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

Optimal. Leaf size=85 \[ \frac{2}{19} b^2 x^{19/2} (3 A c+b B)+\frac{2}{15} A b^3 x^{15/2}+\frac{2}{27} c^2 x^{27/2} (A c+3 b B)+\frac{6}{23} b c x^{23/2} (A c+b B)+\frac{2}{31} B c^3 x^{31/2} \]

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B +
A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31

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Rubi [A]  time = 0.0498296, 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} \[ \frac{2}{19} b^2 x^{19/2} (3 A c+b B)+\frac{2}{15} A b^3 x^{15/2}+\frac{2}{27} c^2 x^{27/2} (A c+3 b B)+\frac{6}{23} b c x^{23/2} (A c+b B)+\frac{2}{31} B c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B +
A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31

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 )^3 \, dx &=\int x^{13/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{13/2}+b^2 (b B+3 A c) x^{17/2}+3 b c (b B+A c) x^{21/2}+c^2 (3 b B+A c) x^{25/2}+B c^3 x^{29/2}\right ) \, dx\\ &=\frac{2}{15} A b^3 x^{15/2}+\frac{2}{19} b^2 (b B+3 A c) x^{19/2}+\frac{6}{23} b c (b B+A c) x^{23/2}+\frac{2}{27} c^2 (3 b B+A c) x^{27/2}+\frac{2}{31} B c^3 x^{31/2}\\ \end{align*}

Mathematica [A]  time = 0.0399064, size = 85, normalized size = 1. \[ \frac{2}{19} b^2 x^{19/2} (3 A c+b B)+\frac{2}{15} A b^3 x^{15/2}+\frac{2}{27} c^2 x^{27/2} (A c+3 b B)+\frac{6}{23} b c x^{23/2} (A c+b B)+\frac{2}{31} B c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B +
A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31

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Maple [A]  time = 0.005, size = 80, normalized size = 0.9 \begin{align*}{\frac{117990\,B{c}^{3}{x}^{8}+135470\,A{c}^{3}{x}^{6}+406410\,B{x}^{6}b{c}^{2}+477090\,Ab{c}^{2}{x}^{4}+477090\,B{x}^{4}{b}^{2}c+577530\,A{b}^{2}c{x}^{2}+192510\,B{x}^{2}{b}^{3}+243846\,A{b}^{3}}{1828845}{x}^{{\frac{15}{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]

2/1828845*x^(15/2)*(58995*B*c^3*x^8+67735*A*c^3*x^6+203205*B*b*c^2*x^6+238545*A*b*c^2*x^4+238545*B*b^2*c*x^4+2
88765*A*b^2*c*x^2+96255*B*b^3*x^2+121923*A*b^3)

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Maxima [A]  time = 1.12662, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{31} \, B c^{3} x^{\frac{31}{2}} + \frac{2}{27} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{27}{2}} + \frac{6}{23} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{23}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{2}{19} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{19}{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]

2/31*B*c^3*x^(31/2) + 2/27*(3*B*b*c^2 + A*c^3)*x^(27/2) + 6/23*(B*b^2*c + A*b*c^2)*x^(23/2) + 2/15*A*b^3*x^(15
/2) + 2/19*(B*b^3 + 3*A*b^2*c)*x^(19/2)

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Fricas [A]  time = 1.82087, size = 211, normalized size = 2.48 \begin{align*} \frac{2}{1828845} \,{\left (58995 \, B c^{3} x^{15} + 67735 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + 238545 \,{\left (B b^{2} c + A b c^{2}\right )} x^{11} + 121923 \, A b^{3} x^{7} + 96255 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{9}\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]

2/1828845*(58995*B*c^3*x^15 + 67735*(3*B*b*c^2 + A*c^3)*x^13 + 238545*(B*b^2*c + A*b*c^2)*x^11 + 121923*A*b^3*
x^7 + 96255*(B*b^3 + 3*A*b^2*c)*x^9)*sqrt(x)

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Sympy [A]  time = 18.2926, size = 95, normalized size = 1.12 \begin{align*} \frac{2 A b^{3} x^{\frac{15}{2}}}{15} + \frac{2 B c^{3} x^{\frac{31}{2}}}{31} + \frac{2 x^{\frac{27}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{27} + \frac{2 x^{\frac{23}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{23} + \frac{2 x^{\frac{19}{2}} \left (3 A b^{2} c + B b^{3}\right )}{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)**3*x**(1/2),x)

[Out]

2*A*b**3*x**(15/2)/15 + 2*B*c**3*x**(31/2)/31 + 2*x**(27/2)*(A*c**3 + 3*B*b*c**2)/27 + 2*x**(23/2)*(3*A*b*c**2
+ 3*B*b**2*c)/23 + 2*x**(19/2)*(3*A*b**2*c + B*b**3)/19

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Giac [A]  time = 1.11954, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{31} \, B c^{3} x^{\frac{31}{2}} + \frac{2}{9} \, B b c^{2} x^{\frac{27}{2}} + \frac{2}{27} \, A c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, B b^{2} c x^{\frac{23}{2}} + \frac{6}{23} \, A b c^{2} x^{\frac{23}{2}} + \frac{2}{19} \, B b^{3} x^{\frac{19}{2}} + \frac{6}{19} \, A b^{2} c x^{\frac{19}{2}} + \frac{2}{15} \, A b^{3} 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)^3*x^(1/2),x, algorithm="giac")

[Out]

2/31*B*c^3*x^(31/2) + 2/9*B*b*c^2*x^(27/2) + 2/27*A*c^3*x^(27/2) + 6/23*B*b^2*c*x^(23/2) + 6/23*A*b*c^2*x^(23/
2) + 2/19*B*b^3*x^(19/2) + 6/19*A*b^2*c*x^(19/2) + 2/15*A*b^3*x^(15/2)