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

Optimal. Leaf size=59 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0681519, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \sqrt{d+e x}}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c*d^2 + a*e^2))/(e^3*Sqrt[d + e*x]) - (4*c*d*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 = 11.2955, size = 58, normalized size = 0.98 \[ - \frac{4 c d \sqrt{d + e x}}{e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} - \frac{2 \left (a e^{2} + c d^{2}\right )}{e^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0428275, size = 43, normalized size = 0.73 \[ \frac{2 \left (c \left (-8 d^2-4 d e x+e^2 x^2\right )-3 a e^2\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a*e^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 = 0.004, size = 41, normalized size = 0.7 \[ -{\frac{-2\,c{e}^{2}{x}^{2}+8\,cdex+6\,a{e}^{2}+16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)/(e*x+d)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.693753, size = 73, normalized size = 1.24 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 6 \, \sqrt{e x + d} c d}{e^{2}} - \frac{3 \,{\left (c d^{2} + a e^{2}\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205488, size = 53, normalized size = 0.9 \[ \frac{2 \,{\left (c e^{2} x^{2} - 4 \, c d e x - 8 \, c d^{2} - 3 \, a e^{2}\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.09834, size = 549, normalized size = 9.31 \[ - \frac{2 a}{e \sqrt{d + e x}} + c \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(e*sqrt(d + e*x)) + 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 = 0.210334, size = 73, normalized size = 1.24 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + 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)*e^(-9) - 2*(c*d^2 + a*e^2)
*e^(-3)/sqrt(x*e + d)