3.1968 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ 2 \sqrt{d+e x} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{3/2}}{3 e^2} \]

[Out]

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

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Rubi [A]  time = 0.0753976, antiderivative size = 41, 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 \[ 2 \sqrt{d+e x} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{3/2}}{3 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.9987, size = 41, normalized size = 1. \[ \frac{2 c d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{e^{2}} \]

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)/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0335905, size = 33, normalized size = 0.8 \[ \frac{2 \sqrt{d+e x} \left (3 a e^2+c d (e x-2 d)\right )}{3 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 31, normalized size = 0.8 \[{\frac{2\,cdex+6\,a{e}^{2}-4\,c{d}^{2}}{3\,{e}^{2}}\sqrt{ex+d}} \]

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)/(e*x+d)^(3/2),x)

[Out]

2/3*(e*x+d)^(1/2)*(c*d*e*x+3*a*e^2-2*c*d^2)/e^2

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Maxima [A]  time = 0.758266, size = 50, normalized size = 1.22 \[ \frac{2 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c d - 3 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}\right )}}{3 \, e^{2}} \]

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)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.217766, size = 41, normalized size = 1. \[ \frac{2 \,{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{3 \, e^{2}} \]

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)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.28187, size = 124, normalized size = 3.02 \[ \begin{cases} - \frac{\frac{2 a d e}{\sqrt{d + e x}} + 2 a e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 c d^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e}}{e} & \text{for}\: e \neq 0 \\\frac{c \sqrt{d} x^{2}}{2} & \text{otherwise} \end{cases} \]

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)/(e*x+d)**(3/2),x)

[Out]

Piecewise((-(2*a*d*e/sqrt(d + e*x) + 2*a*e*(-d/sqrt(d + e*x) - sqrt(d + e*x)) +
2*c*d**2*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*c*d*(d**2/sqrt(d + e*x) + 2*d*
sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e)/e, Ne(e, 0)), (c*sqrt(d)*x**2/2, True))

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GIAC/XCAS [A]  time = 0.200545, size = 63, normalized size = 1.54 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c d e^{4} - 3 \, \sqrt{x e + d} c d^{2} e^{4} + 3 \, \sqrt{x e + d} a e^{6}\right )} e^{\left (-6\right )} \]

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)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c*d*e^4 - 3*sqrt(x*e + d)*c*d^2*e^4 + 3*sqrt(x*e + d)*a*e^6
)*e^(-6)