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

Optimal. Leaf size=85 \[ \frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{13} A b^3 x^{13/2}+\frac{2}{25} c^2 x^{25/2} (A c+3 b B)+\frac{2}{7} b c x^{21/2} (A c+b B)+\frac{2}{29} B c^3 x^{29/2} \]

[Out]

(2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (2*b*c*(b*B + A*c)*x^(21/2))/7 + (2*c^2*(3*b*B + A
*c)*x^(25/2))/25 + (2*B*c^3*x^(29/2))/29

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Rubi [A]  time = 0.0515566, antiderivative size = 85, 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} \[ \frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{13} A b^3 x^{13/2}+\frac{2}{25} c^2 x^{25/2} (A c+3 b B)+\frac{2}{7} b c x^{21/2} (A c+b B)+\frac{2}{29} B c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (2*b*c*(b*B + A*c)*x^(21/2))/7 + (2*c^2*(3*b*B + A
*c)*x^(25/2))/25 + (2*B*c^3*x^(29/2))/29

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 )^3}{\sqrt{x}} \, dx &=\int x^{11/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{11/2}+b^2 (b B+3 A c) x^{15/2}+3 b c (b B+A c) x^{19/2}+c^2 (3 b B+A c) x^{23/2}+B c^3 x^{27/2}\right ) \, dx\\ &=\frac{2}{13} A b^3 x^{13/2}+\frac{2}{17} b^2 (b B+3 A c) x^{17/2}+\frac{2}{7} b c (b B+A c) x^{21/2}+\frac{2}{25} c^2 (3 b B+A c) x^{25/2}+\frac{2}{29} B c^3 x^{29/2}\\ \end{align*}

Mathematica [A]  time = 0.0381778, size = 85, normalized size = 1. \[ \frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{13} A b^3 x^{13/2}+\frac{2}{25} c^2 x^{25/2} (A c+3 b B)+\frac{2}{7} b c x^{21/2} (A c+b B)+\frac{2}{29} B c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (2*b*c*(b*B + A*c)*x^(21/2))/7 + (2*c^2*(3*b*B + A
*c)*x^(25/2))/25 + (2*B*c^3*x^(29/2))/29

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Maple [A]  time = 0.005, size = 80, normalized size = 0.9 \begin{align*}{\frac{77350\,B{c}^{3}{x}^{8}+89726\,A{c}^{3}{x}^{6}+269178\,B{x}^{6}b{c}^{2}+320450\,Ab{c}^{2}{x}^{4}+320450\,B{x}^{4}{b}^{2}c+395850\,A{b}^{2}c{x}^{2}+131950\,B{x}^{2}{b}^{3}+172550\,A{b}^{3}}{1121575}{x}^{{\frac{13}{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)^3/x^(1/2),x)

[Out]

2/1121575*x^(13/2)*(38675*B*c^3*x^8+44863*A*c^3*x^6+134589*B*b*c^2*x^6+160225*A*b*c^2*x^4+160225*B*b^2*c*x^4+1
97925*A*b^2*c*x^2+65975*B*b^3*x^2+86275*A*b^3)

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Maxima [A]  time = 1.16145, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{29} \, B c^{3} x^{\frac{29}{2}} + \frac{2}{25} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{25}{2}} + \frac{2}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{21}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{17}{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)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/29*B*c^3*x^(29/2) + 2/25*(3*B*b*c^2 + A*c^3)*x^(25/2) + 2/7*(B*b^2*c + A*b*c^2)*x^(21/2) + 2/13*A*b^3*x^(13/
2) + 2/17*(B*b^3 + 3*A*b^2*c)*x^(17/2)

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Fricas [A]  time = 1.80342, size = 209, normalized size = 2.46 \begin{align*} \frac{2}{1121575} \,{\left (38675 \, B c^{3} x^{14} + 44863 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 160225 \,{\left (B b^{2} c + A b c^{2}\right )} x^{10} + 86275 \, A b^{3} x^{6} + 65975 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8}\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)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/1121575*(38675*B*c^3*x^14 + 44863*(3*B*b*c^2 + A*c^3)*x^12 + 160225*(B*b^2*c + A*b*c^2)*x^10 + 86275*A*b^3*x
^6 + 65975*(B*b^3 + 3*A*b^2*c)*x^8)*sqrt(x)

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Sympy [A]  time = 56.0395, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{13}{2}}}{13} + \frac{6 A b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 A b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 A c^{3} x^{\frac{25}{2}}}{25} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} c x^{\frac{21}{2}}}{7} + \frac{6 B b c^{2} x^{\frac{25}{2}}}{25} + \frac{2 B c^{3} x^{\frac{29}{2}}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**(1/2),x)

[Out]

2*A*b**3*x**(13/2)/13 + 6*A*b**2*c*x**(17/2)/17 + 2*A*b*c**2*x**(21/2)/7 + 2*A*c**3*x**(25/2)/25 + 2*B*b**3*x*
*(17/2)/17 + 2*B*b**2*c*x**(21/2)/7 + 6*B*b*c**2*x**(25/2)/25 + 2*B*c**3*x**(29/2)/29

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Giac [A]  time = 1.13529, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{29} \, B c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, B b c^{2} x^{\frac{25}{2}} + \frac{2}{25} \, A c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, B b^{2} c x^{\frac{21}{2}} + \frac{2}{7} \, A b c^{2} x^{\frac{21}{2}} + \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{6}{17} \, A b^{2} c x^{\frac{17}{2}} + \frac{2}{13} \, A b^{3} 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)^3/x^(1/2),x, algorithm="giac")

[Out]

2/29*B*c^3*x^(29/2) + 6/25*B*b*c^2*x^(25/2) + 2/25*A*c^3*x^(25/2) + 2/7*B*b^2*c*x^(21/2) + 2/7*A*b*c^2*x^(21/2
) + 2/17*B*b^3*x^(17/2) + 6/17*A*b^2*c*x^(17/2) + 2/13*A*b^3*x^(13/2)