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

Optimal. Leaf size=62 \[ -\frac {2 a^2}{3 x^{3/2}}+\frac {2}{5} x^{5/2} \left (2 a c+b^2\right )+4 a b \sqrt {x}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \]

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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*} -\frac {2 a^2}{3 x^{3/2}}+\frac {2}{5} x^{5/2} \left (2 a c+b^2\right )+4 a b \sqrt {x}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) + 4*a*b*Sqrt[x] + (2*(b^2 + 2*a*c)*x^(5/2))/5 + (4*b*c*x^(9/2))/9 + (2*c^2*x^(13/2))/13

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

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Mathematica [A]  time = 0.07, size = 53, normalized size = 0.85 \begin {gather*} \frac {-390 a^2+468 a \left (5 b x^2+c x^4\right )+234 b^2 x^4+260 b c x^6+90 c^2 x^8}{585 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-390*a^2 + 234*b^2*x^4 + 260*b*c*x^6 + 90*c^2*x^8 + 468*a*(5*b*x^2 + c*x^4))/(585*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 52, normalized size = 0.84 \begin {gather*} \frac {2 \left (-195 a^2+1170 a b x^2+234 a c x^4+117 b^2 x^4+130 b c x^6+45 c^2 x^8\right )}{585 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-195*a^2 + 1170*a*b*x^2 + 117*b^2*x^4 + 234*a*c*x^4 + 130*b*c*x^6 + 45*c^2*x^8))/(585*x^(3/2))

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fricas [A]  time = 0.96, size = 46, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (45 \, c^{2} x^{8} + 130 \, b c x^{6} + 117 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 1170 \, a b x^{2} - 195 \, a^{2}\right )}}{585 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(45*c^2*x^8 + 130*b*c*x^6 + 117*(b^2 + 2*a*c)*x^4 + 1170*a*b*x^2 - 195*a^2)/x^(3/2)

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giac [A]  time = 0.15, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} + \frac {4}{5} \, a c x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 49, normalized size = 0.79 \begin {gather*} -\frac {2 \left (-45 c^{2} x^{8}-130 b c \,x^{6}-234 a c \,x^{4}-117 b^{2} x^{4}-1170 a b \,x^{2}+195 a^{2}\right )}{585 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/585*(-45*c^2*x^8-130*b*c*x^6-234*a*c*x^4-117*b^2*x^4-1170*a*b*x^2+195*a^2)/x^(3/2)

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maxima [A]  time = 1.13, size = 44, normalized size = 0.71 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 45, normalized size = 0.73 \begin {gather*} x^{5/2}\,\left (\frac {2\,b^2}{5}+\frac {4\,a\,c}{5}\right )-\frac {2\,a^2}{3\,x^{3/2}}+\frac {2\,c^2\,x^{13/2}}{13}+4\,a\,b\,\sqrt {x}+\frac {4\,b\,c\,x^{9/2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^(5/2),x)

[Out]

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

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sympy [A]  time = 6.89, size = 68, normalized size = 1.10 \begin {gather*} - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {4 a c x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {5}{2}}}{5} + \frac {4 b c x^{\frac {9}{2}}}{9} + \frac {2 c^{2} x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 4*a*c*x**(5/2)/5 + 2*b**2*x**(5/2)/5 + 4*b*c*x**(9/2)/9 + 2*c**2*x**(13
/2)/13

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