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

Optimal. Leaf size=91 \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]

[Out]

(-2*a^2*(A*b - a*B))/(b^4*Sqrt[a + b*x]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x])/b
^4 + (2*(A*b - 3*a*B)*(a + b*x)^(3/2))/(3*b^4) + (2*B*(a + b*x)^(5/2))/(5*b^4)

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Rubi [A]  time = 0.120091, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a^2 (A b-a B)}{b^4 \sqrt{a+b x}}+\frac{2 (a+b x)^{3/2} (A b-3 a B)}{3 b^4}-\frac{2 a \sqrt{a+b x} (2 A b-3 a B)}{b^4}+\frac{2 B (a+b x)^{5/2}}{5 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*(A*b - a*B))/(b^4*Sqrt[a + b*x]) - (2*a*(2*A*b - 3*a*B)*Sqrt[a + b*x])/b
^4 + (2*(A*b - 3*a*B)*(a + b*x)^(3/2))/(3*b^4) + (2*B*(a + b*x)^(5/2))/(5*b^4)

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Rubi in Sympy [A]  time = 16.1526, size = 88, normalized size = 0.97 \[ \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{2 a^{2} \left (A b - B a\right )}{b^{4} \sqrt{a + b x}} - \frac{2 a \sqrt{a + b x} \left (2 A b - 3 B a\right )}{b^{4}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A b - 3 B a\right )}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x)**(5/2)/(5*b**4) - 2*a**2*(A*b - B*a)/(b**4*sqrt(a + b*x)) - 2*a*sq
rt(a + b*x)*(2*A*b - 3*B*a)/b**4 + 2*(a + b*x)**(3/2)*(A*b - 3*B*a)/(3*b**4)

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Mathematica [A]  time = 0.062989, size = 67, normalized size = 0.74 \[ \frac{2 \left (48 a^3 B-8 a^2 b (5 A-3 B x)-2 a b^2 x (10 A+3 B x)+b^3 x^2 (5 A+3 B x)\right )}{15 b^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(48*a^3*B - 8*a^2*b*(5*A - 3*B*x) + b^3*x^2*(5*A + 3*B*x) - 2*a*b^2*x*(10*A +
3*B*x)))/(15*b^4*Sqrt[a + b*x])

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Maple [A]  time = 0.008, size = 71, normalized size = 0.8 \[ -{\frac{-6\,{b}^{3}B{x}^{3}-10\,A{x}^{2}{b}^{3}+12\,B{x}^{2}a{b}^{2}+40\,Axa{b}^{2}-48\,Bx{a}^{2}b+80\,A{a}^{2}b-96\,B{a}^{3}}{15\,{b}^{4}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15/(b*x+a)^(1/2)*(-3*B*b^3*x^3-5*A*b^3*x^2+6*B*a*b^2*x^2+20*A*a*b^2*x-24*B*a^
2*b*x+40*A*a^2*b-48*B*a^3)/b^4

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Maxima [A]  time = 1.32583, size = 115, normalized size = 1.26 \[ \frac{2 \,{\left (\frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B a^{2} - 2 \, A a b\right )} \sqrt{b x + a}}{b} + \frac{15 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b}\right )}}{15 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*(b*x + a)^(5/2)*B - 5*(3*B*a - A*b)*(b*x + a)^(3/2) + 15*(3*B*a^2 - 2*A
*a*b)*sqrt(b*x + a))/b + 15*(B*a^3 - A*a^2*b)/(sqrt(b*x + a)*b))/b^3

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Fricas [A]  time = 0.213886, size = 97, normalized size = 1.07 \[ \frac{2 \,{\left (3 \, B b^{3} x^{3} + 48 \, B a^{3} - 40 \, A a^{2} b -{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 4 \,{\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )}}{15 \, \sqrt{b x + a} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*B*b^3*x^3 + 48*B*a^3 - 40*A*a^2*b - (6*B*a*b^2 - 5*A*b^3)*x^2 + 4*(6*B*a
^2*b - 5*A*a*b^2)*x)/(sqrt(b*x + a)*b^4)

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Sympy [A]  time = 16.7459, size = 2077, normalized size = 22.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-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**(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) - 30*a**(15/2)*b**2*x**2*s
qrt(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)) + B*(32*a**(45/2)*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**
18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5
*a**14*b**10*x**6) - 32*a**(45/2)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**
6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14
*b**10*x**6) + 176*a**(43/2)*b*x*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x
+ 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9
*x**5 + 5*a**14*b**10*x**6) - 192*a**(43/2)*b*x/(5*a**20*b**4 + 30*a**19*b**5*x
+ 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*
x**5 + 5*a**14*b**10*x**6) + 396*a**(41/2)*b**2*x**2*sqrt(1 + b*x/a)/(5*a**20*b*
*4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*
x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 480*a**(41/2)*b**2*x**2/(5*a**
20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*
b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 462*a**(39/2)*b**3*x**3*s
qrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*
b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 640*
a**(39/2)*b**3*x**3/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a
**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) +
290*a**(37/2)*b**4*x**4*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a*
*18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 +
5*a**14*b**10*x**6) - 480*a**(37/2)*b**4*x**4/(5*a**20*b**4 + 30*a**19*b**5*x +
75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x*
*5 + 5*a**14*b**10*x**6) + 92*a**(35/2)*b**5*x**5*sqrt(1 + b*x/a)/(5*a**20*b**4
+ 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**
4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 192*a**(35/2)*b**5*x**5/(5*a**20*
b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**
8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 16*a**(33/2)*b**6*x**6*sqrt(
1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7
*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) - 32*a**(3
3/2)*b**6*x**6/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**6*x**2 + 100*a**17*
b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14*b**10*x**6) + 6*a*
*(31/2)*b**7*x**7*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b**5*x + 75*a**18*b**
6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*b**9*x**5 + 5*a**14
*b**10*x**6) + 2*a**(29/2)*b**8*x**8*sqrt(1 + b*x/a)/(5*a**20*b**4 + 30*a**19*b*
*5*x + 75*a**18*b**6*x**2 + 100*a**17*b**7*x**3 + 75*a**16*b**8*x**4 + 30*a**15*
b**9*x**5 + 5*a**14*b**10*x**6))

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GIAC/XCAS [A]  time = 0.223806, size = 138, normalized size = 1.52 \[ \frac{2 \,{\left (B a^{3} - A a^{2} b\right )}}{\sqrt{b x + a} b^{4}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{16} - 15 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{16} + 45 \, \sqrt{b x + a} B a^{2} b^{16} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{17} - 30 \, \sqrt{b x + a} A a b^{17}\right )}}{15 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*a^3 - A*a^2*b)/(sqrt(b*x + a)*b^4) + 2/15*(3*(b*x + a)^(5/2)*B*b^16 - 15*(b
*x + a)^(3/2)*B*a*b^16 + 45*sqrt(b*x + a)*B*a^2*b^16 + 5*(b*x + a)^(3/2)*A*b^17
- 30*sqrt(b*x + a)*A*a*b^17)/b^20