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

Optimal. Leaf size=62 \[ 2 a^2 \sqrt {x}+\frac {2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac {4}{5} a b x^{5/2}+\frac {4}{13} b c x^{13/2}+\frac {2}{17} c^2 x^{17/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 62, 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*} 2 a^2 \sqrt {x}+\frac {2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac {4}{5} a b x^{5/2}+\frac {4}{13} b c x^{13/2}+\frac {2}{17} c^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] + (4*a*b*x^(5/2))/5 + (2*(b^2 + 2*a*c)*x^(9/2))/9 + (4*b*c*x^(13/2))/13 + (2*c^2*x^(17/2))/17

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

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Mathematica [A]  time = 0.04, size = 63, normalized size = 1.02 \begin {gather*} 2 \left (a^2 \sqrt {x}+\frac {1}{9} x^{9/2} \left (2 a c+b^2\right )+\frac {2}{5} a b x^{5/2}+\frac {2}{13} b c x^{13/2}+\frac {1}{17} c^2 x^{17/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(a^2*Sqrt[x] + (2*a*b*x^(5/2))/5 + ((b^2 + 2*a*c)*x^(9/2))/9 + (2*b*c*x^(13/2))/13 + (c^2*x^(17/2))/17)

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IntegrateAlgebraic [A]  time = 0.03, size = 62, normalized size = 1.00 \begin {gather*} \frac {2 \left (9945 a^2 \sqrt {x}+3978 a b x^{5/2}+2210 a c x^{9/2}+1105 b^2 x^{9/2}+1530 b c x^{13/2}+585 c^2 x^{17/2}\right )}{9945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(9945*a^2*Sqrt[x] + 3978*a*b*x^(5/2) + 1105*b^2*x^(9/2) + 2210*a*c*x^(9/2) + 1530*b*c*x^(13/2) + 585*c^2*x^
(17/2)))/9945

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fricas [A]  time = 0.97, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{9945} \, {\left (585 \, c^{2} x^{8} + 1530 \, b c x^{6} + 1105 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 3978 \, a b x^{2} + 9945 \, a^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*(585*c^2*x^8 + 1530*b*c*x^6 + 1105*(b^2 + 2*a*c)*x^4 + 3978*a*b*x^2 + 9945*a^2)*sqrt(x)

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giac [A]  time = 0.15, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, b c x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, a c x^{\frac {9}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + 2 \, a^{2} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*c^2*x^(17/2) + 4/13*b*c*x^(13/2) + 2/9*b^2*x^(9/2) + 4/9*a*c*x^(9/2) + 4/5*a*b*x^(5/2) + 2*a^2*sqrt(x)

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maple [A]  time = 0.01, size = 49, normalized size = 0.79 \begin {gather*} \frac {2 \left (585 c^{2} x^{8}+1530 b c \,x^{6}+2210 a c \,x^{4}+1105 b^{2} x^{4}+3978 a b \,x^{2}+9945 a^{2}\right ) \sqrt {x}}{9945} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*x^(1/2)*(585*c^2*x^8+1530*b*c*x^6+2210*a*c*x^4+1105*b^2*x^4+3978*a*b*x^2+9945*a^2)

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maxima [A]  time = 1.14, size = 48, normalized size = 0.77 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, b c x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + 2 \, a^{2} \sqrt {x} + \frac {4}{45} \, {\left (5 \, c x^{\frac {9}{2}} + 9 \, b x^{\frac {5}{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*c^2*x^(17/2) + 4/13*b*c*x^(13/2) + 2/9*b^2*x^(9/2) + 2*a^2*sqrt(x) + 4/45*(5*c*x^(9/2) + 9*b*x^(5/2))*a

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mupad [B]  time = 0.03, size = 45, normalized size = 0.73 \begin {gather*} x^{9/2}\,\left (\frac {2\,b^2}{9}+\frac {4\,a\,c}{9}\right )+2\,a^2\,\sqrt {x}+\frac {2\,c^2\,x^{17/2}}{17}+\frac {4\,a\,b\,x^{5/2}}{5}+\frac {4\,b\,c\,x^{13/2}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^(1/2),x)

[Out]

x^(9/2)*((4*a*c)/9 + (2*b^2)/9) + 2*a^2*x^(1/2) + (2*c^2*x^(17/2))/17 + (4*a*b*x^(5/2))/5 + (4*b*c*x^(13/2))/1
3

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sympy [A]  time = 5.00, size = 68, normalized size = 1.10 \begin {gather*} 2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {4 a c x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {9}{2}}}{9} + \frac {4 b c x^{\frac {13}{2}}}{13} + \frac {2 c^{2} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 4*a*c*x**(9/2)/9 + 2*b**2*x**(9/2)/9 + 4*b*c*x**(13/2)/13 + 2*c**2*x**(17/
2)/17

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