\(\int x^{3/2} (b x^2+c x^4) \, dx\) [294]

Optimal result

Integrand size = 17, antiderivative size = 21 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{9} b x^{9/2}+\frac {2}{13} c x^{13/2} \]

[Out]

2/9*b*x^(9/2)+2/13*c*x^(13/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{9} b x^{9/2}+\frac {2}{13} c x^{13/2} \]

[In]

Int[x^(3/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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*} \text {integral}& = \int \left (b x^{7/2}+c x^{11/2}\right ) \, dx \\ & = \frac {2}{9} b x^{9/2}+\frac {2}{13} c x^{13/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{117} \left (13 b x^{9/2}+9 c x^{13/2}\right ) \]

[In]

Integrate[x^(3/2)*(b*x^2 + c*x^4),x]

[Out]

(2*(13*b*x^(9/2) + 9*c*x^(13/2)))/117

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {2 b \,x^{\frac {9}{2}}}{9}+\frac {2 c \,x^{\frac {13}{2}}}{13}\)	\(14\)
default	\(\frac {2 b \,x^{\frac {9}{2}}}{9}+\frac {2 c \,x^{\frac {13}{2}}}{13}\)	\(14\)
gosper	\(\frac {2 x^{\frac {9}{2}} \left (9 c \,x^{2}+13 b \right )}{117}\)	\(16\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (9 c \,x^{2}+13 b \right )}{117}\)	\(16\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (9 c \,x^{2}+13 b \right )}{117}\)	\(16\)

[In]

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

[Out]

2/9*b*x^(9/2)+2/13*c*x^(13/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{117} \, {\left (9 \, c x^{6} + 13 \, b x^{4}\right )} \sqrt {x} \]

[In]

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

[Out]

2/117*(9*c*x^6 + 13*b*x^4)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2 b x^{\frac {9}{2}}}{9} + \frac {2 c x^{\frac {13}{2}}}{13} \]

[In]

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

[Out]

2*b*x**(9/2)/9 + 2*c*x**(13/2)/13

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{9} \, b x^{\frac {9}{2}} \]

[In]

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

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2}{13} \, c x^{\frac {13}{2}} + \frac {2}{9} \, b x^{\frac {9}{2}} \]

[In]

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

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int x^{3/2} \left (b x^2+c x^4\right ) \, dx=\frac {2\,x^{9/2}\,\left (9\,c\,x^2+13\,b\right )}{117} \]

[In]

int(x^(3/2)*(b*x^2 + c*x^4),x)

[Out]

(2*x^(9/2)*(13*b + 9*c*x^2))/117