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

Optimal. Leaf size=63 \[ \frac{2}{11} A b^2 x^{11/2}+\frac{2}{19} c x^{19/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{23} B c^2 x^{23/2} \]

[Out]

(2*A*b^2*x^(11/2))/11 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(19/2))/19 + (2*B*c^2*x^(23/2))
/23

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Rubi [A]  time = 0.0381059, 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} \[ \frac{2}{11} A b^2 x^{11/2}+\frac{2}{19} c x^{19/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{23} B c^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(11/2))/11 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(19/2))/19 + (2*B*c^2*x^(23/2))
/23

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 \sqrt{x} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{9/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{9/2}+b (b B+2 A c) x^{13/2}+c (2 b B+A c) x^{17/2}+B c^2 x^{21/2}\right ) \, dx\\ &=\frac{2}{11} A b^2 x^{11/2}+\frac{2}{15} b (b B+2 A c) x^{15/2}+\frac{2}{19} c (2 b B+A c) x^{19/2}+\frac{2}{23} B c^2 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.0288025, size = 53, normalized size = 0.84 \[ \frac{2 x^{11/2} \left (6555 A b^2+3795 c x^4 (A c+2 b B)+4807 b x^2 (2 A c+b B)+3135 B c^2 x^6\right )}{72105} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(11/2)*(6555*A*b^2 + 4807*b*(b*B + 2*A*c)*x^2 + 3795*c*(2*b*B + A*c)*x^4 + 3135*B*c^2*x^6))/72105

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{6270\,B{c}^{2}{x}^{6}+7590\,A{c}^{2}{x}^{4}+15180\,B{x}^{4}bc+19228\,Abc{x}^{2}+9614\,B{x}^{2}{b}^{2}+13110\,A{b}^{2}}{72105}{x}^{{\frac{11}{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]

2/72105*x^(11/2)*(3135*B*c^2*x^6+3795*A*c^2*x^4+7590*B*b*c*x^4+9614*A*b*c*x^2+4807*B*b^2*x^2+6555*A*b^2)

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Maxima [A]  time = 1.12733, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{23} \, B c^{2} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{19}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{15} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{15}{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]

2/23*B*c^2*x^(23/2) + 2/19*(2*B*b*c + A*c^2)*x^(19/2) + 2/11*A*b^2*x^(11/2) + 2/15*(B*b^2 + 2*A*b*c)*x^(15/2)

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Fricas [A]  time = 1.5609, size = 149, normalized size = 2.37 \begin{align*} \frac{2}{72105} \,{\left (3135 \, B c^{2} x^{11} + 3795 \,{\left (2 \, B b c + A c^{2}\right )} x^{9} + 6555 \, A b^{2} x^{5} + 4807 \,{\left (B b^{2} + 2 \, A b c\right )} x^{7}\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]

2/72105*(3135*B*c^2*x^11 + 3795*(2*B*b*c + A*c^2)*x^9 + 6555*A*b^2*x^5 + 4807*(B*b^2 + 2*A*b*c)*x^7)*sqrt(x)

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Sympy [A]  time = 6.59938, size = 66, normalized size = 1.05 \begin{align*} \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{23}{2}}}{23} + \frac{2 x^{\frac{19}{2}} \left (A c^{2} + 2 B b c\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (2 A b c + B b^{2}\right )}{15} \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**(11/2)/11 + 2*B*c**2*x**(23/2)/23 + 2*x**(19/2)*(A*c**2 + 2*B*b*c)/19 + 2*x**(15/2)*(2*A*b*c + B*b
**2)/15

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Giac [A]  time = 1.17093, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{23} \, B c^{2} x^{\frac{23}{2}} + \frac{4}{19} \, B b c x^{\frac{19}{2}} + \frac{2}{19} \, A c^{2} x^{\frac{19}{2}} + \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A b c x^{\frac{15}{2}} + \frac{2}{11} \, A b^{2} 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^(1/2),x, algorithm="giac")

[Out]

2/23*B*c^2*x^(23/2) + 4/19*B*b*c*x^(19/2) + 2/19*A*c^2*x^(19/2) + 2/15*B*b^2*x^(15/2) + 4/15*A*b*c*x^(15/2) +
2/11*A*b^2*x^(11/2)