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

Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{7/2}}{7 b^3}+\frac{2 (a+b x)^{11/2}}{11 b^3}-\frac{4 a (a+b x)^{9/2}}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0382252, antiderivative size = 53, 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 \[ \frac{2 a^2 (a+b x)^{7/2}}{7 b^3}+\frac{2 (a+b x)^{11/2}}{11 b^3}-\frac{4 a (a+b x)^{9/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.09474, size = 49, normalized size = 0.92 \[ \frac{2 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3}} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{11}{2}}}{11 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0304784, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{7/2} \left (8 a^2-28 a b x+63 b^2 x^2\right )}{693 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(7/2)*(8*a^2 - 28*a*b*x + 63*b^2*x^2))/(693*b^3)

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Maple [A]  time = 0.008, size = 32, normalized size = 0.6 \[{\frac{126\,{b}^{2}{x}^{2}-56\,abx+16\,{a}^{2}}{693\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(5/2),x)

[Out]

2/693*(b*x+a)^(7/2)*(63*b^2*x^2-28*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 1.34435, size = 55, normalized size = 1.04 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}}}{11 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{9}{2}} a}{9 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215194, size = 86, normalized size = 1.62 \[ \frac{2 \,{\left (63 \, b^{5} x^{5} + 161 \, a b^{4} x^{4} + 113 \, a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x + 8 \, a^{5}\right )} \sqrt{b x + a}}{693 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(63*b^5*x^5 + 161*a*b^4*x^4 + 113*a^2*b^3*x^3 + 3*a^3*b^2*x^2 - 4*a^4*b*x
+ 8*a^5)*sqrt(b*x + a)/b^3

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Sympy [A]  time = 9.33297, size = 124, normalized size = 2.34 \[ \begin{cases} \frac{16 a^{5} \sqrt{a + b x}}{693 b^{3}} - \frac{8 a^{4} x \sqrt{a + b x}}{693 b^{2}} + \frac{2 a^{3} x^{2} \sqrt{a + b x}}{231 b} + \frac{226 a^{2} x^{3} \sqrt{a + b x}}{693} + \frac{46 a b x^{4} \sqrt{a + b x}}{99} + \frac{2 b^{2} x^{5} \sqrt{a + b x}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((16*a**5*sqrt(a + b*x)/(693*b**3) - 8*a**4*x*sqrt(a + b*x)/(693*b**2)
+ 2*a**3*x**2*sqrt(a + b*x)/(231*b) + 226*a**2*x**3*sqrt(a + b*x)/693 + 46*a*b*x
**4*sqrt(a + b*x)/99 + 2*b**2*x**5*sqrt(a + b*x)/11, Ne(b, 0)), (a**(5/2)*x**3/3
, True))

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GIAC/XCAS [A]  time = 0.208088, size = 259, normalized size = 4.89 \[ \frac{2 \,{\left (\frac{33 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} a^{2}}{b^{14}} + \frac{22 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} a}{b^{26}} + \frac{315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}}{b^{42}}\right )}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(33*(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(
3/2)*a^2*b^12)*a^2/b^14 + 22*(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^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)/b^42)/b