3.300 \(\int x^3 (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac{2 a^2 (a+b x)^{9/2}}{3 b^4}+\frac{2 (a+b x)^{13/2}}{13 b^4}-\frac{6 a (a+b x)^{11/2}}{11 b^4} \]

[Out]

(-2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a + b
*x)^(11/2))/(11*b^4) + (2*(a + b*x)^(13/2))/(13*b^4)

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Rubi [A]  time = 0.0505628, 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 \[ -\frac{2 a^3 (a+b x)^{7/2}}{7 b^4}+\frac{2 a^2 (a+b x)^{9/2}}{3 b^4}+\frac{2 (a+b x)^{13/2}}{13 b^4}-\frac{6 a (a+b x)^{11/2}}{11 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^(5/2),x]

[Out]

(-2*a^3*(a + b*x)^(7/2))/(7*b^4) + (2*a^2*(a + b*x)^(9/2))/(3*b^4) - (6*a*(a + b
*x)^(11/2))/(11*b^4) + (2*(a + b*x)^(13/2))/(13*b^4)

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Rubi in Sympy [A]  time = 11.2714, size = 68, normalized size = 0.94 \[ - \frac{2 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{3 b^{4}} - \frac{6 a \left (a + b x\right )^{\frac{11}{2}}}{11 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{13}{2}}}{13 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*(a + b*x)**(7/2)/(7*b**4) + 2*a**2*(a + b*x)**(9/2)/(3*b**4) - 6*a*(a +
b*x)**(11/2)/(11*b**4) + 2*(a + b*x)**(13/2)/(13*b**4)

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Mathematica [A]  time = 0.0417677, size = 46, normalized size = 0.64 \[ \frac{2 (a+b x)^{7/2} \left (-16 a^3+56 a^2 b x-126 a b^2 x^2+231 b^3 x^3\right )}{3003 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(7/2)*(-16*a^3 + 56*a^2*b*x - 126*a*b^2*x^2 + 231*b^3*x^3))/(3003*b
^4)

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{-462\,{b}^{3}{x}^{3}+252\,a{b}^{2}{x}^{2}-112\,{a}^{2}bx+32\,{a}^{3}}{3003\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(5/2),x)

[Out]

-2/3003*(b*x+a)^(7/2)*(-231*b^3*x^3+126*a*b^2*x^2-56*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.33702, size = 76, normalized size = 1.06 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{13}{2}}}{13 \, b^{4}} - \frac{6 \,{\left (b x + a\right )}^{\frac{11}{2}} a}{11 \, b^{4}} + \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2}}{3 \, b^{4}} - \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*(b*x + a)^(13/2)/b^4 - 6/11*(b*x + a)^(11/2)*a/b^4 + 2/3*(b*x + a)^(9/2)*a^
2/b^4 - 2/7*(b*x + a)^(7/2)*a^3/b^4

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Fricas [A]  time = 0.227565, size = 101, normalized size = 1.4 \[ \frac{2 \,{\left (231 \, b^{6} x^{6} + 567 \, a b^{5} x^{5} + 371 \, a^{2} b^{4} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{4} b^{2} x^{2} + 8 \, a^{5} b x - 16 \, a^{6}\right )} \sqrt{b x + a}}{3003 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3003*(231*b^6*x^6 + 567*a*b^5*x^5 + 371*a^2*b^4*x^4 + 5*a^3*b^3*x^3 - 6*a^4*b^
2*x^2 + 8*a^5*b*x - 16*a^6)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 11.7951, size = 146, normalized size = 2.03 \[ \begin{cases} - \frac{32 a^{6} \sqrt{a + b x}}{3003 b^{4}} + \frac{16 a^{5} x \sqrt{a + b x}}{3003 b^{3}} - \frac{4 a^{4} x^{2} \sqrt{a + b x}}{1001 b^{2}} + \frac{10 a^{3} x^{3} \sqrt{a + b x}}{3003 b} + \frac{106 a^{2} x^{4} \sqrt{a + b x}}{429} + \frac{54 a b x^{5} \sqrt{a + b x}}{143} + \frac{2 b^{2} x^{6} \sqrt{a + b x}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-32*a**6*sqrt(a + b*x)/(3003*b**4) + 16*a**5*x*sqrt(a + b*x)/(3003*b*
*3) - 4*a**4*x**2*sqrt(a + b*x)/(1001*b**2) + 10*a**3*x**3*sqrt(a + b*x)/(3003*b
) + 106*a**2*x**4*sqrt(a + b*x)/429 + 54*a*b*x**5*sqrt(a + b*x)/143 + 2*b**2*x**
6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(5/2)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.209442, size = 321, normalized size = 4.46 \[ \frac{2 \,{\left (\frac{143 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} a^{2}}{b^{27}} + \frac{26 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} a}{b^{43}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{60} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{60} + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{60} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{60} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{60} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{60}\right )}}{b^{63}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(143*(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^2/b^27 + 26*(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)*a/b^43 + 5*(693*(b*x +
a)^(13/2)*b^60 - 4095*(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^(9/2)*a^2*b^60
- 12870*(b*x + a)^(7/2)*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60 - 3003*(b*x + a
)^(3/2)*a^5*b^60)/b^63)/b