3.435 \(\int \frac{x (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0816219, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9819, size = 60, normalized size = 0.95 \[ \frac{2 B \sqrt{a + b x}}{b^{3}} + \frac{2 a \left (A b - B a\right )}{3 b^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (A b - 2 B a\right )}{b^{3} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0534743, size = 46, normalized size = 0.73 \[ \frac{16 a^2 B-4 a b (A-6 B x)+6 b^2 x (B x-A)}{3 b^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(16*a^2*B - 4*a*b*(A - 6*B*x) + 6*b^2*x*(-A + B*x))/(3*b^3*(a + b*x)^(3/2))

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Maple [A]  time = 0.006, size = 47, normalized size = 0.8 \[ -{\frac{-6\,{b}^{2}B{x}^{2}+6\,Ax{b}^{2}-24\,Bxab+4\,Aab-16\,B{a}^{2}}{3\,{b}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(b*x+a)^(5/2),x)

[Out]

-2/3/(b*x+a)^(3/2)*(-3*B*b^2*x^2+3*A*b^2*x-12*B*a*b*x+2*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.33996, size = 78, normalized size = 1.24 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{b x + a} B}{b} - \frac{B a^{2} - A a b - 3 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214182, size = 78, normalized size = 1.24 \[ \frac{2 \,{\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 2 \, A a b + 3 \,{\left (4 \, B a b - A b^{2}\right )} x\right )}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*B*b^2*x^2 + 8*B*a^2 - 2*A*a*b + 3*(4*B*a*b - A*b^2)*x)/((b^4*x + a*b^3)*s
qrt(b*x + a))

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Sympy [A]  time = 4.07984, size = 211, normalized size = 3.35 \[ \begin{cases} - \frac{4 A a b}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} - \frac{6 A b^{2} x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{16 B a^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{24 B a b x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{6 B b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{3}}{3}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-4*A*a*b/(3*a*b**3*sqrt(a + b*x) + 3*b**4*x*sqrt(a + b*x)) - 6*A*b**2
*x/(3*a*b**3*sqrt(a + b*x) + 3*b**4*x*sqrt(a + b*x)) + 16*B*a**2/(3*a*b**3*sqrt(
a + b*x) + 3*b**4*x*sqrt(a + b*x)) + 24*B*a*b*x/(3*a*b**3*sqrt(a + b*x) + 3*b**4
*x*sqrt(a + b*x)) + 6*B*b**2*x**2/(3*a*b**3*sqrt(a + b*x) + 3*b**4*x*sqrt(a + b*
x)), Ne(b, 0)), ((A*x**2/2 + B*x**3/3)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.213045, size = 74, normalized size = 1.17 \[ \frac{2 \, \sqrt{b x + a} B}{b^{3}} + \frac{2 \,{\left (6 \,{\left (b x + a\right )} B a - B a^{2} - 3 \,{\left (b x + a\right )} A b + A a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x + a)*B/b^3 + 2/3*(6*(b*x + a)*B*a - B*a^2 - 3*(b*x + a)*A*b + A*a*b)/
((b*x + a)^(3/2)*b^3)