3.425 \(\int \frac{x (A+B x)}{(a+b x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0789107, 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 \[ \frac{2 \sqrt{a+b x} (A b-2 a B)}{b^3}+\frac{2 a (A b-a B)}{b^3 \sqrt{a+b x}}+\frac{2 B (a+b x)^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6889, size = 60, normalized size = 0.95 \[ \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{2 a \left (A b - B a\right )}{b^{3} \sqrt{a + b x}} + \frac{2 \sqrt{a + b x} \left (A b - 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0447637, size = 47, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a (6 A b-4 b B x)+b^2 x (3 A+B x)\right )}{3 b^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(-8*a^2*B + b^2*x*(3*A + B*x) + a*(6*A*b - 4*b*B*x)))/(3*b^3*Sqrt[a + b*x])

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Maple [A]  time = 0.006, size = 46, normalized size = 0.7 \[{\frac{2\,{b}^{2}B{x}^{2}+6\,Ax{b}^{2}-8\,Bxab+12\,Aab-16\,B{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

2/3/(b*x+a)^(1/2)*(B*b^2*x^2+3*A*b^2*x-4*B*a*b*x+6*A*a*b-8*B*a^2)/b^3

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Maxima [A]  time = 1.34032, size = 82, normalized size = 1.3 \[ \frac{2 \,{\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}} B - 3 \,{\left (2 \, B a - A b\right )} \sqrt{b x + a}}{b} - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b}\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209956, size = 63, normalized size = 1. \[ \frac{2 \,{\left (B b^{2} x^{2} - 8 \, B a^{2} + 6 \, A a b -{\left (4 \, B a b - 3 \, A b^{2}\right )} x\right )}}{3 \, \sqrt{b x + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

2/3*(B*b^2*x^2 - 8*B*a^2 + 6*A*a*b - (4*B*a*b - 3*A*b^2)*x)/(sqrt(b*x + a)*b^3)

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Sympy [A]  time = 10.7912, size = 576, normalized size = 9.14 \[ A \left (\begin{cases} \frac{4 a}{b^{2} \sqrt{a + b x}} + \frac{2 x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (- \frac{16 a^{\frac{19}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{40 a^{\frac{17}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{30 a^{\frac{15}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{4 a^{\frac{13}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{2 a^{\frac{11}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

A*Piecewise((4*a/(b**2*sqrt(a + b*x)) + 2*x/(b*sqrt(a + b*x)), Ne(b, 0)), (x**2/
(2*a**(3/2)), True)) + B*(-16*a**(19/2)*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b*
*4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 16*a**(19/2)/(3*a**8*b**3 + 9*a**7
*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 40*a**(17/2)*b*x*sqrt(1 + b*x/a
)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(1
7/2)*b*x/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 3
0*a**(15/2)*b**2*x**2*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5
*x**2 + 3*a**5*b**6*x**3) + 48*a**(15/2)*b**2*x**2/(3*a**8*b**3 + 9*a**7*b**4*x
+ 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 4*a**(13/2)*b**3*x**3*sqrt(1 + b*x/a)/(
3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 16*a**(13/2
)*b**3*x**3/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3)
+ 2*a**(11/2)*b**4*x**4*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b*
*5*x**2 + 3*a**5*b**6*x**3))

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GIAC/XCAS [A]  time = 0.22947, size = 93, normalized size = 1.48 \[ -\frac{2 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} B b^{6} - 6 \, \sqrt{b x + a} B a b^{6} + 3 \, \sqrt{b x + a} A b^{7}\right )}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

-2*(B*a^2 - A*a*b)/(sqrt(b*x + a)*b^3) + 2/3*((b*x + a)^(3/2)*B*b^6 - 6*sqrt(b*x
+ a)*B*a*b^6 + 3*sqrt(b*x + a)*A*b^7)/b^9