\(\int x^{3/2} (a+b x^2+c x^4)^2 \, dx\) [1049]

Optimal result

Integrand size = 20, antiderivative size = 64 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{5} a^2 x^{5/2}+\frac {4}{9} a b x^{9/2}+\frac {2}{13} \left (b^2+2 a c\right ) x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \]

[Out]

2/5*a^2*x^(5/2)+4/9*a*b*x^(9/2)+2/13*(2*a*c+b^2)*x^(13/2)+4/17*b*c*x^(17/2)+2/21*c^2*x^(21/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122} \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{5} a^2 x^{5/2}+\frac {2}{13} x^{13/2} \left (2 a c+b^2\right )+\frac {4}{9} a b x^{9/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \]

[In]

Int[x^(3/2)*(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*b*x^(9/2))/9 + (2*(b^2 + 2*a*c)*x^(13/2))/13 + (4*b*c*x^(17/2))/17 + (2*c^2*x^(21/2))
/21

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x^{3/2}+2 a b x^{7/2}+\left (b^2+2 a c\right ) x^{11/2}+2 b c x^{15/2}+c^2 x^{19/2}\right ) \, dx \\ & = \frac {2}{5} a^2 x^{5/2}+\frac {4}{9} a b x^{9/2}+\frac {2}{13} \left (b^2+2 a c\right ) x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2 x^{5/2} \left (13923 a^2+1190 a \left (13 b x^2+9 c x^4\right )+15 x^4 \left (357 b^2+546 b c x^2+221 c^2 x^4\right )\right )}{69615} \]

[In]

Integrate[x^(3/2)*(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*x^(5/2)*(13923*a^2 + 1190*a*(13*b*x^2 + 9*c*x^4) + 15*x^4*(357*b^2 + 546*b*c*x^2 + 221*c^2*x^4)))/69615

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {17}{2}}}{17}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\)	\(45\)
default	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {17}{2}}}{17}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\)	\(45\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (3315 c^{2} x^{8}+8190 b c \,x^{6}+10710 a c \,x^{4}+5355 b^{2} x^{4}+15470 a b \,x^{2}+13923 a^{2}\right )}{69615}\)	\(49\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (3315 c^{2} x^{8}+8190 b c \,x^{6}+10710 a c \,x^{4}+5355 b^{2} x^{4}+15470 a b \,x^{2}+13923 a^{2}\right )}{69615}\)	\(49\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (3315 c^{2} x^{8}+8190 b c \,x^{6}+10710 a c \,x^{4}+5355 b^{2} x^{4}+15470 a b \,x^{2}+13923 a^{2}\right )}{69615}\)	\(49\)

[In]

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

[Out]

2/5*a^2*x^(5/2)+4/9*a*b*x^(9/2)+2/13*(2*a*c+b^2)*x^(13/2)+4/17*b*c*x^(17/2)+2/21*c^2*x^(21/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.77 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{69615} \, {\left (3315 \, c^{2} x^{10} + 8190 \, b c x^{8} + 5355 \, {\left (b^{2} + 2 \, a c\right )} x^{6} + 15470 \, a b x^{4} + 13923 \, a^{2} x^{2}\right )} \sqrt {x} \]

[In]

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

[Out]

2/69615*(3315*c^2*x^10 + 8190*b*c*x^8 + 5355*(b^2 + 2*a*c)*x^6 + 15470*a*b*x^4 + 13923*a^2*x^2)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {4 a c x^{\frac {13}{2}}}{13} + \frac {2 b^{2} x^{\frac {13}{2}}}{13} + \frac {4 b c x^{\frac {17}{2}}}{17} + \frac {2 c^{2} x^{\frac {21}{2}}}{21} \]

[In]

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

[Out]

2*a**2*x**(5/2)/5 + 4*a*b*x**(9/2)/9 + 4*a*c*x**(13/2)/13 + 2*b**2*x**(13/2)/13 + 4*b*c*x**(17/2)/17 + 2*c**2*
x**(21/2)/21

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.69 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {13}{2}} + \frac {4}{9} \, a b x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*(b^2 + 2*a*c)*x^(13/2) + 4/9*a*b*x^(9/2) + 2/5*a^2*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=\frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} x^{\frac {13}{2}} + \frac {4}{13} \, a c x^{\frac {13}{2}} + \frac {4}{9} \, a b x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*b^2*x^(13/2) + 4/13*a*c*x^(13/2) + 4/9*a*b*x^(9/2) + 2/5*a^2*x^(5
/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.70 \[ \int x^{3/2} \left (a+b x^2+c x^4\right )^2 \, dx=x^{13/2}\,\left (\frac {2\,b^2}{13}+\frac {4\,a\,c}{13}\right )+\frac {2\,a^2\,x^{5/2}}{5}+\frac {2\,c^2\,x^{21/2}}{21}+\frac {4\,a\,b\,x^{9/2}}{9}+\frac {4\,b\,c\,x^{17/2}}{17} \]

[In]

int(x^(3/2)*(a + b*x^2 + c*x^4)^2,x)

[Out]

x^(13/2)*((4*a*c)/13 + (2*b^2)/13) + (2*a^2*x^(5/2))/5 + (2*c^2*x^(21/2))/21 + (4*a*b*x^(9/2))/9 + (4*b*c*x^(1
7/2))/17