3.1048 \(\int x^{5/2} (a+b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=64 \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

[Out]

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

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Rubi [A]  time = 0.0229386, antiderivative size = 64, 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} \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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

Mathematica [A]  time = 3.54703, size = 64, normalized size = 1. \[ \frac{2}{7} a^2 x^{7/2}+\frac{2}{15} x^{15/2} \left (2 a c+b^2\right )+\frac{4}{11} a b x^{11/2}+\frac{4}{19} b c x^{19/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 49, normalized size = 0.8 \begin{align*}{\frac{43890\,{c}^{2}{x}^{8}+106260\,bc{x}^{6}+134596\,{x}^{4}ac+67298\,{b}^{2}{x}^{4}+183540\,ab{x}^{2}+144210\,{a}^{2}}{504735}{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)^2,x)

[Out]

2/504735*x^(7/2)*(21945*c^2*x^8+53130*b*c*x^6+67298*a*c*x^4+33649*b^2*x^4+91770*a*b*x^2+72105*a^2)

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Maxima [A]  time = 0.974124, size = 59, normalized size = 0.92 \begin{align*} \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b c x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.20814, size = 147, normalized size = 2.3 \begin{align*} \frac{2}{504735} \,{\left (21945 \, c^{2} x^{11} + 53130 \, b c x^{9} + 33649 \,{\left (b^{2} + 2 \, a c\right )} x^{7} + 91770 \, a b x^{5} + 72105 \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

2/504735*(21945*c^2*x^11 + 53130*b*c*x^9 + 33649*(b^2 + 2*a*c)*x^7 + 91770*a*b*x^5 + 72105*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 21.3145, size = 70, normalized size = 1.09 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{4 a c x^{\frac{15}{2}}}{15} + \frac{2 b^{2} x^{\frac{15}{2}}}{15} + \frac{4 b c x^{\frac{19}{2}}}{19} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(11/2)/11 + 4*a*c*x**(15/2)/15 + 2*b**2*x**(15/2)/15 + 4*b*c*x**(19/2)/19 + 2*c**
2*x**(23/2)/23

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Giac [A]  time = 1.18456, size = 62, normalized size = 0.97 \begin{align*} \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b c x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

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