3.2261 \(\int \frac{a+b x+c x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0870152, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 \left (a e^2-b d e+c d^2\right )}{e^3 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4806, size = 70, normalized size = 0.99 \[ \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right )}{e^{3}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )}{e^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.061295, size = 54, normalized size = 0.76 \[ \frac{6 e (-a e+2 b d+b e x)+2 c \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*e*(2*b*d - a*e + b*e*x) + 2*c*(-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 = 0.005, size = 53, normalized size = 0.8 \[ -{\frac{-2\,c{e}^{2}{x}^{2}-6\,b{e}^{2}x+8\,cdex+6\,a{e}^{2}-12\,bde+16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.67811, size = 89, normalized size = 1.25 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205087, size = 72, normalized size = 1.01 \[ \frac{2 \,{\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e - 3 \, a e^{2} -{\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.0244, size = 590, normalized size = 8.31 \[ - \frac{2 a}{e \sqrt{d + e x}} + b \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + c \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(e*sqrt(d + e*x)) + b*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/(e*sqrt(d +
e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + c*(-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 = 0.203539, size = 99, normalized size = 1.39 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6} + 3 \, \sqrt{x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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