3.17.48 \(\int (a+\frac {b}{x}) x^{3/2} \, dx\) [1648]

Optimal. Leaf size=21 \[ \frac {2}{3} b x^{3/2}+\frac {2}{5} a x^{5/2} \]

[Out]

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

________________________________________________________________________________________

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} \begin {gather*} \frac {2}{5} a x^{5/2}+\frac {2}{3} b x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x^(3/2),x]

[Out]

(2*b*x^(3/2))/3 + (2*a*x^(5/2))/5

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{15} x^{3/2} (5 b+3 a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x^(3/2),x]

[Out]

(2*x^(3/2)*(5*b + 3*a*x))/15

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 14, normalized size = 0.67

method	result	size
gosper	\(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\)	\(14\)
derivativedivides	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 a \,x^{\frac {5}{2}}}{5}\)	\(14\)
default	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 a \,x^{\frac {5}{2}}}{5}\)	\(14\)
trager	\(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\)	\(14\)
risch	\(\frac {2 \left (3 a x +5 b \right ) x^{\frac {3}{2}}}{15}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x*b)*x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 15, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, {\left (3 \, a + \frac {5 \, b}{x}\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a + 5*b/x)*x^(5/2)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 16, normalized size = 0.76 \begin {gather*} \frac {2}{15} \, {\left (3 \, a x^{2} + 5 \, b x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a*x^2 + 5*b*x)*sqrt(x)

________________________________________________________________________________________

Sympy [A]
time = 0.15, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 a x^{\frac {5}{2}}}{5} + \frac {2 b x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(3/2)/3

________________________________________________________________________________________

Giac [A]
time = 0.55, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, a x^{\frac {5}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*a*x^(5/2) + 2/3*b*x^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 13, normalized size = 0.62 \begin {gather*} \frac {2\,x^{3/2}\,\left (5\,b+3\,a\,x\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + b/x),x)

[Out]

(2*x^(3/2)*(5*b + 3*a*x))/15

________________________________________________________________________________________