3.172 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15/2))/15 + (2*B*c^2*x^(19/2))/1
9

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Rubi [A]  time = 0.0393163, 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} \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(3/2),x]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15/2))/15 + (2*B*c^2*x^(19/2))/1
9

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

Mathematica [A]  time = 0.0278801, size = 63, normalized size = 1. \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15/2))/15 + (2*B*c^2*x^(19/2))/1
9

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{2310\,B{c}^{2}{x}^{6}+2926\,A{c}^{2}{x}^{4}+5852\,B{x}^{4}bc+7980\,Abc{x}^{2}+3990\,B{x}^{2}{b}^{2}+6270\,A{b}^{2}}{21945}{x}^{{\frac{7}{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^(3/2),x)

[Out]

2/21945*x^(7/2)*(1155*B*c^2*x^6+1463*A*c^2*x^4+2926*B*b*c*x^4+3990*A*b*c*x^2+1995*B*b^2*x^2+3135*A*b^2)

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Maxima [A]  time = 1.17092, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{11}{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^(3/2),x, algorithm="maxima")

[Out]

2/19*B*c^2*x^(19/2) + 2/15*(2*B*b*c + A*c^2)*x^(15/2) + 2/7*A*b^2*x^(7/2) + 2/11*(B*b^2 + 2*A*b*c)*x^(11/2)

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Fricas [A]  time = 1.56349, size = 147, normalized size = 2.33 \begin{align*} \frac{2}{21945} \,{\left (1155 \, B c^{2} x^{9} + 1463 \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + 3135 \, A b^{2} x^{3} + 1995 \,{\left (B b^{2} + 2 \, A b c\right )} x^{5}\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^(3/2),x, algorithm="fricas")

[Out]

2/21945*(1155*B*c^2*x^9 + 1463*(2*B*b*c + A*c^2)*x^7 + 3135*A*b^2*x^3 + 1995*(B*b^2 + 2*A*b*c)*x^5)*sqrt(x)

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Sympy [A]  time = 13.4522, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \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**(3/2),x)

[Out]

2*A*b**2*x**(7/2)/7 + 4*A*b*c*x**(11/2)/11 + 2*A*c**2*x**(15/2)/15 + 2*B*b**2*x**(11/2)/11 + 4*B*b*c*x**(15/2)
/15 + 2*B*c**2*x**(19/2)/19

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Giac [A]  time = 1.12225, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{15} \, B b c x^{\frac{15}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{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^(3/2),x, algorithm="giac")

[Out]

2/19*B*c^2*x^(19/2) + 4/15*B*b*c*x^(15/2) + 2/15*A*c^2*x^(15/2) + 2/11*B*b^2*x^(11/2) + 4/11*A*b*c*x^(11/2) +
2/7*A*b^2*x^(7/2)