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

Optimal. Leaf size=21 \[ \frac {2}{5} a x^{5/2}+\frac {2}{9} b x^{9/2} \]

[Out]

2/5*a*x^(5/2)+2/9*b*x^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 21, 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 = {14} \[ \frac {2}{5} a x^{5/2}+\frac {2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {2}{5} a x^{5/2}+\frac {2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9

________________________________________________________________________________________

fricas [A]  time = 1.29, size = 18, normalized size = 0.86 \[ \frac {2}{45} \, {\left (5 \, b x^{4} + 9 \, a x^{2}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*b*x^4 + 9*a*x^2)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.63, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b*x^(9/2) + 2/5*a*x^(5/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (5 b \,x^{2}+9 a \right ) x^{\frac {5}{2}}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*x^(5/2)*(5*b*x^2+9*a)

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, b x^{\frac {9}{2}} + \frac {2}{5} \, a x^{\frac {5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b*x^(9/2) + 2/5*a*x^(5/2)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 15, normalized size = 0.71 \[ \frac {2\,x^{5/2}\,\left (5\,b\,x^2+9\,a\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + b*x^2),x)

[Out]

(2*x^(5/2)*(9*a + 5*b*x^2))/45

________________________________________________________________________________________

sympy [A]  time = 0.95, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 b x^{\frac {9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(9/2)/9

________________________________________________________________________________________