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

Optimal. Leaf size=116 \[ \frac{2 a^3 (A b-a B)}{b^5 \sqrt{a+b x}}+\frac{2 a^2 \sqrt{a+b x} (3 A b-4 a B)}{b^5}+\frac{2 (a+b x)^{5/2} (A b-4 a B)}{5 b^5}-\frac{2 a (a+b x)^{3/2} (A b-2 a B)}{b^5}+\frac{2 B (a+b x)^{7/2}}{7 b^5} \]

[Out]

(2*a^3*(A*b - a*B))/(b^5*Sqrt[a + b*x]) + (2*a^2*(3*A*b - 4*a*B)*Sqrt[a + b*x])/
b^5 - (2*a*(A*b - 2*a*B)*(a + b*x)^(3/2))/b^5 + (2*(A*b - 4*a*B)*(a + b*x)^(5/2)
)/(5*b^5) + (2*B*(a + b*x)^(7/2))/(7*b^5)

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Rubi [A]  time = 0.150607, antiderivative size = 116, 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 \[ \frac{2 a^3 (A b-a B)}{b^5 \sqrt{a+b x}}+\frac{2 a^2 \sqrt{a+b x} (3 A b-4 a B)}{b^5}+\frac{2 (a+b x)^{5/2} (A b-4 a B)}{5 b^5}-\frac{2 a (a+b x)^{3/2} (A b-2 a B)}{b^5}+\frac{2 B (a+b x)^{7/2}}{7 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*(A*b - a*B))/(b^5*Sqrt[a + b*x]) + (2*a^2*(3*A*b - 4*a*B)*Sqrt[a + b*x])/
b^5 - (2*a*(A*b - 2*a*B)*(a + b*x)^(3/2))/b^5 + (2*(A*b - 4*a*B)*(a + b*x)^(5/2)
)/(5*b^5) + (2*B*(a + b*x)^(7/2))/(7*b^5)

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Rubi in Sympy [A]  time = 21.7531, size = 114, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{5}} + \frac{2 a^{3} \left (A b - B a\right )}{b^{5} \sqrt{a + b x}} + \frac{2 a^{2} \sqrt{a + b x} \left (3 A b - 4 B a\right )}{b^{5}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}} \left (A b - 2 B a\right )}{b^{5}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A b - 4 B a\right )}{5 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x)**(7/2)/(7*b**5) + 2*a**3*(A*b - B*a)/(b**5*sqrt(a + b*x)) + 2*a**2
*sqrt(a + b*x)*(3*A*b - 4*B*a)/b**5 - 2*a*(a + b*x)**(3/2)*(A*b - 2*B*a)/b**5 +
2*(a + b*x)**(5/2)*(A*b - 4*B*a)/(5*b**5)

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Mathematica [A]  time = 0.0772596, size = 86, normalized size = 0.74 \[ \frac{2 \left (-128 a^4 B+16 a^3 b (7 A-4 B x)+8 a^2 b^2 x (7 A+2 B x)-2 a b^3 x^2 (7 A+4 B x)+b^4 x^3 (7 A+5 B x)\right )}{35 b^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-128*a^4*B + 16*a^3*b*(7*A - 4*B*x) + 8*a^2*b^2*x*(7*A + 2*B*x) - 2*a*b^3*x^
2*(7*A + 4*B*x) + b^4*x^3*(7*A + 5*B*x)))/(35*b^5*Sqrt[a + b*x])

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Maple [A]  time = 0.009, size = 95, normalized size = 0.8 \[{\frac{10\,B{x}^{4}{b}^{4}+14\,A{b}^{4}{x}^{3}-16\,Ba{b}^{3}{x}^{3}-28\,Aa{b}^{3}{x}^{2}+32\,B{a}^{2}{b}^{2}{x}^{2}+112\,A{a}^{2}{b}^{2}x-128\,B{a}^{3}bx+224\,A{a}^{3}b-256\,B{a}^{4}}{35\,{b}^{5}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35/(b*x+a)^(1/2)*(5*B*b^4*x^4+7*A*b^4*x^3-8*B*a*b^3*x^3-14*A*a*b^3*x^2+16*B*a^
2*b^2*x^2+56*A*a^2*b^2*x-64*B*a^3*b*x+112*A*a^3*b-128*B*a^4)/b^5

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Maxima [A]  time = 1.33945, size = 146, normalized size = 1.26 \[ \frac{2 \,{\left (\frac{5 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 7 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 35 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{3}{2}} - 35 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt{b x + a}}{b} - \frac{35 \,{\left (B a^{4} - A a^{3} b\right )}}{\sqrt{b x + a} b}\right )}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*((5*(b*x + a)^(7/2)*B - 7*(4*B*a - A*b)*(b*x + a)^(5/2) + 35*(2*B*a^2 - A*a
*b)*(b*x + a)^(3/2) - 35*(4*B*a^3 - 3*A*a^2*b)*sqrt(b*x + a))/b - 35*(B*a^4 - A*
a^3*b)/(sqrt(b*x + a)*b))/b^4

