3.171 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{21} B c^2 x^{21/2} \]

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(13/2))/13 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(21/2))/2
1

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Rubi [A]  time = 0.0392227, 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} \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{21} B c^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(13/2))/13 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(21/2))/2
1

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}{\sqrt{x}} \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{7/2}+b (b B+2 A c) x^{11/2}+c (2 b B+A c) x^{15/2}+B c^2 x^{19/2}\right ) \, dx\\ &=\frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} b (b B+2 A c) x^{13/2}+\frac{2}{17} c (2 b B+A c) x^{17/2}+\frac{2}{21} B c^2 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0288147, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 A b^2+819 c x^4 (A c+2 b B)+1071 b x^2 (2 A c+b B)+663 B c^2 x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(1547*A*b^2 + 1071*b*(b*B + 2*A*c)*x^2 + 819*c*(2*b*B + A*c)*x^4 + 663*B*c^2*x^6))/13923

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*}{\frac{1326\,B{c}^{2}{x}^{6}+1638\,A{c}^{2}{x}^{4}+3276\,B{x}^{4}bc+4284\,Abc{x}^{2}+2142\,B{x}^{2}{b}^{2}+3094\,A{b}^{2}}{13923}{x}^{{\frac{9}{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^(1/2),x)

[Out]

2/13923*x^(9/2)*(663*B*c^2*x^6+819*A*c^2*x^4+1638*B*b*c*x^4+2142*A*b*c*x^2+1071*B*b^2*x^2+1547*A*b^2)

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Maxima [A]  time = 1.17118, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{21} \, B c^{2} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{13}{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^(1/2),x, algorithm="maxima")

[Out]

2/21*B*c^2*x^(21/2) + 2/17*(2*B*b*c + A*c^2)*x^(17/2) + 2/9*A*b^2*x^(9/2) + 2/13*(B*b^2 + 2*A*b*c)*x^(13/2)

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Fricas [A]  time = 1.54821, size = 146, normalized size = 2.32 \begin{align*} \frac{2}{13923} \,{\left (663 \, B c^{2} x^{10} + 819 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 1547 \, A b^{2} x^{4} + 1071 \,{\left (B b^{2} + 2 \, A b c\right )} x^{6}\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^(1/2),x, algorithm="fricas")

[Out]

2/13923*(663*B*c^2*x^10 + 819*(2*B*b*c + A*c^2)*x^8 + 1547*A*b^2*x^4 + 1071*(B*b^2 + 2*A*b*c)*x^6)*sqrt(x)

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Sympy [A]  time = 11.9179, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{21}{2}}}{21} \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**(1/2),x)

[Out]

2*A*b**2*x**(9/2)/9 + 4*A*b*c*x**(13/2)/13 + 2*A*c**2*x**(17/2)/17 + 2*B*b**2*x**(13/2)/13 + 4*B*b*c*x**(17/2)
/17 + 2*B*c**2*x**(21/2)/21

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Giac [A]  time = 1.16155, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{21} \, B c^{2} x^{\frac{21}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A b c x^{\frac{13}{2}} + \frac{2}{9} \, A b^{2} 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^(1/2),x, algorithm="giac")

[Out]

2/21*B*c^2*x^(21/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/13*B*b^2*x^(13/2) + 4/13*A*b*c*x^(13/2) +
2/9*A*b^2*x^(9/2)