\(\int \frac {(a+b x^2+c x^4)^3}{\sqrt {x}} \, dx\) [1058]

Optimal result

Integrand size = 20, antiderivative size = 101 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a \left (b^2+a c\right ) x^{9/2}+\frac {2}{13} b \left (b^2+6 a c\right ) x^{13/2}+\frac {6}{17} c \left (b^2+a c\right ) x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \]

[Out]

6/5*a^2*b*x^(5/2)+2/3*a*(a*c+b^2)*x^(9/2)+2/13*b*(6*a*c+b^2)*x^(13/2)+6/17*c*(a*c+b^2)*x^(17/2)+2/7*b*c^2*x^(2
1/2)+2/25*c^3*x^(25/2)+2*a^3*x^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 101, 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} \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{17} c x^{17/2} \left (a c+b^2\right )+\frac {2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac {2}{3} a x^{9/2} \left (a c+b^2\right )+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \]

[In]

Int[(a + b*x^2 + c*x^4)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*(b^2 + a*c)*x^(9/2))/3 + (2*b*(b^2 + 6*a*c)*x^(13/2))/13 + (6*c*(b^
2 + a*c)*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*c^3*x^(25/2))/25

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 (\frac {a^3}{\sqrt {x}}+3 a^2 b x^{3/2}+3 a \left (b^2+a c\right ) x^{7/2}+b \left (b^2+6 a c\right ) x^{11/2}+3 c \left (b^2+a c\right ) x^{15/2}+3 b c^2 x^{19/2}+c^3 x^{23/2}\right ) \, dx \\ & = 2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a \left (b^2+a c\right ) x^{9/2}+\frac {2}{13} b \left (b^2+6 a c\right ) x^{13/2}+\frac {6}{17} c \left (b^2+a c\right ) x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (116025 a^3+7735 a^2 \left (9 b x^2+5 c x^4\right )+3 x^6 \left (2975 b^3+6825 b^2 c x^2+5525 b c^2 x^4+1547 c^3 x^6\right )+175 a \left (221 b^2 x^4+306 b c x^6+117 c^2 x^8\right )\right )}{116025} \]

[In]

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

[Out]

(2*Sqrt[x]*(116025*a^3 + 7735*a^2*(9*b*x^2 + 5*c*x^4) + 3*x^6*(2975*b^3 + 6825*b^2*c*x^2 + 5525*b*c^2*x^4 + 15
47*c^3*x^6) + 175*a*(221*b^2*x^4 + 306*b*c*x^6 + 117*c^2*x^8)))/116025

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88

method	result	size
trager	\(\left (\frac {2}{25} c^{3} x^{12}+\frac {2}{7} b \,c^{2} x^{10}+\frac {6}{17} a \,c^{2} x^{8}+\frac {6}{17} b^{2} c \,x^{8}+\frac {12}{13} a b c \,x^{6}+\frac {2}{13} b^{3} x^{6}+\frac {2}{3} a^{2} c \,x^{4}+\frac {2}{3} b^{2} x^{4} a +\frac {6}{5} a^{2} b \,x^{2}+2 a^{3}\right ) \sqrt {x}\)	\(89\)
gosper	\(\frac {2 \sqrt {x}\, \left (4641 c^{3} x^{12}+16575 b \,c^{2} x^{10}+20475 a \,c^{2} x^{8}+20475 b^{2} c \,x^{8}+53550 a b c \,x^{6}+8925 b^{3} x^{6}+38675 a^{2} c \,x^{4}+38675 b^{2} x^{4} a +69615 a^{2} b \,x^{2}+116025 a^{3}\right )}{116025}\)	\(90\)
risch	\(\frac {2 \sqrt {x}\, \left (4641 c^{3} x^{12}+16575 b \,c^{2} x^{10}+20475 a \,c^{2} x^{8}+20475 b^{2} c \,x^{8}+53550 a b c \,x^{6}+8925 b^{3} x^{6}+38675 a^{2} c \,x^{4}+38675 b^{2} x^{4} a +69615 a^{2} b \,x^{2}+116025 a^{3}\right )}{116025}\)	\(90\)
derivativedivides	\(\frac {2 c^{3} x^{\frac {25}{2}}}{25}+\frac {2 b \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a^{3} \sqrt {x}\)	\(111\)
default	\(\frac {2 c^{3} x^{\frac {25}{2}}}{25}+\frac {2 b \,c^{2} x^{\frac {21}{2}}}{7}+\frac {2 \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {6 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a^{3} \sqrt {x}\)	\(111\)

