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

Optimal. Leaf size=51 \[ \frac{2 A x}{3 a^2 \sqrt{a+c x^2}}-\frac{a B-A c x}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0338404, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 A x}{3 a^2 \sqrt{a+c x^2}}-\frac{a B-A c x}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.83681, size = 42, normalized size = 0.82 \[ \frac{2 A x}{3 a^{2} \sqrt{a + c x^{2}}} - \frac{- A c x + B a}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0371263, size = 43, normalized size = 0.84 \[ \frac{-a^2 B+3 a A c x+2 A c^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 0.8 \[{\frac{2\,A{c}^{2}{x}^{3}+3\,aAcx-{a}^{2}B}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.677874, size = 65, normalized size = 1.27 \[ \frac{2 \, A x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{A x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{B}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292393, size = 84, normalized size = 1.65 \[ \frac{{\left (2 \, A c^{2} x^{3} + 3 \, A a c x - B a^{2}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 36.7273, size = 146, normalized size = 2.86 \[ A \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + B \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(3*a*x/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a))
+ 2*c*x**3/(3*a**(7/2)*sqrt(1 + c*x**2/a) + 3*a**(5/2)*c*x**2*sqrt(1 + c*x**2/a)
)) + B*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))

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GIAC/XCAS [A]  time = 0.274694, size = 50, normalized size = 0.98 \[ \frac{{\left (\frac{2 \, A c x^{2}}{a^{2}} + \frac{3 \, A}{a}\right )} x - \frac{B}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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