3.1615 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4 b \sqrt{d+e x} (b d-a e)}{e^3}-\frac{2 (b d-a e)^2}{e^3 \sqrt{d+e x}}+\frac{2 b^2 (d+e x)^{3/2}}{3 e^3} \]

[Out]

(-2*(b*d - a*e)^2)/(e^3*Sqrt[d + e*x]) - (4*b*(b*d - a*e)*Sqrt[d + e*x])/e^3 + (
2*b^2*(d + e*x)^(3/2))/(3*e^3)

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Rubi [A]  time = 0.0776724, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b \sqrt{d+e x} (b d-a e)}{e^3}-\frac{2 (b d-a e)^2}{e^3 \sqrt{d+e x}}+\frac{2 b^2 (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^2)/(e^3*Sqrt[d + e*x]) - (4*b*(b*d - a*e)*Sqrt[d + e*x])/e^3 + (
2*b^2*(d + e*x)^(3/2))/(3*e^3)

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Rubi in Sympy [A]  time = 29.7541, size = 61, normalized size = 0.91 \[ \frac{2 b^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{4 b \sqrt{d + e x} \left (a e - b d\right )}{e^{3}} - \frac{2 \left (a e - b d\right )^{2}}{e^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(d + e*x)**(3/2)/(3*e**3) + 4*b*sqrt(d + e*x)*(a*e - b*d)/e**3 - 2*(a*e -
b*d)**2/(e**3*sqrt(d + e*x))

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Mathematica [A]  time = 0.0635979, size = 59, normalized size = 0.88 \[ \frac{2 \left (-3 a^2 e^2+6 a b e (2 d+e x)+b^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a^2*e^2 + 6*a*b*e*(2*d + e*x) + b^2*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3
*Sqrt[d + e*x])

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Maple [A]  time = 0.012, size = 63, normalized size = 0.9 \[ -{\frac{-2\,{x}^{2}{b}^{2}{e}^{2}-12\,xab{e}^{2}+8\,x{b}^{2}de+6\,{a}^{2}{e}^{2}-24\,abde+16\,{b}^{2}{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-b^2*e^2*x^2-6*a*b*e^2*x+4*b^2*d*e*x+3*a^2*e^2-12*a*b*d*e+8*b^2*d^2)/(e*x+
d)^(1/2)/e^3

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Maxima [A]  time = 0.733582, size = 101, normalized size = 1.51 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} b^{2} - 6 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206157, size = 85, normalized size = 1.27 \[ \frac{2 \,{\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 3 \, a b e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^2*e^2*x^2 - 8*b^2*d^2 + 12*a*b*d*e - 3*a^2*e^2 - 2*(2*b^2*d*e - 3*a*b*e^2
)*x)/(sqrt(e*x + d)*e^3)

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Sympy [A]  time = 11.8307, size = 597, normalized size = 8.91 \[ - \frac{2 a^{2}}{e \sqrt{d + e x}} + 2 a b \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + b^{2} \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(e*sqrt(d + e*x)) + 2*a*b*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/(e*s
qrt(d + e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + b**2*(-16*d**(19/2)*sqrt(
1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) +
16*d**(19/2)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3
) - 40*d**(17/2)*e*x*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*
x**2 + 3*d**5*e**6*x**3) + 48*d**(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**
6*e**5*x**2 + 3*d**5*e**6*x**3) - 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(3*d**8
*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2)*e**2
*x**2/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 4*d*
*(13/2)*e**3*x**3*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**
2 + 3*d**5*e**6*x**3) + 16*d**(13/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*x + 9*
d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)/(3*d*
*8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3))

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GIAC/XCAS [A]  time = 0.218916, size = 112, normalized size = 1.67 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{6} - 6 \, \sqrt{x e + d} b^{2} d e^{6} + 6 \, \sqrt{x e + d} a b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((x*e + d)^(3/2)*b^2*e^6 - 6*sqrt(x*e + d)*b^2*d*e^6 + 6*sqrt(x*e + d)*a*b*e
^7)*e^(-9) - 2*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*e^(-3)/sqrt(x*e + d)