3.167 \(\int x^{7/2} (A+B x^2) (b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{17} A b^2 x^{17/2}+\frac{2}{25} c x^{25/2} (A c+2 b B)+\frac{2}{21} b x^{21/2} (2 A c+b B)+\frac{2}{29} B c^2 x^{29/2} \]

[Out]

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

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Rubi [A]  time = 0.0412536, 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} \[ \frac{2}{17} A b^2 x^{17/2}+\frac{2}{25} c x^{25/2} (A c+2 b B)+\frac{2}{21} b x^{21/2} (2 A c+b B)+\frac{2}{29} B c^2 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]

[Out]

(2*A*b^2*x^(17/2))/17 + (2*b*(b*B + 2*A*c)*x^(21/2))/21 + (2*c*(2*b*B + A*c)*x^(25/2))/25 + (2*B*c^2*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 x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{15/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{15/2}+b (b B+2 A c) x^{19/2}+c (2 b B+A c) x^{23/2}+B c^2 x^{27/2}\right ) \, dx\\ &=\frac{2}{17} A b^2 x^{17/2}+\frac{2}{21} b (b B+2 A c) x^{21/2}+\frac{2}{25} c (2 b B+A c) x^{25/2}+\frac{2}{29} B c^2 x^{29/2}\\ \end{align*}

Mathematica [A]  time = 0.0320792, size = 63, normalized size = 1. \[ \frac{2}{17} A b^2 x^{17/2}+\frac{2}{25} c x^{25/2} (A c+2 b B)+\frac{2}{21} b x^{21/2} (2 A c+b B)+\frac{2}{29} B c^2 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{17850\,B{c}^{2}{x}^{6}+20706\,A{c}^{2}{x}^{4}+41412\,B{x}^{4}bc+49300\,Abc{x}^{2}+24650\,B{x}^{2}{b}^{2}+30450\,A{b}^{2}}{258825}{x}^{{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^2,x)

[Out]

2/258825*x^(17/2)*(8925*B*c^2*x^6+10353*A*c^2*x^4+20706*B*b*c*x^4+24650*A*b*c*x^2+12325*B*b^2*x^2+15225*A*b^2)

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Maxima [A]  time = 1.09971, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{29} \, B c^{2} x^{\frac{29}{2}} + \frac{2}{25} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{25}{2}} + \frac{2}{17} \, A b^{2} x^{\frac{17}{2}} + \frac{2}{21} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{21}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57489, size = 157, normalized size = 2.49 \begin{align*} \frac{2}{258825} \,{\left (8925 \, B c^{2} x^{14} + 10353 \,{\left (2 \, B b c + A c^{2}\right )} x^{12} + 15225 \, A b^{2} x^{8} + 12325 \,{\left (B b^{2} + 2 \, A b c\right )} x^{10}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

2/258825*(8925*B*c^2*x^14 + 10353*(2*B*b*c + A*c^2)*x^12 + 15225*A*b^2*x^8 + 12325*(B*b^2 + 2*A*b*c)*x^10)*sqr
t(x)

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Sympy [A]  time = 96.2608, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{17}{2}}}{17} + \frac{4 A b c x^{\frac{21}{2}}}{21} + \frac{2 A c^{2} x^{\frac{25}{2}}}{25} + \frac{2 B b^{2} x^{\frac{21}{2}}}{21} + \frac{4 B b c x^{\frac{25}{2}}}{25} + \frac{2 B c^{2} x^{\frac{29}{2}}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x**2+A)*(c*x**4+b*x**2)**2,x)

[Out]

2*A*b**2*x**(17/2)/17 + 4*A*b*c*x**(21/2)/21 + 2*A*c**2*x**(25/2)/25 + 2*B*b**2*x**(21/2)/21 + 4*B*b*c*x**(25/
2)/25 + 2*B*c**2*x**(29/2)/29

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Giac [A]  time = 1.17831, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{29} \, B c^{2} x^{\frac{29}{2}} + \frac{4}{25} \, B b c x^{\frac{25}{2}} + \frac{2}{25} \, A c^{2} x^{\frac{25}{2}} + \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{4}{21} \, A b c x^{\frac{21}{2}} + \frac{2}{17} \, A b^{2} x^{\frac{17}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

2/29*B*c^2*x^(29/2) + 4/25*B*b*c*x^(25/2) + 2/25*A*c^2*x^(25/2) + 2/21*B*b^2*x^(21/2) + 4/21*A*b*c*x^(21/2) +
2/17*A*b^2*x^(17/2)