3.1060 \(\int \frac{(a+b x^2+c x^4)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{13} c x^{13/2} \left (a c+b^2\right )+\frac{2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{6}{5} a x^{5/2} \left (a c+b^2\right )+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2 + 6*a*c)*x^(9/2))/9 + (6*c*(b
^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

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Rubi [A]  time = 0.0439858, antiderivative size = 101, 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} \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{13} c x^{13/2} \left (a c+b^2\right )+\frac{2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{6}{5} a x^{5/2} \left (a c+b^2\right )+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2 + 6*a*c)*x^(9/2))/9 + (6*c*(b
^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

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

Mathematica [A]  time = 0.0718501, size = 103, normalized size = 1.02 \[ 2 \left (3 a^2 b \sqrt{x}-\frac{a^3}{3 x^{3/2}}+\frac{3}{13} c x^{13/2} \left (a c+b^2\right )+\frac{1}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac{3}{5} a x^{5/2} \left (a c+b^2\right )+\frac{3}{17} b c^2 x^{17/2}+\frac{1}{21} c^3 x^{21/2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-a^3/(3*x^(3/2)) + 3*a^2*b*Sqrt[x] + (3*a*(b^2 + a*c)*x^(5/2))/5 + (b*(b^2 + 6*a*c)*x^(9/2))/9 + (3*c*(b^2
+ a*c)*x^(13/2))/13 + (3*b*c^2*x^(17/2))/17 + (c^3*x^(21/2))/21)

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Maple [A]  time = 0.045, size = 90, normalized size = 0.9 \begin{align*} -{\frac{-6630\,{c}^{3}{x}^{12}-24570\,b{c}^{2}{x}^{10}-32130\,{x}^{8}a{c}^{2}-32130\,{x}^{8}{b}^{2}c-92820\,{x}^{6}abc-15470\,{x}^{6}{b}^{3}-83538\,{a}^{2}c{x}^{4}-83538\,{x}^{4}{b}^{2}a-417690\,{a}^{2}b{x}^{2}+46410\,{a}^{3}}{69615}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/69615*(-3315*c^3*x^12-12285*b*c^2*x^10-16065*a*c^2*x^8-16065*b^2*c*x^8-46410*a*b*c*x^6-7735*b^3*x^6-41769*a
^2*c*x^4-41769*a*b^2*x^4-208845*a^2*b*x^2+23205*a^3)/x^(3/2)

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Maxima [A]  time = 0.967803, size = 109, normalized size = 1.08 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{9}{2}} + 6 \, a^{2} b \sqrt{x} + \frac{6}{5} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{5}{2}} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*(b^2*c + a*c^2)*x^(13/2) + 2/9*(b^3 + 6*a*b*c)*x^(9/2) + 6*a^2*
b*sqrt(x) + 6/5*(a*b^2 + a^2*c)*x^(5/2) - 2/3*a^3/x^(3/2)

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Fricas [A]  time = 1.25003, size = 224, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (3315 \, c^{3} x^{12} + 12285 \, b c^{2} x^{10} + 16065 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 7735 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 208845 \, a^{2} b x^{2} + 41769 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 23205 \, a^{3}\right )}}{69615 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/69615*(3315*c^3*x^12 + 12285*b*c^2*x^10 + 16065*(b^2*c + a*c^2)*x^8 + 7735*(b^3 + 6*a*b*c)*x^6 + 208845*a^2*
b*x^2 + 41769*(a*b^2 + a^2*c)*x^4 - 23205*a^3)/x^(3/2)

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Sympy [A]  time = 23.6497, size = 128, normalized size = 1.27 \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} + 6 a^{2} b \sqrt{x} + \frac{6 a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b c x^{\frac{9}{2}}}{3} + \frac{6 a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a**2*c*x**(5/2)/5 + 6*a*b**2*x**(5/2)/5 + 4*a*b*c*x**(9/2)/3 + 6*a
*c**2*x**(13/2)/13 + 2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21

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Giac [A]  time = 1.12786, size = 117, normalized size = 1.16 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, a c^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{4}{3} \, a b c x^{\frac{9}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + \frac{6}{5} \, a^{2} c x^{\frac{5}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 6/13*a*c^2*x^(13/2) + 2/9*b^3*x^(9/2) + 4/3*a*
b*c*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6/5*a^2*c*x^(5/2) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)