3.1059 \(\int \frac{(a+b x^2+c x^4)^3}{x^{3/2}} \, dx\)

Optimal. Leaf size=99 \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} c x^{15/2} \left (a c+b^2\right )+\frac{2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{6}{7} a x^{7/2} \left (a c+b^2\right )+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2} \]

[Out]

(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*(b^2 + a*c)*x^(7/2))/7 + (2*b*(b^2 + 6*a*c)*x^(11/2))/11 + (2*c*(b^2
+ a*c)*x^(15/2))/5 + (6*b*c^2*x^(19/2))/19 + (2*c^3*x^(23/2))/23

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Rubi [A]  time = 0.0425691, antiderivative size = 99, 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} \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} c x^{15/2} \left (a c+b^2\right )+\frac{2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{6}{7} a x^{7/2} \left (a c+b^2\right )+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*(b^2 + a*c)*x^(7/2))/7 + (2*b*(b^2 + 6*a*c)*x^(11/2))/11 + (2*c*(b^2
+ a*c)*x^(15/2))/5 + (6*b*c^2*x^(19/2))/19 + (2*c^3*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 \frac{\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx &=\int \left (\frac{a^3}{x^{3/2}}+3 a^2 b \sqrt{x}+3 a \left (b^2+a c\right ) x^{5/2}+b \left (b^2+6 a c\right ) x^{9/2}+3 c \left (b^2+a c\right ) x^{13/2}+3 b c^2 x^{17/2}+c^3 x^{21/2}\right ) \, dx\\ &=-\frac{2 a^3}{\sqrt{x}}+2 a^2 b x^{3/2}+\frac{6}{7} a \left (b^2+a c\right ) x^{7/2}+\frac{2}{11} b \left (b^2+6 a c\right ) x^{11/2}+\frac{2}{5} c \left (b^2+a c\right ) x^{15/2}+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.0810406, size = 100, normalized size = 1.01 \[ 2 \left (a^2 b x^{3/2}-\frac{a^3}{\sqrt{x}}+\frac{1}{5} c x^{15/2} \left (a c+b^2\right )+\frac{1}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{3}{7} a x^{7/2} \left (a c+b^2\right )+\frac{3}{19} b c^2 x^{19/2}+\frac{1}{23} c^3 x^{23/2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-(a^3/Sqrt[x]) + a^2*b*x^(3/2) + (3*a*(b^2 + a*c)*x^(7/2))/7 + (b*(b^2 + 6*a*c)*x^(11/2))/11 + (c*(b^2 + a*
c)*x^(15/2))/5 + (3*b*c^2*x^(19/2))/19 + (c^3*x^(23/2))/23)

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Maple [A]  time = 0.046, size = 90, normalized size = 0.9 \begin{align*} -{\frac{-14630\,{c}^{3}{x}^{12}-53130\,b{c}^{2}{x}^{10}-67298\,{x}^{8}a{c}^{2}-67298\,{x}^{8}{b}^{2}c-183540\,{x}^{6}abc-30590\,{x}^{6}{b}^{3}-144210\,{a}^{2}c{x}^{4}-144210\,{x}^{4}{b}^{2}a-336490\,{a}^{2}b{x}^{2}+336490\,{a}^{3}}{168245}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/168245*(-7315*c^3*x^12-26565*b*c^2*x^10-33649*a*c^2*x^8-33649*b^2*c*x^8-91770*a*b*c*x^6-15295*b^3*x^6-72105
*a^2*c*x^4-72105*a*b^2*x^4-168245*a^2*b*x^2+168245*a^3)/x^(1/2)

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Maxima [A]  time = 0.961496, size = 109, normalized size = 1.1 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b c^{2} x^{\frac{19}{2}} + \frac{2}{5} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{11}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + \frac{6}{7} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*(b^2*c + a*c^2)*x^(15/2) + 2/11*(b^3 + 6*a*b*c)*x^(11/2) + 2*a^2
*b*x^(3/2) + 6/7*(a*b^2 + a^2*c)*x^(7/2) - 2*a^3/sqrt(x)

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Fricas [A]  time = 1.19299, size = 228, normalized size = 2.3 \begin{align*} \frac{2 \,{\left (7315 \, c^{3} x^{12} + 26565 \, b c^{2} x^{10} + 33649 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 15295 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 168245 \, a^{2} b x^{2} + 72105 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 168245 \, a^{3}\right )}}{168245 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/168245*(7315*c^3*x^12 + 26565*b*c^2*x^10 + 33649*(b^2*c + a*c^2)*x^8 + 15295*(b^3 + 6*a*b*c)*x^6 + 168245*a^
2*b*x^2 + 72105*(a*b^2 + a^2*c)*x^4 - 168245*a^3)/sqrt(x)

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Sympy [A]  time = 20.8602, size = 126, normalized size = 1.27 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 a b^{2} x^{\frac{7}{2}}}{7} + \frac{12 a b c x^{\frac{11}{2}}}{11} + \frac{2 a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c x^{\frac{15}{2}}}{5} + \frac{6 b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 c^{3} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/sqrt(x) + 2*a**2*b*x**(3/2) + 6*a**2*c*x**(7/2)/7 + 6*a*b**2*x**(7/2)/7 + 12*a*b*c*x**(11/2)/11 + 2*a*
c**2*x**(15/2)/5 + 2*b**3*x**(11/2)/11 + 2*b**2*c*x**(15/2)/5 + 6*b*c**2*x**(19/2)/19 + 2*c**3*x**(23/2)/23

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Giac [A]  time = 1.13689, size = 117, normalized size = 1.18 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b c^{2} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c x^{\frac{15}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{12}{11} \, a b c x^{\frac{11}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*b^2*c*x^(15/2) + 2/5*a*c^2*x^(15/2) + 2/11*b^3*x^(11/2) + 12/11*
a*b*c*x^(11/2) + 6/7*a*b^2*x^(7/2) + 6/7*a^2*c*x^(7/2) + 2*a^2*b*x^(3/2) - 2*a^3/sqrt(x)