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Fricas [A]  time = 0.216857, size = 130, normalized size = 1.12 \[ \frac{2 \,{\left (5 \, B b^{4} x^{4} - 128 \, B a^{4} + 112 \, A a^{3} b -{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )}}{35 \, \sqrt{b x + a} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*B*b^4*x^4 - 128*B*a^4 + 112*A*a^3*b - (8*B*a*b^3 - 7*A*b^4)*x^3 + 2*(8*B
*a^2*b^2 - 7*A*a*b^3)*x^2 - 8*(8*B*a^3*b - 7*A*a^2*b^2)*x)/(sqrt(b*x + a)*b^5)

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Sympy [A]  time = 25.5408, size = 5149, normalized size = 44.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(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 + 3
0*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*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) - 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 + 3
0*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**(33/2)*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 + 7
5*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)) + B*(-256*a**(87/2)*sqrt(1 + b*x/a)/(35*a**40*b**5 + 35
0*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x
**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 15
75*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 256*a**(87/
2)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x*
*3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200
*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**1
5*x**10) - 2432*a**(85/2)*b*x*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x
+ 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**3
5*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*
x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2560*a**(85/2)*b*x/(35*a**
40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*
a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**
12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) -
10336*a**(83/2)*b**2*x**2*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1
575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b
**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**
8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 11520*a**(83/2)*b**2*x**2/(35
*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7
350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33
*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**1
0) - 25840*a**(81/2)*b**3*x**3*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x
+ 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**
35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13
*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 30720*a**(81/2)*b**3*x**3
/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3
+ 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a
**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*
x**10) - 41990*a**(79/2)*b**4*x**4*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b*
*6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820
*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b
**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 53760*a**(79/2)*b**4*
x**4/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*
x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 42
00*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b*
*15*x**10) - 46182*a**(77/2)*b**5*x**5*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**3
9*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 +
8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**
32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 64512*a**(77/2)*b
**5*x**5/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b
**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6
+ 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**3
0*b**15*x**10) - 34584*a**(75/2)*b**6*x**6*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*
a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**
4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575
*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 53760*a**(75/
2)*b**6*x**6/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**
37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*
a**30*b**15*x**10) - 17112*a**(73/2)*b**7*x**7*sqrt(1 + b*x/a)/(35*a**40*b**5 +
350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9
*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 +
1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 30720*a**
(73/2)*b**7*x**7/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200
*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**
11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 +
35*a**30*b**15*x**10) - 4980*a**(71/2)*b**8*x**8*sqrt(1 + b*x/a)/(35*a**40*b**5
+ 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b
**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7
+ 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 11520*
a**(71/2)*b**8*x**8/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4
200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*
b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**
9 + 35*a**30*b**15*x**10) - 340*a**(69/2)*b**9*x**9*sqrt(1 + b*x/a)/(35*a**40*b*
*5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36
*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x*
*7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2560
*a**(69/2)*b**9*x**9/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 +
4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34
*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x*
*9 + 35*a**30*b**15*x**10) + 424*a**(67/2)*b**10*x**10*sqrt(1 + b*x/a)/(35*a**40
*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a*
*36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12
*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 2
56*a**(67/2)*b**10*x**10/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**
2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a
**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**1
4*x**9 + 35*a**30*b**15*x**10) + 248*a**(65/2)*b**11*x**11*sqrt(1 + b*x/a)/(35*a
**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 735
0*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b
**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10)
+ 74*a**(63/2)*b**12*x**12*sqrt(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x +
1575*a**38*b**7*x**2 + 4200*a**37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*
b**10*x**5 + 7350*a**34*b**11*x**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x*
*8 + 350*a**31*b**14*x**9 + 35*a**30*b**15*x**10) + 10*a**(61/2)*b**13*x**13*sqr
t(1 + b*x/a)/(35*a**40*b**5 + 350*a**39*b**6*x + 1575*a**38*b**7*x**2 + 4200*a**
37*b**8*x**3 + 7350*a**36*b**9*x**4 + 8820*a**35*b**10*x**5 + 7350*a**34*b**11*x
**6 + 4200*a**33*b**12*x**7 + 1575*a**32*b**13*x**8 + 350*a**31*b**14*x**9 + 35*
a**30*b**15*x**10))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215253, size = 181, normalized size = 1.56 \[ -\frac{2 \,{\left (B a^{4} - A a^{3} b\right )}}{\sqrt{b x + a} b^{5}} + \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} B b^{30} - 28 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{30} + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{30} - 140 \, \sqrt{b x + a} B a^{3} b^{30} + 7 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{31} - 35 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{31} + 105 \, \sqrt{b x + a} A a^{2} b^{31}\right )}}{35 \, b^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a^4 - A*a^3*b)/(sqrt(b*x + a)*b^5) + 2/35*(5*(b*x + a)^(7/2)*B*b^30 - 28*(
b*x + a)^(5/2)*B*a*b^30 + 70*(b*x + a)^(3/2)*B*a^2*b^30 - 140*sqrt(b*x + a)*B*a^
3*b^30 + 7*(b*x + a)^(5/2)*A*b^31 - 35*(b*x + a)^(3/2)*A*a*b^31 + 105*sqrt(b*x +
a)*A*a^2*b^31)/b^35