3.9.24 \(\int \sqrt {x} (a+b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=103 \[ \frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {6}{19} c x^{19/2} \left (a c+b^2\right )+\frac {2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac {6}{11} a x^{11/2} \left (a c+b^2\right )+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1108} \begin {gather*} \frac {6}{7} a^2 b x^{7/2}+\frac {2}{3} a^3 x^{3/2}+\frac {6}{19} c x^{19/2} \left (a c+b^2\right )+\frac {2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac {6}{11} a x^{11/2} \left (a c+b^2\right )+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2 + c*x^4)^3,x]

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(b^2 + 6*a*c)*x^(15/2))/15 + (6
*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/23 + (2*c^3*x^(27/2))/27

Rule 1108

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*} \int \sqrt {x} \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 \sqrt {x}+3 a^2 b x^{5/2}+3 a \left (b^2+a c\right ) x^{9/2}+b \left (b^2+6 a c\right ) x^{13/2}+3 c \left (b^2+a c\right ) x^{17/2}+3 b c^2 x^{21/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {6}{11} a \left (b^2+a c\right ) x^{11/2}+\frac {2}{15} b \left (b^2+6 a c\right ) x^{15/2}+\frac {6}{19} c \left (b^2+a c\right ) x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.38, size = 103, normalized size = 1.00 \begin {gather*} \frac {2}{3} a^3 x^{3/2}+\frac {6}{7} a^2 b x^{7/2}+\frac {6}{19} c x^{19/2} \left (a c+b^2\right )+\frac {2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac {6}{11} a x^{11/2} \left (a c+b^2\right )+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*(b^2 + a*c)*x^(11/2))/11 + (2*b*(b^2 + 6*a*c)*x^(15/2))/15 + (6
*c*(b^2 + a*c)*x^(19/2))/19 + (6*b*c^2*x^(23/2))/23 + (2*c^3*x^(27/2))/27

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 111, normalized size = 1.08 \begin {gather*} \frac {2 \left (1514205 a^3 x^{3/2}+1946835 a^2 b x^{7/2}+1238895 a^2 c x^{11/2}+1238895 a b^2 x^{11/2}+1817046 a b c x^{15/2}+717255 a c^2 x^{19/2}+302841 b^3 x^{15/2}+717255 b^2 c x^{19/2}+592515 b c^2 x^{23/2}+168245 c^3 x^{27/2}\right )}{4542615} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1514205*a^3*x^(3/2) + 1946835*a^2*b*x^(7/2) + 1238895*a*b^2*x^(11/2) + 1238895*a^2*c*x^(11/2) + 302841*b^3
*x^(15/2) + 1817046*a*b*c*x^(15/2) + 717255*b^2*c*x^(19/2) + 717255*a*c^2*x^(19/2) + 592515*b*c^2*x^(23/2) + 1
68245*c^3*x^(27/2)))/4542615

