x])

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Rubi [A]  time = 0.335979, 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 \[ \frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 c^3 d^3 (d+e x)^{5/2}}+\frac{8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*(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*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*d^
3*(d + e*x)^(5/2)) + (8*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^
(5/2))/(63*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)
^(5/2))/(9*c*d*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 52.9849, size = 160, normalized size = 0.94 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 c d \sqrt{d + e x}} - \frac{8 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{315 c^{3} d^{3} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(9*c*d*sqrt(d + e*x)) - 8*(a
*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(63*c**2*d**2*
(d + e*x)**(3/2)) + 16*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(5/2)/(315*c**3*d**3*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.150563, size = 88, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 e^4-4 a c d e^2 (9 d+5 e x)+c^2 d^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )}{315 c^3 d^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(8*a^2*e^4 - 4*a*c*d*e^2*(9*d + 5*e*x) + c^2*
d^2*(63*d^2 + 90*d*e*x + 35*e^2*x^2)))/(315*c^3*d^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.011, size = 110, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}-20\,xacd{e}^{3}+90\,x{c}^{2}{d}^{3}e+8\,{a}^{2}{e}^{4}-36\,ac{d}^{2}{e}^{2}+63\,{c}^{2}{d}^{4} \right ) }{315\,{c}^{3}{d}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(c*d*x+a*e)*(35*c^2*d^2*e^2*x^2-20*a*c*d*e^3*x+90*c^2*d^3*e*x+8*a^2*e^4-36
*a*c*d^2*e^2+63*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/c^3/d^3/(e*x+d)
^(3/2)

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Maxima [A]  time = 0.766991, size = 255, normalized size = 1.49 \[ \frac{2 \,{\left (35 \, c^{4} d^{4} e^{2} x^{4} + 63 \, a^{2} c^{2} d^{4} e^{2} - 36 \, a^{3} c d^{2} e^{4} + 8 \, a^{4} e^{6} + 10 \,{\left (9 \, c^{4} d^{5} e + 5 \, a c^{3} d^{3} e^{3}\right )} x^{3} + 3 \,{\left (21 \, c^{4} d^{6} + 48 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (63 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 2 \, a^{3} c d e^{5}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{315 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \]

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]

2/315*(35*c^4*d^4*e^2*x^4 + 63*a^2*c^2*d^4*e^2 - 36*a^3*c*d^2*e^4 + 8*a^4*e^6 +
10*(9*c^4*d^5*e + 5*a*c^3*d^3*e^3)*x^3 + 3*(21*c^4*d^6 + 48*a*c^3*d^4*e^2 + a^2*
c^2*d^2*e^4)*x^2 + 2*(63*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 2*a^3*c*d*e^5)*x)*sqr
t(c*d*x + a*e)*(e*x + d)/(c^3*d^3*e*x + c^3*d^4)

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Fricas [A]  time = 0.21615, size = 429, normalized size = 2.51 \[ \frac{2 \,{\left (35 \, c^{5} d^{5} e^{3} x^{6} + 63 \, a^{3} c^{2} d^{5} e^{3} - 36 \, a^{4} c d^{3} e^{5} + 8 \, a^{5} d e^{7} + 5 \,{\left (25 \, c^{5} d^{6} e^{2} + 17 \, a c^{4} d^{4} e^{4}\right )} x^{5} +{\left (153 \, c^{5} d^{7} e + 319 \, a c^{4} d^{5} e^{3} + 53 \, a^{2} c^{3} d^{3} e^{5}\right )} x^{4} +{\left (63 \, c^{5} d^{8} + 423 \, a c^{4} d^{6} e^{2} + 215 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{3} +{\left (189 \, a c^{4} d^{7} e + 351 \, a^{2} c^{3} d^{5} e^{3} - 19 \, a^{3} c^{2} d^{3} e^{5} + 4 \, a^{4} c d e^{7}\right )} x^{2} +{\left (189 \, a^{2} c^{3} d^{6} e^{2} + 45 \, a^{3} c^{2} d^{4} e^{4} - 32 \, a^{4} c d^{2} e^{6} + 8 \, a^{5} e^{8}\right )} x\right )}}{315 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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]

2/315*(35*c^5*d^5*e^3*x^6 + 63*a^3*c^2*d^5*e^3 - 36*a^4*c*d^3*e^5 + 8*a^5*d*e^7
+ 5*(25*c^5*d^6*e^2 + 17*a*c^4*d^4*e^4)*x^5 + (153*c^5*d^7*e + 319*a*c^4*d^5*e^3
+ 53*a^2*c^3*d^3*e^5)*x^4 + (63*c^5*d^8 + 423*a*c^4*d^6*e^2 + 215*a^2*c^3*d^4*e
^4 - a^3*c^2*d^2*e^6)*x^3 + (189*a*c^4*d^7*e + 351*a^2*c^3*d^5*e^3 - 19*a^3*c^2*
d^3*e^5 + 4*a^4*c*d*e^7)*x^2 + (189*a^2*c^3*d^6*e^2 + 45*a^3*c^2*d^4*e^4 - 32*a^
4*c*d^2*e^6 + 8*a^5*e^8)*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)**(1/2)*(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 [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