3.1061 \(\int \frac{(a+b x^2+c x^4)^3}{x^{7/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0416287, 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} \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a^3}{5 x^{5/2}}+\frac{6}{11} c x^{11/2} \left (a c+b^2\right )+\frac{2}{7} b x^{7/2} \left (6 a c+b^2\right )+2 a x^{3/2} \left (a c+b^2\right )+\frac{2}{5} b c^2 x^{15/2}+\frac{2}{19} c^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0733973, size = 100, normalized size = 1.01 \[ 2 \left (-\frac{3 a^2 b}{\sqrt{x}}-\frac{a^3}{5 x^{5/2}}+\frac{3}{11} c x^{11/2} \left (a c+b^2\right )+\frac{1}{7} b x^{7/2} \left (6 a c+b^2\right )+a x^{3/2} \left (a c+b^2\right )+\frac{1}{5} b c^2 x^{15/2}+\frac{1}{19} c^3 x^{19/2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x^(7/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 90, normalized size = 0.9 \begin{align*} -{\frac{-770\,{c}^{3}{x}^{12}-2926\,b{c}^{2}{x}^{10}-3990\,{x}^{8}a{c}^{2}-3990\,{x}^{8}{b}^{2}c-12540\,{x}^{6}abc-2090\,{x}^{6}{b}^{3}-14630\,{a}^{2}c{x}^{4}-14630\,a{b}^{2}{x}^{4}+43890\,{a}^{2}b{x}^{2}+2926\,{a}^{3}}{7315}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^3/x^(7/2),x)

[Out]

-2/7315*(-385*c^3*x^12-1463*b*c^2*x^10-1995*a*c^2*x^8-1995*b^2*c*x^8-6270*a*b*c*x^6-1045*b^3*x^6-7315*a^2*c*x^
4-7315*a*b^2*x^4+21945*a^2*b*x^2+1463*a^3)/x^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 0.968151, size = 111, normalized size = 1.12 \begin{align*} \frac{2}{19} \, c^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b c^{2} x^{\frac{15}{2}} + \frac{6}{11} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{7}{2}} + 2 \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.26604, size = 215, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (385 \, c^{3} x^{12} + 1463 \, b c^{2} x^{10} + 1995 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 1045 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} - 21945 \, a^{2} b x^{2} + 7315 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 1463 \, a^{3}\right )}}{7315 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/7315*(385*c^3*x^12 + 1463*b*c^2*x^10 + 1995*(b^2*c + a*c^2)*x^8 + 1045*(b^3 + 6*a*b*c)*x^6 - 21945*a^2*b*x^2
+ 7315*(a*b^2 + a^2*c)*x^4 - 1463*a^3)/x^(5/2)

________________________________________________________________________________________

Sympy [A]  time = 29.167, size = 124, normalized size = 1.25 \begin{align*} - \frac{2 a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 a^{2} b}{\sqrt{x}} + 2 a^{2} c x^{\frac{3}{2}} + 2 a b^{2} x^{\frac{3}{2}} + \frac{12 a b c x^{\frac{7}{2}}}{7} + \frac{6 a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} + \frac{6 b^{2} c x^{\frac{11}{2}}}{11} + \frac{2 b c^{2} x^{\frac{15}{2}}}{5} + \frac{2 c^{3} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**3/x**(7/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19079, size = 119, normalized size = 1.2 \begin{align*} \frac{2}{19} \, c^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b c^{2} x^{\frac{15}{2}} + \frac{6}{11} \, b^{2} c x^{\frac{11}{2}} + \frac{6}{11} \, a c^{2} x^{\frac{11}{2}} + \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{12}{7} \, a b c x^{\frac{7}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 2 \, a^{2} c x^{\frac{3}{2}} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="giac")

[Out]

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