3.1616 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0750267, 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 \[ \frac{4 b (b d-a e)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 b^2 \sqrt{d+e x}}{e^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.9099, size = 61, normalized size = 0.91 \[ \frac{2 b^{2} \sqrt{d + e x}}{e^{3}} - \frac{4 b \left (a e - b d\right )}{e^{3} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{2}}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0701556, size = 64, normalized size = 0.96 \[ \sqrt{d+e x} \left (\frac{4 b (b d-a e)}{e^3 (d+e x)}-\frac{2 (a e-b d)^2}{3 e^3 (d+e x)^2}+\frac{2 b^2}{e^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 62, normalized size = 0.9 \[ -{\frac{-6\,{x}^{2}{b}^{2}{e}^{2}+12\,xab{e}^{2}-24\,x{b}^{2}de+2\,{a}^{2}{e}^{2}+8\,abde-16\,{b}^{2}{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.73656, size = 97, normalized size = 1.45 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} b^{2}}{e^{2}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} - 6 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \]

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.206638, size = 100, normalized size = 1.49 \[ \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 4 \, a b d e - a^{2} e^{2} + 6 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.93692, size = 265, normalized size = 3.96 \[ \begin{cases} - \frac{2 a^{2} e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{8 a b d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{12 a b e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 b^{2} d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 b^{2} d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 b^{2} e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

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)**(5/2),x)

[Out]

Piecewise((-2*a**2*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 8*a*
b*d*e/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 12*a*b*e**2*x/(3*d*e**
3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*b**2*d**2/(3*d*e**3*sqrt(d + e*x)
+ 3*e**4*x*sqrt(d + e*x)) + 24*b**2*d*e*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sq
rt(d + e*x)) + 6*b**2*e**2*x**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)
), Ne(e, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/d**(5/2), True))

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GIAC/XCAS [A]  time = 0.219813, size = 101, normalized size = 1.51 \[ 2 \, \sqrt{x e + d} b^{2} e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} b^{2} d - b^{2} d^{2} - 6 \,{\left (x e + d\right )} a b e + 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

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