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

Optimal. Leaf size=63 \[ \frac{2}{15} A b^2 x^{15/2}+\frac{2}{23} c x^{23/2} (A c+2 b B)+\frac{2}{19} b x^{19/2} (2 A c+b B)+\frac{2}{27} B c^2 x^{27/2} \]

[Out]

(2*A*b^2*x^(15/2))/15 + (2*b*(b*B + 2*A*c)*x^(19/2))/19 + (2*c*(2*b*B + A*c)*x^(23/2))/23 + (2*B*c^2*x^(27/2))
/27

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Rubi [A]  time = 0.0388137, 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} \[ \frac{2}{15} A b^2 x^{15/2}+\frac{2}{23} c x^{23/2} (A c+2 b B)+\frac{2}{19} b x^{19/2} (2 A c+b B)+\frac{2}{27} B c^2 x^{27/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(15/2))/15 + (2*b*(b*B + 2*A*c)*x^(19/2))/19 + (2*c*(2*b*B + A*c)*x^(23/2))/23 + (2*B*c^2*x^(27/2))
/27

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 x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{13/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{13/2}+b (b B+2 A c) x^{17/2}+c (2 b B+A c) x^{21/2}+B c^2 x^{25/2}\right ) \, dx\\ &=\frac{2}{15} A b^2 x^{15/2}+\frac{2}{19} b (b B+2 A c) x^{19/2}+\frac{2}{23} c (2 b B+A c) x^{23/2}+\frac{2}{27} B c^2 x^{27/2}\\ \end{align*}

Mathematica [A]  time = 0.0297814, size = 53, normalized size = 0.84 \[ \frac{2 x^{15/2} \left (3933 A b^2+2565 c x^4 (A c+2 b B)+3105 b x^2 (2 A c+b B)+2185 B c^2 x^6\right )}{58995} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(15/2)*(3933*A*b^2 + 3105*b*(b*B + 2*A*c)*x^2 + 2565*c*(2*b*B + A*c)*x^4 + 2185*B*c^2*x^6))/58995

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{4370\,B{c}^{2}{x}^{6}+5130\,A{c}^{2}{x}^{4}+10260\,B{x}^{4}bc+12420\,Abc{x}^{2}+6210\,B{x}^{2}{b}^{2}+7866\,A{b}^{2}}{58995}{x}^{{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/58995*x^(15/2)*(2185*B*c^2*x^6+2565*A*c^2*x^4+5130*B*b*c*x^4+6210*A*b*c*x^2+3105*B*b^2*x^2+3933*A*b^2)

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Maxima [A]  time = 1.15168, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{27} \, B c^{2} x^{\frac{27}{2}} + \frac{2}{23} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{23}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} + \frac{2}{19} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{19}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*B*c^2*x^(27/2) + 2/23*(2*B*b*c + A*c^2)*x^(23/2) + 2/15*A*b^2*x^(15/2) + 2/19*(B*b^2 + 2*A*b*c)*x^(19/2)

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Fricas [A]  time = 1.70015, size = 150, normalized size = 2.38 \begin{align*} \frac{2}{58995} \,{\left (2185 \, B c^{2} x^{13} + 2565 \,{\left (2 \, B b c + A c^{2}\right )} x^{11} + 3933 \, A b^{2} x^{7} + 3105 \,{\left (B b^{2} + 2 \, A b c\right )} x^{9}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/58995*(2185*B*c^2*x^13 + 2565*(2*B*b*c + A*c^2)*x^11 + 3933*A*b^2*x^7 + 3105*(B*b^2 + 2*A*b*c)*x^9)*sqrt(x)

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Sympy [A]  time = 53.7629, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{15}{2}}}{15} + \frac{4 A b c x^{\frac{19}{2}}}{19} + \frac{2 A c^{2} x^{\frac{23}{2}}}{23} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} + \frac{4 B b c x^{\frac{23}{2}}}{23} + \frac{2 B c^{2} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(15/2)/15 + 4*A*b*c*x**(19/2)/19 + 2*A*c**2*x**(23/2)/23 + 2*B*b**2*x**(19/2)/19 + 4*B*b*c*x**(23/
2)/23 + 2*B*c**2*x**(27/2)/27

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Giac [A]  time = 1.16581, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{27} \, B c^{2} x^{\frac{27}{2}} + \frac{4}{23} \, B b c x^{\frac{23}{2}} + \frac{2}{23} \, A c^{2} x^{\frac{23}{2}} + \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{4}{19} \, A b c x^{\frac{19}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*B*c^2*x^(27/2) + 4/23*B*b*c*x^(23/2) + 2/23*A*c^2*x^(23/2) + 2/19*B*b^2*x^(19/2) + 4/19*A*b*c*x^(19/2) +
2/15*A*b^2*x^(15/2)