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3.433 \(\int \frac{x^3 (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{2 a^3 (A b-a B)}{3 b^5 (a+b x)^{3/2}}-\frac{2 a^2 (3 A b-4 a B)}{b^5 \sqrt{a+b x}}-\frac{6 a \sqrt{a+b x} (A b-2 a B)}{b^5}+\frac{2 (a+b x)^{3/2} (A b-4 a B)}{3 b^5}+\frac{2 B (a+b x)^{5/2}}{5 b^5} \]

[Out]

(2*a^3*(A*b - a*B))/(3*b^5*(a + b*x)^(3/2)) - (2*a^2*(3*A*b - 4*a*B))/(b^5*Sqrt[
a + b*x]) - (6*a*(A*b - 2*a*B)*Sqrt[a + b*x])/b^5 + (2*(A*b - 4*a*B)*(a + b*x)^(
3/2))/(3*b^5) + (2*B*(a + b*x)^(5/2))/(5*b^5)

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Rubi [A]  time = 0.154055, antiderivative size = 118, 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 \[ \frac{2 a^3 (A b-a B)}{3 b^5 (a+b x)^{3/2}}-\frac{2 a^2 (3 A b-4 a B)}{b^5 \sqrt{a+b x}}-\frac{6 a \sqrt{a+b x} (A b-2 a B)}{b^5}+\frac{2 (a+b x)^{3/2} (A b-4 a B)}{3 b^5}+\frac{2 B (a+b x)^{5/2}}{5 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(2*a^3*(A*b - a*B))/(3*b^5*(a + b*x)^(3/2)) - (2*a^2*(3*A*b - 4*a*B))/(b^5*Sqrt[
a + b*x]) - (6*a*(A*b - 2*a*B)*Sqrt[a + b*x])/b^5 + (2*(A*b - 4*a*B)*(a + b*x)^(
3/2))/(3*b^5) + (2*B*(a + b*x)^(5/2))/(5*b^5)

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Rubi in Sympy [A]  time = 21.9477, size = 116, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b^{5}} + \frac{2 a^{3} \left (A b - B a\right )}{3 b^{5} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 a^{2} \left (3 A b - 4 B a\right )}{b^{5} \sqrt{a + b x}} - \frac{6 a \sqrt{a + b x} \left (A b - 2 B a\right )}{b^{5}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A b - 4 B a\right )}{3 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0912995, size = 86, normalized size = 0.73 \[ \frac{2 \left (128 a^4 B+a^3 (192 b B x-80 A b)+24 a^2 b^2 x (2 B x-5 A)-2 a b^3 x^2 (15 A+4 B x)+b^4 x^3 (5 A+3 B x)\right )}{15 b^5 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(2*(128*a^4*B + 24*a^2*b^2*x*(-5*A + 2*B*x) + b^4*x^3*(5*A + 3*B*x) - 2*a*b^3*x^
2*(15*A + 4*B*x) + a^3*(-80*A*b + 192*b*B*x)))/(15*b^5*(a + b*x)^(3/2))

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Maple [A]  time = 0.007, size = 95, normalized size = 0.8 \[ -{\frac{-6\,B{x}^{4}{b}^{4}-10\,A{b}^{4}{x}^{3}+16\,Ba{b}^{3}{x}^{3}+60\,Aa{b}^{3}{x}^{2}-96\,B{a}^{2}{b}^{2}{x}^{2}+240\,A{a}^{2}{b}^{2}x-384\,B{a}^{3}bx+160\,A{a}^{3}b-256\,B{a}^{4}}{15\,{b}^{5}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)/(b*x+a)^(5/2),x)

[Out]

-2/15/(b*x+a)^(3/2)*(-3*B*b^4*x^4-5*A*b^4*x^3+8*B*a*b^3*x^3+30*A*a*b^3*x^2-48*B*
a^2*b^2*x^2+120*A*a^2*b^2*x-192*B*a^3*b*x+80*A*a^3*b-128*B*a^4)/b^5

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Maxima [A]  time = 1.34063, size = 143, normalized size = 1.21 \[ \frac{2 \,{\left (\frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} B - 5 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 45 \,{\left (2 \, B a^{2} - A a b\right )} \sqrt{b x + a}}{b} - \frac{5 \,{\left (B a^{4} - A a^{3} b - 3 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215124, size = 143, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, B b^{4} x^{4} + 128 \, B a^{4} - 80 \, A a^{3} b -{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 24 \,{\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )}}{15 \,{\left (b^{6} x + a b^{5}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^4*x^4 + 128*B*a^4 - 80*A*a^3*b - (8*B*a*b^3 - 5*A*b^4)*x^3 + 6*(8*B*
a^2*b^2 - 5*A*a*b^3)*x^2 + 24*(8*B*a^3*b - 5*A*a^2*b^2)*x)/((b^6*x + a*b^5)*sqrt
(b*x + a))

