3.292 \(\int x^3 (a+b x)^{3/2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac{6 a^2 (a+b x)^{7/2}}{7 b^4}+\frac{2 (a+b x)^{11/2}}{11 b^4}-\frac{2 a (a+b x)^{9/2}}{3 b^4} \]

[Out]

(-2*a^3*(a + b*x)^(5/2))/(5*b^4) + (6*a^2*(a + b*x)^(7/2))/(7*b^4) - (2*a*(a + b
*x)^(9/2))/(3*b^4) + (2*(a + b*x)^(11/2))/(11*b^4)

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Rubi [A]  time = 0.0507967, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 a^3 (a+b x)^{5/2}}{5 b^4}+\frac{6 a^2 (a+b x)^{7/2}}{7 b^4}+\frac{2 (a+b x)^{11/2}}{11 b^4}-\frac{2 a (a+b x)^{9/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^(3/2),x]

[Out]

(-2*a^3*(a + b*x)^(5/2))/(5*b^4) + (6*a^2*(a + b*x)^(7/2))/(7*b^4) - (2*a*(a + b
*x)^(9/2))/(3*b^4) + (2*(a + b*x)^(11/2))/(11*b^4)

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Rubi in Sympy [A]  time = 11.1589, size = 68, normalized size = 0.94 \[ - \frac{2 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{2 a \left (a + b x\right )^{\frac{9}{2}}}{3 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{11}{2}}}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*(a + b*x)**(5/2)/(5*b**4) + 6*a**2*(a + b*x)**(7/2)/(7*b**4) - 2*a*(a +
b*x)**(9/2)/(3*b**4) + 2*(a + b*x)**(11/2)/(11*b**4)

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Mathematica [A]  time = 0.035504, size = 46, normalized size = 0.64 \[ \frac{2 (a+b x)^{5/2} \left (-16 a^3+40 a^2 b x-70 a b^2 x^2+105 b^3 x^3\right )}{1155 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(5/2)*(-16*a^3 + 40*a^2*b*x - 70*a*b^2*x^2 + 105*b^3*x^3))/(1155*b^
4)

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Maple [A]  time = 0.009, size = 43, normalized size = 0.6 \[ -{\frac{-210\,{b}^{3}{x}^{3}+140\,a{b}^{2}{x}^{2}-80\,{a}^{2}bx+32\,{a}^{3}}{1155\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(3/2),x)

[Out]

-2/1155*(b*x+a)^(5/2)*(-105*b^3*x^3+70*a*b^2*x^2-40*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.34908, size = 76, normalized size = 1.06 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}}}{11 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}} a}{3 \, b^{4}} + \frac{6 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3}}{5 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b*x + a)^(11/2)/b^4 - 2/3*(b*x + a)^(9/2)*a/b^4 + 6/7*(b*x + a)^(7/2)*a^2/
b^4 - 2/5*(b*x + a)^(5/2)*a^3/b^4

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Fricas [A]  time = 0.215592, size = 86, normalized size = 1.19 \[ \frac{2 \,{\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt{b x + a}}{1155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(105*b^5*x^5 + 140*a*b^4*x^4 + 5*a^2*b^3*x^3 - 6*a^3*b^2*x^2 + 8*a^4*b*x
- 16*a^5)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 9.47027, size = 1742, normalized size = 24.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*a**(51/2)*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18
*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**
5 + 1155*a**14*b**10*x**6) + 32*a**(51/2)/(1155*a**20*b**4 + 6930*a**19*b**5*x +
17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a*
*15*b**9*x**5 + 1155*a**14*b**10*x**6) - 176*a**(49/2)*b*x*sqrt(1 + b*x/a)/(1155
*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3
+ 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 192*a*
*(49/2)*b*x/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100
*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**
10*x**6) - 396*a**(47/2)*b**2*x**2*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19
*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4
+ 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 480*a**(47/2)*b**2*x**2/(1155*
a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 +
17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) - 462*a**
(45/2)*b**3*x**3*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a*
*18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*
x**5 + 1155*a**14*b**10*x**6) + 640*a**(45/2)*b**3*x**3/(1155*a**20*b**4 + 6930*
a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*
x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 480*a**(43/2)*b**4*x**4/(
1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x
**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 18
48*a**(41/2)*b**5*x**5*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17
325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15
*b**9*x**5 + 1155*a**14*b**10*x**6) + 192*a**(41/2)*b**5*x**5/(1155*a**20*b**4 +
6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16
*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 5544*a**(39/2)*b**6
*x**6*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x*
*2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155
*a**14*b**10*x**6) + 32*a**(39/2)*b**6*x**6/(1155*a**20*b**4 + 6930*a**19*b**5*x
+ 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*
a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 8844*a**(37/2)*b**7*x**7*sqrt(1 + b*x
/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b
**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6)
+ 8448*a**(35/2)*b**8*x**8*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x
+ 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*
a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 4840*a**(33/2)*b**9*x**9*sqrt(1 + b*x
/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**17*b
**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x**6)
+ 1540*a**(31/2)*b**10*x**10*sqrt(1 + b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5
*x + 17325*a**18*b**6*x**2 + 23100*a**17*b**7*x**3 + 17325*a**16*b**8*x**4 + 693
0*a**15*b**9*x**5 + 1155*a**14*b**10*x**6) + 210*a**(29/2)*b**11*x**11*sqrt(1 +
b*x/a)/(1155*a**20*b**4 + 6930*a**19*b**5*x + 17325*a**18*b**6*x**2 + 23100*a**1
7*b**7*x**3 + 17325*a**16*b**8*x**4 + 6930*a**15*b**9*x**5 + 1155*a**14*b**10*x*
*6)

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GIAC/XCAS [A]  time = 0.213821, size = 193, normalized size = 2.68 \[ \frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} a}{b^{27}} + \frac{315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}}{b^{43}}\right )}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(11*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 189*(b*x + a)
^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*a/b^27 + (315*(b*x + a)^(11/2)*b
^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 2772*(b*x +
a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)/b^43)/b