3.179 \(\int x^2 \sqrt {a+b x} \, dx\)

Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*x],x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int x^2 \sqrt {a+b x} \, dx &=\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{3/2}}{3 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 35, normalized size = 0.66 \[ \frac {2 (a+b x)^{3/2} \left (8 a^2-12 a b x+15 b^2 x^2\right )}{105 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2)*(8*a^2 - 12*a*b*x + 15*b^2*x^2))/(105*b^3)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 42, normalized size = 0.79 \[ \frac {2 \, {\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x + a)/b^3

________________________________________________________________________________________

giac [B]  time = 1.03, size = 93, normalized size = 1.75 \[ \frac {2 \, {\left (\frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a}{b^{2}} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )}}{b^{2}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/105*(7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*a/b^2 + 3*(5*(b*x + a)^(7/2) - 21*(
b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)/b^2)/b

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.60 \[ \frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (15 b^{2} x^{2}-12 a x b +8 a^{2}\right )}{105 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^(1/2),x)

[Out]

2/105*(b*x+a)^(3/2)*(15*b^2*x^2-12*a*b*x+8*a^2)/b^3

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 41, normalized size = 0.77 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 37, normalized size = 0.70 \[ \frac {30\,{\left (a+b\,x\right )}^{7/2}-84\,a\,{\left (a+b\,x\right )}^{5/2}+70\,a^2\,{\left (a+b\,x\right )}^{3/2}}{105\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(1/2),x)

[Out]

(30*(a + b*x)^(7/2) - 84*a*(a + b*x)^(5/2) + 70*a^2*(a + b*x)^(3/2))/(105*b^3)

________________________________________________________________________________________

sympy [B]  time = 1.75, size = 666, normalized size = 12.57 \[ \frac {16 a^{\frac {23}{2}} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {23}{2}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {21}{2}} b x \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b x}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {19}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b^{2} x^{2}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {17}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {17}{2}} b^{3} x^{3}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {100 a^{\frac {15}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {96 a^{\frac {13}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {11}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)**(1/2),x)

[Out]

16*a**(23/2)*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 16*
a**(23/2)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(21/2)*b*x*sqrt(
1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(21/2)*b*x/(105
*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 30*a**(19/2)*b**2*x**2*sqrt(1 + b*x/
a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(19/2)*b**2*x**2/(105*a
**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(17/2)*b**3*x**3*sqrt(1 + b*x/a)
/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 16*a**(17/2)*b**3*x**3/(105*a**
8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 100*a**(15/2)*b**4*x**4*sqrt(1 + b*x/a)/
(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 96*a**(13/2)*b**5*x**5*sqrt(1 +
b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 30*a**(11/2)*b**6*x**6*sq
rt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3)

________________________________________________________________________________________