[In]

int((c*x^4+b*x^2+a)^3/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

(2/25*c^3*x^12+2/7*b*c^2*x^10+6/17*a*c^2*x^8+6/17*b^2*c*x^8+12/13*a*b*c*x^6+2/13*b^3*x^6+2/3*a^2*c*x^4+2/3*b^2
*x^4*a+6/5*a^2*b*x^2+2*a^3)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{116025} \, {\left (4641 \, c^{3} x^{12} + 16575 \, b c^{2} x^{10} + 20475 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 8925 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 69615 \, a^{2} b x^{2} + 38675 \, {\left (a b^{2} + a^{2} c\right )} x^{4} + 116025 \, a^{3}\right )} \sqrt {x} \]

[In]

integrate((c*x^4+b*x^2+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/116025*(4641*c^3*x^12 + 16575*b*c^2*x^10 + 20475*(b^2*c + a*c^2)*x^8 + 8925*(b^3 + 6*a*b*c)*x^6 + 69615*a^2*
b*x^2 + 38675*(a*b^2 + a^2*c)*x^4 + 116025*a^3)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=2 a^{3} \sqrt {x} + \frac {6 a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 a^{2} c x^{\frac {9}{2}}}{3} + \frac {2 a b^{2} x^{\frac {9}{2}}}{3} + \frac {12 a b c x^{\frac {13}{2}}}{13} + \frac {6 a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 b^{3} x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \]

[In]

integrate((c*x**4+b*x**2+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a**2*c*x**(9/2)/3 + 2*a*b**2*x**(9/2)/3 + 12*a*b*c*x**(13/2)/13 + 6*a
*c**2*x**(17/2)/17 + 2*b**3*x**(13/2)/13 + 6*b**2*c*x**(17/2)/17 + 2*b*c**2*x**(21/2)/7 + 2*c**3*x**(25/2)/25

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + 2 \, a^{3} \sqrt {x} + \frac {2}{15} \, {\left (5 \, c x^{\frac {9}{2}} + 9 \, b x^{\frac {5}{2}}\right )} a^{2} + \frac {2}{663} \, {\left (117 \, c^{2} x^{\frac {17}{2}} + 306 \, b c x^{\frac {13}{2}} + 221 \, b^{2} x^{\frac {9}{2}}\right )} a \]

[In]

integrate((c*x^4+b*x^2+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 2/13*b^3*x^(13/2) + 2*a^3*sqrt(x) + 2/15*(5*c*x
^(9/2) + 9*b*x^(5/2))*a^2 + 2/663*(117*c^2*x^(17/2) + 306*b*c*x^(13/2) + 221*b^2*x^(9/2))*a

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=\frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {12}{13} \, a b c x^{\frac {13}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + 2 \, a^{3} \sqrt {x} \]

[In]

integrate((c*x^4+b*x^2+a)^3/x^(1/2),x, algorithm="giac")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 6/17*a*c^2*x^(17/2) + 2/13*b^3*x^(13/2) + 12/13
*a*b*c*x^(13/2) + 2/3*a*b^2*x^(9/2) + 2/3*a^2*c*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2+c x^4\right )^3}{\sqrt {x}} \, dx=x^{13/2}\,\left (\frac {2\,b^3}{13}+\frac {12\,a\,c\,b}{13}\right )+2\,a^3\,\sqrt {x}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {2\,b\,c^2\,x^{21/2}}{7}+\frac {2\,a\,x^{9/2}\,\left (b^2+a\,c\right )}{3}+\frac {6\,c\,x^{17/2}\,\left (b^2+a\,c\right )}{17} \]

[In]

int((a + b*x^2 + c*x^4)^3/x^(1/2),x)

[Out]

x^(13/2)*((2*b^3)/13 + (12*a*b*c)/13) + 2*a^3*x^(1/2) + (2*c^3*x^(25/2))/25 + (6*a^2*b*x^(5/2))/5 + (2*b*c^2*x
^(21/2))/7 + (2*a*x^(9/2)*(a*c + b^2))/3 + (6*c*x^(17/2)*(a*c + b^2))/17