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

Optimal. Leaf size=64 \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

(-2*d*(c*d - b*e))/(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 = 0.0945966, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*(c*d - b*e))/(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 = 13.351, size = 60, normalized size = 0.94 \[ \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 d \left (b e - c d\right )}{e^{3} \sqrt{d + e x}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0531498, size = 48, normalized size = 0.75 \[ \frac{2 \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(3*b*e*(2*d + e*x) + 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 = 0.006, size = 46, normalized size = 0.7 \[{\frac{2\,c{e}^{2}{x}^{2}+6\,b{e}^{2}x-8\,cdex+12\,bde-16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.695724, size = 82, normalized size = 1.28 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c d^{2} - b d e\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)/(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)
/(sqrt(e*x + d)*e^2))/e

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Fricas [A]  time = 0.217156, size = 63, normalized size = 0.98 \[ \frac{2 \,{\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e -{\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.731, size = 576, normalized size = 9. \[ b \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + c \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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**(1
7/2)*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) - 3
0*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 = 0.205924, size = 93, normalized size = 1.45 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6} + 3 \, \sqrt{x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} - b d e\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)/(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)*e^(-3)/sqrt(x*e + d)