________________________________________________________________________________________

fricas [A]  time = 2.69, size = 84, normalized size = 0.82 \begin {gather*} \frac {2}{4542615} \, {\left (168245 \, c^{3} x^{13} + 592515 \, b c^{2} x^{11} + 717255 \, {\left (b^{2} c + a c^{2}\right )} x^{9} + 302841 \, {\left (b^{3} + 6 \, a b c\right )} x^{7} + 1946835 \, a^{2} b x^{3} + 1238895 \, {\left (a b^{2} + a^{2} c\right )} x^{5} + 1514205 \, a^{3} x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4542615*(168245*c^3*x^13 + 592515*b*c^2*x^11 + 717255*(b^2*c + a*c^2)*x^9 + 302841*(b^3 + 6*a*b*c)*x^7 + 194
6835*a^2*b*x^3 + 1238895*(a*b^2 + a^2*c)*x^5 + 1514205*a^3*x)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 87, normalized size = 0.84 \begin {gather*} \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, a c^{2} x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} + \frac {4}{5} \, a b c x^{\frac {15}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c x^{\frac {11}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*b^2*c*x^(19/2) + 6/19*a*c^2*x^(19/2) + 2/15*b^3*x^(15/2) + 4/5*
a*b*c*x^(15/2) + 6/11*a*b^2*x^(11/2) + 6/11*a^2*c*x^(11/2) + 6/7*a^2*b*x^(7/2) + 2/3*a^3*x^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 90, normalized size = 0.87 \begin {gather*} \frac {2 \left (168245 c^{3} x^{12}+592515 b \,c^{2} x^{10}+717255 a \,c^{2} x^{8}+717255 b^{2} c \,x^{8}+1817046 a b c \,x^{6}+302841 b^{3} x^{6}+1238895 a^{2} c \,x^{4}+1238895 a \,b^{2} x^{4}+1946835 a^{2} b \,x^{2}+1514205 a^{3}\right ) x^{\frac {3}{2}}}{4542615} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4542615*x^(3/2)*(168245*c^3*x^12+592515*b*c^2*x^10+717255*a*c^2*x^8+717255*b^2*c*x^8+1817046*a*b*c*x^6+30284
1*b^3*x^6+1238895*a^2*c*x^4+1238895*a*b^2*x^4+1946835*a^2*b*x^2+1514205*a^3)

________________________________________________________________________________________

maxima [A]  time = 1.04, size = 81, normalized size = 0.79 \begin {gather*} \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {15}{2}} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {6}{11} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{3} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*c^3*x^(27/2) + 6/23*b*c^2*x^(23/2) + 6/19*(b^2*c + a*c^2)*x^(19/2) + 2/15*(b^3 + 6*a*b*c)*x^(15/2) + 6/7*
a^2*b*x^(7/2) + 6/11*(a*b^2 + a^2*c)*x^(11/2) + 2/3*a^3*x^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 76, normalized size = 0.74 \begin {gather*} x^{15/2}\,\left (\frac {2\,b^3}{15}+\frac {4\,a\,c\,b}{5}\right )+\frac {2\,a^3\,x^{3/2}}{3}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,a^2\,b\,x^{7/2}}{7}+\frac {6\,b\,c^2\,x^{23/2}}{23}+\frac {6\,a\,x^{11/2}\,\left (b^2+a\,c\right )}{11}+\frac {6\,c\,x^{19/2}\,\left (b^2+a\,c\right )}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x^2 + c*x^4)^3,x)

[Out]

x^(15/2)*((2*b^3)/15 + (4*a*b*c)/5) + (2*a^3*x^(3/2))/3 + (2*c^3*x^(27/2))/27 + (6*a^2*b*x^(7/2))/7 + (6*b*c^2
*x^(23/2))/23 + (6*a*x^(11/2)*(a*c + b^2))/11 + (6*c*x^(19/2)*(a*c + b^2))/19

________________________________________________________________________________________

sympy [A]  time = 6.05, size = 112, normalized size = 1.09 \begin {gather*} \frac {2 a^{3} x^{\frac {3}{2}}}{3} + \frac {6 a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 b c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} + \frac {2 x^{\frac {19}{2}} \left (3 a c^{2} + 3 b^{2} c\right )}{19} + \frac {2 x^{\frac {15}{2}} \left (6 a b c + b^{3}\right )}{15} + \frac {2 x^{\frac {11}{2}} \left (3 a^{2} c + 3 a b^{2}\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*x**(3/2)/3 + 6*a**2*b*x**(7/2)/7 + 6*b*c**2*x**(23/2)/23 + 2*c**3*x**(27/2)/27 + 2*x**(19/2)*(3*a*c**2
+ 3*b**2*c)/19 + 2*x**(15/2)*(6*a*b*c + b**3)/15 + 2*x**(11/2)*(3*a**2*c + 3*a*b**2)/11

________________________________________________________________________________________