3.1974 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \]

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)
^(7/2))/(7*e^3) + (2*c^2*d^2*(d + e*x)^(9/2))/(9*e^3)

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Rubi [A]  time = 0.120077, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ -\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)
^(7/2))/(7*e^3) + (2*c^2*d^2*(d + e*x)^(9/2))/(9*e^3)

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Rubi in Sympy [A]  time = 35.28, size = 76, normalized size = 0.92 \[ \frac{2 c^{2} d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{4 c d \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )}{7 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{5 e^{3}} \]

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)**2/(e*x+d)**(1/2),x)

[Out]

2*c**2*d**2*(d + e*x)**(9/2)/(9*e**3) + 4*c*d*(d + e*x)**(7/2)*(a*e**2 - c*d**2)
/(7*e**3) + 2*(d + e*x)**(5/2)*(a*e**2 - c*d**2)**2/(5*e**3)

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Mathematica [A]  time = 0.0927192, size = 67, normalized size = 0.81 \[ \frac{2 (d+e x)^{5/2} \left (63 a^2 e^4+18 a c d e^2 (5 e x-2 d)+c^2 d^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(63*a^2*e^4 + 18*a*c*d*e^2*(-2*d + 5*e*x) + c^2*d^2*(8*d^2 -
20*d*e*x + 35*e^2*x^2)))/(315*e^3)

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Maple [A]  time = 0.01, size = 73, normalized size = 0.9 \[{\frac{70\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+180\,xacd{e}^{3}-40\,x{c}^{2}{d}^{3}e+126\,{a}^{2}{e}^{4}-72\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

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)^2/(e*x+d)^(1/2),x)

[Out]

2/315*(e*x+d)^(5/2)*(35*c^2*d^2*e^2*x^2+90*a*c*d*e^3*x-20*c^2*d^3*e*x+63*a^2*e^4
-36*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [A]  time = 0.775433, size = 378, normalized size = 4.55 \[ \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} d^{2} e^{2} + 42 \,{\left (\frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c d}{e} + \frac{5 \,{\left (c d^{2} + a e^{2}\right )}{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )}}{e}\right )} a d e + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )}{\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac{21 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )}}{e^{2}}\right )}}{315 \, e} \]

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)^2/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(e*x + d)*a^2*d^2*e^2 + 42*((3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/
2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e + 5*(c*d^2 + a*e^2)*((e*x + d)^(3/2) - 3*sqrt
(e*x + d)*d)/e)*a*d*e + (35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x +
d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^2/e^2 + 1
8*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e
*x + d)*d^3)*(c*d^2 + a*e^2)*c*d/e^2 + 21*(c*d^2 + a*e^2)^2*(3*(e*x + d)^(5/2) -
10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)/e^2)/e

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Fricas [A]  time = 0.218677, size = 198, normalized size = 2.39 \[ \frac{2 \,{\left (35 \, c^{2} d^{2} e^{4} x^{4} + 8 \, c^{2} d^{6} - 36 \, a c d^{4} e^{2} + 63 \, a^{2} d^{2} e^{4} + 10 \,{\left (5 \, c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (c^{2} d^{4} e^{2} + 48 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (2 \, c^{2} d^{5} e - 9 \, a c d^{3} e^{3} - 63 \, a^{2} d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \]

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)^2/sqrt(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 60.043, size = 631, normalized size = 7.6 \[ \begin{cases} - \frac{\frac{2 a^{2} d^{3} e^{2}}{\sqrt{d + e x}} + 6 a^{2} d^{2} e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} d e^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + 2 a^{2} e^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 4 a c d^{4} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 12 a c d^{3} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + 12 a c d^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 4 a c d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right ) + \frac{2 c^{2} d^{5} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{6 c^{2} d^{4} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 c^{2} d^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 c^{2} d^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{7}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \]

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)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*a**2*d**3*e**2/sqrt(d + e*x) + 6*a**2*d**2*e**2*(-d/sqrt(d + e*x)
- sqrt(d + e*x)) + 6*a**2*d*e**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d +
e*x)**(3/2)/3) + 2*a**2*e**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d
+ e*x)**(3/2) - (d + e*x)**(5/2)/5) + 4*a*c*d**4*(-d/sqrt(d + e*x) - sqrt(d + e
*x)) + 12*a*c*d**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)
+ 12*a*c*d**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2)
- (d + e*x)**(5/2)/5) + 4*a*c*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d
**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7) + 2*c**2*d**
5*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 6*c**2*d*
*4*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)*
*(5/2)/5)/e**2 + 6*c**2*d**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2
*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**2 + 2*c**2*d
**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2
*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**2)/e, N
e(e, 0)), (c**2*d**(7/2)*x**3/3, True))

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GIAC/XCAS [A]  time = 0.207393, size = 589, normalized size = 7.1 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} c^{2} d^{4} e^{\left (-10\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} c^{2} d^{3} e^{\left (-20\right )} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a c d^{3} + 84 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a c d^{2} e^{\left (-8\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} c^{2} d^{2} e^{\left (-34\right )} + 315 \, \sqrt{x e + d} a^{2} d^{2} e^{2} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} d e^{2} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a c d e^{\left (-18\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{2} e^{\left (-6\right )}\right )} e^{\left (-1\right )} \]

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)^2/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/315*(21*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d
^2*e^8)*c^2*d^4*e^(-10) + 18*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18
+ 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*c^2*d^3*e^(-20) + 21
0*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*c*d^3 + 84*(3*(x*e + d)^(5/2)*e^8 - 10
*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*a*c*d^2*e^(-8) + (35*(x*e + d
)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(
x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d^4*e^32)*c^2*d^2*e^(-34) + 315*sqrt
(x*e + d)*a^2*d^2*e^2 + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d*e^2 + 18
*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^
18 - 35*sqrt(x*e + d)*d^3*e^18)*a*c*d*e^(-18) + 21*(3*(x*e + d)^(5/2)*e^8 - 10*(
x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*a^2*e^(-6))*e^(-1)