3.1966 \(\int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{5} (d+e x)^{5/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{7/2}}{7 e^2} \]

[Out]

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

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Rubi [A]  time = 0.0671011, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{2}{5} (d+e x)^{5/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{7/2}}{7 e^2} \]

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),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.0692, size = 42, normalized size = 0.98 \[ \frac{2 c d \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0642692, size = 34, normalized size = 0.79 \[ \frac{2 (d+e x)^{5/2} \left (7 a e^2+c d (5 e x-2 d)\right )}{35 e^2} \]

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),x]

[Out]

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

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Maple [A]  time = 0.006, size = 32, normalized size = 0.7 \[{\frac{10\,cdex+14\,a{e}^{2}-4\,c{d}^{2}}{35\,{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.74846, size = 51, normalized size = 1.19 \[ \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} c d - 7 \,{\left (c d^{2} - a e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{35 \, e^{2}} \]

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

[Out]

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

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Fricas [A]  time = 0.221486, size = 100, normalized size = 2.33 \[ \frac{2 \,{\left (5 \, c d e^{3} x^{3} - 2 \, c d^{4} + 7 \, a d^{2} e^{2} +{\left (8 \, c d^{2} e^{2} + 7 \, a e^{4}\right )} x^{2} +{\left (c d^{3} e + 14 \, a d e^{3}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{2}} \]

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

[Out]

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

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Sympy [A]  time = 4.63391, size = 41, normalized size = 0.95 \[ \frac{2 \left (\frac{c d \left (d + e x\right )^{\frac{7}{2}}}{7 e} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.200952, size = 165, normalized size = 3.84 \[ \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} c d^{2} e^{\left (-1\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a d e +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c d e^{\left (-13\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a e\right )} e^{\left (-1\right )} \]

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)*sqrt(e*x + d),x, algorithm="giac")

[Out]

2/105*(7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*c*d^2*e^(-1) + 35*(x*e + d)^(
3/2)*a*d*e + (15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)
^(3/2)*d^2*e^12)*c*d*e^(-13) + 7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*e)*
e^(-1)