3.173 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{17} B c^2 x^{17/2} \]

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(17/2))/17

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Rubi [A]  time = 0.038272, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1584, 448} \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{17} B c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(5/2),x]

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(17/2))/17

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.0298686, size = 53, normalized size = 0.84 \[ \frac{2 x^{5/2} \left (1989 A b^2+765 c x^4 (A c+2 b B)+1105 b x^2 (2 A c+b B)+585 B c^2 x^6\right )}{9945} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(1989*A*b^2 + 1105*b*(b*B + 2*A*c)*x^2 + 765*c*(2*b*B + A*c)*x^4 + 585*B*c^2*x^6))/9945

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{1170\,B{c}^{2}{x}^{6}+1530\,A{c}^{2}{x}^{4}+3060\,B{x}^{4}bc+4420\,Abc{x}^{2}+2210\,B{x}^{2}{b}^{2}+3978\,A{b}^{2}}{9945}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*x^(5/2)*(585*B*c^2*x^6+765*A*c^2*x^4+1530*B*b*c*x^4+2210*A*b*c*x^2+1105*B*b^2*x^2+1989*A*b^2)

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Maxima [A]  time = 1.1389, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 2/13*(2*B*b*c + A*c^2)*x^(13/2) + 2/5*A*b^2*x^(5/2) + 2/9*(B*b^2 + 2*A*b*c)*x^(9/2)

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Fricas [A]  time = 1.59709, size = 143, normalized size = 2.27 \begin{align*} \frac{2}{9945} \,{\left (585 \, B c^{2} x^{8} + 765 \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + 1989 \, A b^{2} x^{2} + 1105 \,{\left (B b^{2} + 2 \, A b c\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*(585*B*c^2*x^8 + 765*(2*B*b*c + A*c^2)*x^6 + 1989*A*b^2*x^2 + 1105*(B*b^2 + 2*A*b*c)*x^4)*sqrt(x)

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Sympy [A]  time = 16.4983, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{4 A b c x^{\frac{9}{2}}}{9} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} + \frac{4 B b c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(5/2)/5 + 4*A*b*c*x**(9/2)/9 + 2*A*c**2*x**(13/2)/13 + 2*B*b**2*x**(9/2)/9 + 4*B*b*c*x**(13/2)/13
+ 2*B*c**2*x**(17/2)/17

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Giac [A]  time = 1.13223, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, B b c x^{\frac{13}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A b c x^{\frac{9}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 4/13*B*b*c*x^(13/2) + 2/13*A*c^2*x^(13/2) + 2/9*B*b^2*x^(9/2) + 4/9*A*b*c*x^(9/2) + 2/5*
A*b^2*x^(5/2)