3.1050 \(\int \sqrt{x} (a+b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=64 \[ \frac{2}{3} a^2 x^{3/2}+\frac{2}{11} x^{11/2} \left (2 a c+b^2\right )+\frac{4}{7} a b x^{7/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2} \]

[Out]

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

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Rubi [A]  time = 0.0230685, 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} \[ \frac{2}{3} a^2 x^{3/2}+\frac{2}{11} x^{11/2} \left (2 a c+b^2\right )+\frac{4}{7} a b x^{7/2}+\frac{4}{15} b c x^{15/2}+\frac{2}{19} c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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

Mathematica [A]  time = 3.32368, size = 50, normalized size = 0.78 \[ \frac{2 x^{3/2} \left (7315 a^2+1995 x^4 \left (2 a c+b^2\right )+6270 a b x^2+2926 b c x^6+1155 c^2 x^8\right )}{21945} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(7315*a^2 + 6270*a*b*x^2 + 1995*(b^2 + 2*a*c)*x^4 + 2926*b*c*x^6 + 1155*c^2*x^8))/21945

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Maple [A]  time = 0.045, size = 49, normalized size = 0.8 \begin{align*}{\frac{2310\,{c}^{2}{x}^{8}+5852\,bc{x}^{6}+7980\,{x}^{4}ac+3990\,{b}^{2}{x}^{4}+12540\,ab{x}^{2}+14630\,{a}^{2}}{21945}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21945*x^(3/2)*(1155*c^2*x^8+2926*b*c*x^6+3990*a*c*x^4+1995*b^2*x^4+6270*a*b*x^2+7315*a^2)

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Maxima [A]  time = 0.956327, size = 59, normalized size = 0.92 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b c x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{11}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.24976, size = 135, normalized size = 2.11 \begin{align*} \frac{2}{21945} \,{\left (1155 \, c^{2} x^{9} + 2926 \, b c x^{7} + 1995 \,{\left (b^{2} + 2 \, a c\right )} x^{5} + 6270 \, a b x^{3} + 7315 \, a^{2} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21945*(1155*c^2*x^9 + 2926*b*c*x^7 + 1995*(b^2 + 2*a*c)*x^5 + 6270*a*b*x^3 + 7315*a^2*x)*sqrt(x)

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Sympy [A]  time = 3.66458, size = 63, normalized size = 0.98 \begin{align*} \frac{2 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 a b x^{\frac{7}{2}}}{7} + \frac{4 b c x^{\frac{15}{2}}}{15} + \frac{2 c^{2} x^{\frac{19}{2}}}{19} + \frac{2 x^{\frac{11}{2}} \left (2 a c + b^{2}\right )}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*b*x**(7/2)/7 + 4*b*c*x**(15/2)/15 + 2*c**2*x**(19/2)/19 + 2*x**(11/2)*(2*a*c + b**2)/1
1

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Giac [A]  time = 1.14697, size = 62, normalized size = 0.97 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b c x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, a c x^{\frac{11}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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