3.1617 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}} \]

[Out]

(-2*(b*d - a*e)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*b*(b*d - a*e))/(3*e^3*(d + e*x)^
(3/2)) - (2*b^2)/(e^3*Sqrt[d + e*x])

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Rubi [A]  time = 0.0760856, antiderivative size = 69, 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 \[ \frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*b*(b*d - a*e))/(3*e^3*(d + e*x)^
(3/2)) - (2*b^2)/(e^3*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 29.7854, size = 65, normalized size = 0.94 \[ - \frac{2 b^{2}}{e^{3} \sqrt{d + e x}} - \frac{4 b \left (a e - b d\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e - b d\right )^{2}}{5 e^{3} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

-2*b**2/(e**3*sqrt(d + e*x)) - 4*b*(a*e - b*d)/(3*e**3*(d + e*x)**(3/2)) - 2*(a*
e - b*d)**2/(5*e**3*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.0693611, size = 61, normalized size = 0.88 \[ -\frac{2 \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(3*a^2*e^2 + 2*a*b*e*(2*d + 5*e*x) + b^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(
15*e^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.011, size = 63, normalized size = 0.9 \[ -{\frac{30\,{x}^{2}{b}^{2}{e}^{2}+20\,xab{e}^{2}+40\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+8\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(7/2),x)

[Out]

-2/15*(15*b^2*e^2*x^2+10*a*b*e^2*x+20*b^2*d*e*x+3*a^2*e^2+4*a*b*d*e+8*b^2*d^2)/(
e*x+d)^(5/2)/e^3

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Maxima [A]  time = 0.731483, size = 88, normalized size = 1.28 \[ -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} b^{2} + 3 \, b^{2} d^{2} - 6 \, a b d e + 3 \, a^{2} e^{2} - 10 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*b^2 + 3*b^2*d^2 - 6*a*b*d*e + 3*a^2*e^2 - 10*(b^2*d - a*b*
e)*(e*x + d))/((e*x + d)^(5/2)*e^3)

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Fricas [A]  time = 0.209327, size = 113, normalized size = 1.64 \[ -\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 4 \, a b d e + 3 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(15*b^2*e^2*x^2 + 8*b^2*d^2 + 4*a*b*d*e + 3*a^2*e^2 + 10*(2*b^2*d*e + a*b*
e^2)*x)/((e^5*x^2 + 2*d*e^4*x + d^2*e^3)*sqrt(e*x + d))

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Sympy [A]  time = 8.60519, size = 389, normalized size = 5.64 \[ \begin{cases} - \frac{6 a^{2} e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{8 a b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{20 a b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 b^{2} d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 b^{2} d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 b^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

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)**(7/2),x)

[Out]

Piecewise((-6*a**2*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x)
+ 15*e**5*x**2*sqrt(d + e*x)) - 8*a*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**
4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 20*a*b*e**2*x/(15*d**2*e**3*sq
rt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*b**2*
d**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt
(d + e*x)) - 40*b**2*d*e*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*
x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*b**2*e**2*x**2/(15*d**2*e**3*sqrt(d + e*x)
+ 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x
+ a*b*x**2 + b**2*x**3/3)/d**(7/2), True))

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GIAC/XCAS [A]  time = 0.215599, size = 97, normalized size = 1.41 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{2} - 10 \,{\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} + 10 \,{\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

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)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^2*b^2 - 10*(x*e + d)*b^2*d + 3*b^2*d^2 + 10*(x*e + d)*a*b*e
- 6*a*b*d*e + 3*a^2*e^2)*e^(-3)/(x*e + d)^(5/2)