3.1055 \(\int x^{5/2} (a+b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=103 \[ \frac{6}{11} a^2 b x^{11/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} c x^{23/2} \left (a c+b^2\right )+\frac{2}{19} b x^{19/2} \left (6 a c+b^2\right )+\frac{2}{5} a x^{15/2} \left (a c+b^2\right )+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*(b^2 + a*c)*x^(15/2))/5 + (2*b*(b^2 + 6*a*c)*x^(19/2))/19 + (
6*c*(b^2 + a*c)*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

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Rubi [A]  time = 0.0484514, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1108} \[ \frac{6}{11} a^2 b x^{11/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} c x^{23/2} \left (a c+b^2\right )+\frac{2}{19} b x^{19/2} \left (6 a c+b^2\right )+\frac{2}{5} a x^{15/2} \left (a c+b^2\right )+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*(b^2 + a*c)*x^(15/2))/5 + (2*b*(b^2 + 6*a*c)*x^(19/2))/19 + (
6*c*(b^2 + a*c)*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

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 x^{5/2} \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{9/2}+3 a \left (b^2+a c\right ) x^{13/2}+b \left (b^2+6 a c\right ) x^{17/2}+3 c \left (b^2+a c\right ) x^{21/2}+3 b c^2 x^{25/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a^2 b x^{11/2}+\frac{2}{5} a \left (b^2+a c\right ) x^{15/2}+\frac{2}{19} b \left (b^2+6 a c\right ) x^{19/2}+\frac{6}{23} c \left (b^2+a c\right ) x^{23/2}+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2}\\ \end{align*}

Mathematica [A]  time = 3.5143, size = 103, normalized size = 1. \[ \frac{6}{11} a^2 b x^{11/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} c x^{23/2} \left (a c+b^2\right )+\frac{2}{19} b x^{19/2} \left (6 a c+b^2\right )+\frac{2}{5} a x^{15/2} \left (a c+b^2\right )+\frac{2}{9} b c^2 x^{27/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*(b^2 + a*c)*x^(15/2))/5 + (2*b*(b^2 + 6*a*c)*x^(19/2))/19 + (
6*c*(b^2 + a*c)*x^(23/2))/23 + (2*b*c^2*x^(27/2))/9 + (2*c^3*x^(31/2))/31

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Maple [A]  time = 0.046, size = 90, normalized size = 0.9 \begin{align*}{\frac{3028410\,{c}^{3}{x}^{12}[phone]\,b{c}^{2}{x}^{10}[phone]\,{x}^{8}a{c}^{2}[phone]\,{x}^{8}{b}^{2}c+29646540\,{x}^{6}abc+4941090\,{x}^{6}{b}^{3}[phone]\,{a}^{2}c{x}^{4}[phone]\,{x}^{4}{b}^{2}a+25603830\,{a}^{2}b{x}^{2}[phone]\,{a}^{3}}{46940355}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+b*x^2+a)^3,x)

[Out]

2/46940355*x^(7/2)*(1514205*c^3*x^12+5215595*b*c^2*x^10+6122655*a*c^2*x^8+6122655*b^2*c*x^8+14823270*a*b*c*x^6
+2470545*b^3*x^6+9388071*a^2*c*x^4+9388071*a*b^2*x^4+12801915*a^2*b*x^2+6705765*a^3)

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Maxima [A]  time = 0.96827, size = 109, normalized size = 1.06 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, b c^{2} x^{\frac{27}{2}} + \frac{6}{23} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{19}{2}} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{2}{5} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{15}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

2/31*c^3*x^(31/2) + 2/9*b*c^2*x^(27/2) + 6/23*(b^2*c + a*c^2)*x^(23/2) + 2/19*(b^3 + 6*a*b*c)*x^(19/2) + 6/11*
a^2*b*x^(11/2) + 2/5*(a*b^2 + a^2*c)*x^(15/2) + 2/7*a^3*x^(7/2)

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Fricas [A]  time = 1.36986, size = 257, normalized size = 2.5 \begin{align*} \frac{2}{46940355} \,{\left (1514205 \, c^{3} x^{15} + 5215595 \, b c^{2} x^{13} + 6122655 \,{\left (b^{2} c + a c^{2}\right )} x^{11} + 2470545 \,{\left (b^{3} + 6 \, a b c\right )} x^{9} + 12801915 \, a^{2} b x^{5} + 9388071 \,{\left (a b^{2} + a^{2} c\right )} x^{7} + 6705765 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

2/46940355*(1514205*c^3*x^15 + 5215595*b*c^2*x^13 + 6122655*(b^2*c + a*c^2)*x^11 + 2470545*(b^3 + 6*a*b*c)*x^9
+ 12801915*a^2*b*x^5 + 9388071*(a*b^2 + a^2*c)*x^7 + 6705765*a^3*x^3)*sqrt(x)

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Sympy [A]  time = 63.1449, size = 129, normalized size = 1.25 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} b x^{\frac{11}{2}}}{11} + \frac{2 a^{2} c x^{\frac{15}{2}}}{5} + \frac{2 a b^{2} x^{\frac{15}{2}}}{5} + \frac{12 a b c x^{\frac{19}{2}}}{19} + \frac{6 a c^{2} x^{\frac{23}{2}}}{23} + \frac{2 b^{3} x^{\frac{19}{2}}}{19} + \frac{6 b^{2} c x^{\frac{23}{2}}}{23} + \frac{2 b c^{2} x^{\frac{27}{2}}}{9} + \frac{2 c^{3} x^{\frac{31}{2}}}{31} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**4+b*x**2+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 6*a**2*b*x**(11/2)/11 + 2*a**2*c*x**(15/2)/5 + 2*a*b**2*x**(15/2)/5 + 12*a*b*c*x**(19/2)/1
9 + 6*a*c**2*x**(23/2)/23 + 2*b**3*x**(19/2)/19 + 6*b**2*c*x**(23/2)/23 + 2*b*c**2*x**(27/2)/9 + 2*c**3*x**(31
/2)/31

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Giac [A]  time = 1.15364, size = 117, normalized size = 1.14 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, b c^{2} x^{\frac{27}{2}} + \frac{6}{23} \, b^{2} c x^{\frac{23}{2}} + \frac{6}{23} \, a c^{2} x^{\frac{23}{2}} + \frac{2}{19} \, b^{3} x^{\frac{19}{2}} + \frac{12}{19} \, a b c x^{\frac{19}{2}} + \frac{2}{5} \, a b^{2} x^{\frac{15}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{15}{2}} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

2/31*c^3*x^(31/2) + 2/9*b*c^2*x^(27/2) + 6/23*b^2*c*x^(23/2) + 6/23*a*c^2*x^(23/2) + 2/19*b^3*x^(19/2) + 12/19
*a*b*c*x^(19/2) + 2/5*a*b^2*x^(15/2) + 2/5*a^2*c*x^(15/2) + 6/11*a^2*b*x^(11/2) + 2/7*a^3*x^(7/2)