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Sympy [A]  time = 24.0131, size = 3624, normalized size = 30.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

A*Piecewise((-32*a**3/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 48*a**
2*b*x/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 12*a*b**2*x**2/(3*a*b*
*4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 2*b**3*x**3/(3*a*b**4*sqrt(a + b*x)
 + 3*b**5*x*sqrt(a + b*x)), Ne(b, 0)), (x**4/(4*a**(5/2)), True)) + B*(256*a**(8
5/2)*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1
800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*
b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9
 + 15*a**30*b**15*x**10) - 256*a**(85/2)/(15*a**40*b**5 + 150*a**39*b**6*x + 675
*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**1
0*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 +
150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 2432*a**(83/2)*b*x*sqrt(1 + b*x/a
)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3
 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a
**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x
**10) - 2560*a**(83/2)*b*x/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 10336*a**(81/2)*b**2*x**2*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 11520*a**(81/2)*b**2*x**2/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 25840*a**(79/2)*b**3*x**3*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 30720*a**(79/2)*b**3*x**3/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 41990*a**(77/2)*b**4*x**4*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 53760*a**(77/2)*b**4*x**4/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 46192*a**(75/2)*b**5*x**5*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 64512*a**(75/2)*b**5*x**5/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 34664*a**(73/2)*b**6*x**6*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 53760*a**(73/2)*b**6*x**6/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 17392*a**(71/2)*b**7*x**7*sqrt(1 + b*x/a)/(15*a
**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150
*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b*
*12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
 30720*a**(71/2)*b**7*x**7/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x*
*2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*
a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**1
4*x**9 + 15*a**30*b**15*x**10) + 5540*a**(69/2)*b**8*x**8*sqrt(1 + b*x/a)/(15*a*
*40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*
a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**
12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) -
11520*a**(69/2)*b**8*x**8/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**
2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a
**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14
*x**9 + 15*a**30*b**15*x**10) + 1040*a**(67/2)*b**9*x**9*sqrt(1 + b*x/a)/(15*a**
40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a
**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**1
2*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 2
560*a**(67/2)*b**9*x**9/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2
+ 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**
34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x
**9 + 15*a**30*b**15*x**10) + 136*a**(65/2)*b**10*x**10*sqrt(1 + b*x/a)/(15*a**4
0*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a*
*36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12
*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) - 25
6*a**(65/2)*b**10*x**10/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2
+ 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**
34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x
**9 + 15*a**30*b**15*x**10) + 32*a**(63/2)*b**11*x**11*sqrt(1 + b*x/a)/(15*a**40
*b**5 + 150*a**39*b**6*x + 675*a**38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**
36*b**9*x**4 + 3780*a**35*b**10*x**5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*
x**7 + 675*a**32*b**13*x**8 + 150*a**31*b**14*x**9 + 15*a**30*b**15*x**10) + 6*a
**(61/2)*b**12*x**12*sqrt(1 + b*x/a)/(15*a**40*b**5 + 150*a**39*b**6*x + 675*a**
38*b**7*x**2 + 1800*a**37*b**8*x**3 + 3150*a**36*b**9*x**4 + 3780*a**35*b**10*x*
*5 + 3150*a**34*b**11*x**6 + 1800*a**33*b**12*x**7 + 675*a**32*b**13*x**8 + 150*
a**31*b**14*x**9 + 15*a**30*b**15*x**10))

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GIAC/XCAS [A]  time = 0.21942, size = 169, normalized size = 1.43 \[ \frac{2 \,{\left (12 \,{\left (b x + a\right )} B a^{3} - B a^{4} - 9 \,{\left (b x + a\right )} A a^{2} b + A a^{3} b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{5}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{20} - 20 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{20} + 90 \, \sqrt{b x + a} B a^{2} b^{20} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{21} - 45 \, \sqrt{b x + a} A a b^{21}\right )}}{15 \, b^{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

2/3*(12*(b*x + a)*B*a^3 - B*a^4 - 9*(b*x + a)*A*a^2*b + A*a^3*b)/((b*x + a)^(3/2
)*b^5) + 2/15*(3*(b*x + a)^(5/2)*B*b^20 - 20*(b*x + a)^(3/2)*B*a*b^20 + 90*sqrt(
b*x + a)*B*a^2*b^20 + 5*(b*x + a)^(3/2)*A*b^21 - 45*sqrt(b*x + a)*A*a*b^21)/b^25

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