3.297 \(\int \frac {b x^2+c x^4}{x^{3/2}} \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

[Out]

2/3*b*x^(3/2)+2/7*c*x^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ \frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 16, normalized size = 0.76 \[ \frac {2}{21} \, {\left (3 \, c x^{3} + 7 \, b x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*c*x^3 + 7*b*x)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 13, normalized size = 0.62 \[ \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(3/2),x, algorithm="giac")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (3 c \,x^{2}+7 b \right ) x^{\frac {3}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^(3/2),x)

[Out]

2/21*x^(3/2)*(3*c*x^2+7*b)

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 13, normalized size = 0.62 \[ \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(3/2),x, algorithm="maxima")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 15, normalized size = 0.71 \[ \frac {2\,x^{3/2}\,\left (3\,c\,x^2+7\,b\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^(3/2),x)

[Out]

(2*x^(3/2)*(7*b + 3*c*x^2))/21

________________________________________________________________________________________

sympy [A]  time = 0.79, size = 19, normalized size = 0.90 \[ \frac {2 b x^{\frac {3}{2}}}{3} + \frac {2 c x^{\frac {7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**(3/2),x)

[Out]

2*b*x**(3/2)/3 + 2*c*x**(7/2)/7

________________________________________________________________________________________