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

Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3} \]

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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} \begin {gather*} \frac {2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(3/2),x]

[Out]

(2*a^2*(a + b*x)^(5/2))/(5*b^3) - (4*a*(a + b*x)^(7/2))/(7*b^3) + (2*(a + b*x)^(9/2))/(9*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 (a+b x)^{3/2} \, dx &=\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{7/2}}{7 b^3}+\frac {2 (a+b x)^{9/2}}{9 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2)*(8*a^2 - 20*a*b*x + 35*b^2*x^2))/(315*b^3)

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IntegrateAlgebraic [A]  time = 0.02, size = 45, normalized size = 0.85 \begin {gather*} \frac {2 \left (63 a^2 (a+b x)^{5/2}+35 (a+b x)^{9/2}-90 a (a+b x)^{7/2}\right )}{315 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(63*a^2*(a + b*x)^(5/2) - 90*a*(a + b*x)^(7/2) + 35*(a + b*x)^(9/2)))/(315*b^3)

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fricas [A]  time = 0.75, size = 53, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^4*x^4 + 50*a*b^3*x^3 + 3*a^2*b^2*x^2 - 4*a^3*b*x + 8*a^4)*sqrt(b*x + a)/b^3

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giac [B]  time = 0.94, size = 156, normalized size = 2.94 \begin {gather*} \frac {2 \, {\left (\frac {21 \, {\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^{2}}{b^{2}} + \frac {18 \, {\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 )} a}{b^{2}} + \frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}}{b^{2}}\right )}}{315 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*a^2/b^2 + 18*(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)*a/b^2 + (35*(b*x + a)^(9/2) - 180*(b*x +
a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)/b^2)/b

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maple [A]  time = 0.01, size = 32, normalized size = 0.60 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (35 b^{2} x^{2}-20 a b x +8 a^{2}\right )}{315 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^(3/2),x)

[Out]

2/315*(b*x+a)^(5/2)*(35*b^2*x^2-20*a*b*x+8*a^2)/b^3

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maxima [A]  time = 1.35, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}}}{9 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {7}{2}} a}{7 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {70\,{\left (a+b\,x\right )}^{9/2}-180\,a\,{\left (a+b\,x\right )}^{7/2}+126\,a^2\,{\left (a+b\,x\right )}^{5/2}}{315\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(3/2),x)

[Out]

(70*(a + b*x)^(9/2) - 180*a*(a + b*x)^(7/2) + 126*a^2*(a + b*x)^(5/2))/(315*b^3)

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sympy [B]  time = 2.17, size = 733, normalized size = 13.83 \begin {gather*} \frac {16 a^{\frac {25}{2}} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {25}{2}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {23}{2}} b x \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {23}{2}} b x}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {21}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b^{2} x^{2}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {110 a^{\frac {19}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {19}{2}} b^{3} x^{3}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {380 a^{\frac {17}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {516 a^{\frac {15}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {310 a^{\frac {13}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac {70 a^{\frac {11}{2}} b^{7} x^{7} \sqrt {1 + \frac {b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*a**(25/2)*sqrt(1 + b*x/a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) - 16*
a**(25/2)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 40*a**(23/2)*b*x*sqrt(
1 + b*x/a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) - 48*a**(23/2)*b*x/(315
*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 30*a**(21/2)*b**2*x**2*sqrt(1 + b*x/
a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) - 48*a**(21/2)*b**2*x**2/(315*a
**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 110*a**(19/2)*b**3*x**3*sqrt(1 + b*x/a
)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) - 16*a**(19/2)*b**3*x**3/(315*a*
*8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 380*a**(17/2)*b**4*x**4*sqrt(1 + b*x/a)
/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 516*a**(15/2)*b**5*x**5*sqrt(1
+ b*x/a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 310*a**(13/2)*b**6*x**6
*sqrt(1 + b*x/a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3) + 70*a**(11/2)*b*
*7*x**7*sqrt(1 + b*x/a)/(315*a**8*b**3 + 945*a**7*b**4*x + 945*a**6*b**5*x**2 + 315*a**5*b**6*x**3)

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