3.377 \(\int \frac{x (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{B x}{3 a c \sqrt{a+c x^2}}-\frac{A+B x}{3 c \left (a+c x^2\right )^{3/2}} \]

[Out]

-(A + B*x)/(3*c*(a + c*x^2)^(3/2)) + (B*x)/(3*a*c*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.0600122, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{B x}{3 a c \sqrt{a+c x^2}}-\frac{A+B x}{3 c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-(A + B*x)/(3*c*(a + c*x^2)^(3/2)) + (B*x)/(3*a*c*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 5.6007, size = 36, normalized size = 0.77 \[ \frac{B x}{3 a c \sqrt{a + c x^{2}}} - \frac{A + B x}{3 c \left (a + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

B*x/(3*a*c*sqrt(a + c*x**2)) - (A + B*x)/(3*c*(a + c*x**2)**(3/2))

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Mathematica [A]  time = 0.0396312, size = 32, normalized size = 0.68 \[ \frac{B c x^3-a A}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(-(a*A) + B*c*x^3)/(3*a*c*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 29, normalized size = 0.6 \[ -{\frac{-Bc{x}^{3}+aA}{3\,ac} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(c*x^2+a)^(5/2),x)

[Out]

-1/3*(-B*c*x^3+A*a)/(c*x^2+a)^(3/2)/a/c

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Maxima [A]  time = 0.673348, size = 69, normalized size = 1.47 \[ -\frac{B x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{B x}{3 \, \sqrt{c x^{2} + a} a c} - \frac{A}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

-1/3*B*x/((c*x^2 + a)^(3/2)*c) + 1/3*B*x/(sqrt(c*x^2 + a)*a*c) - 1/3*A/((c*x^2 +
a)^(3/2)*c)

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Fricas [A]  time = 0.291123, size = 66, normalized size = 1.4 \[ \frac{{\left (B c x^{3} - A a\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(B*c*x^3 - A*a)*sqrt(c*x^2 + a)/(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)

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Sympy [A]  time = 37.0968, size = 95, normalized size = 2.02 \[ A \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{B x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{3}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

A*Piecewise((-1/(3*a*c*sqrt(a + c*x**2) + 3*c**2*x**2*sqrt(a + c*x**2)), Ne(c, 0
)), (x**2/(2*a**(5/2)), True)) + B*x**3/(3*a**(5/2)*sqrt(1 + c*x**2/a) + 3*a**(3
/2)*c*x**2*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.277117, size = 35, normalized size = 0.74 \[ \frac{\frac{B x^{3}}{a} - \frac{A}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

1/3*(B*x^3/a - A/c)/(c*x^2 + a)^(3/2)