3.361 \(\int x^{3/2} (a+b x^2)^3 (A+B x^2) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)
*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21

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Rubi [A]  time = 0.0403044, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)
*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21

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^{3/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{11/2}+b^2 (A b+3 a B) x^{15/2}+b^3 B x^{19/2}\right ) \, dx\\ &=\frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 (3 A b+a B) x^{9/2}+\frac{6}{13} a b (A b+a B) x^{13/2}+\frac{2}{17} b^2 (A b+3 a B) x^{17/2}+\frac{2}{21} b^3 B x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0360561, size = 85, normalized size = 1. \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)
*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21

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Maple [A]  time = 0.005, size = 80, normalized size = 0.9 \begin{align*}{\frac{6630\,B{b}^{3}{x}^{8}+8190\,{x}^{6}A{b}^{3}+24570\,{x}^{6}Ba{b}^{2}+32130\,{x}^{4}Aa{b}^{2}+32130\,{x}^{4}B{a}^{2}b+46410\,{x}^{2}A{a}^{2}b+15470\,{x}^{2}B{a}^{3}+27846\,A{a}^{3}}{69615}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)^3*(B*x^2+A),x)

[Out]

2/69615*x^(5/2)*(3315*B*b^3*x^8+4095*A*b^3*x^6+12285*B*a*b^2*x^6+16065*A*a*b^2*x^4+16065*B*a^2*b*x^4+23205*A*a
^2*b*x^2+7735*B*a^3*x^2+13923*A*a^3)

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Maxima [A]  time = 1.1162, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{17}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="maxima")

[Out]

2/21*B*b^3*x^(21/2) + 2/17*(3*B*a*b^2 + A*b^3)*x^(17/2) + 6/13*(B*a^2*b + A*a*b^2)*x^(13/2) + 2/5*A*a^3*x^(5/2
) + 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]  time = 0.82957, size = 198, normalized size = 2.33 \begin{align*} \frac{2}{69615} \,{\left (3315 \, B b^{3} x^{10} + 4095 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + 16065 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 13923 \, A a^{3} x^{2} + 7735 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="fricas")

[Out]

2/69615*(3315*B*b^3*x^10 + 4095*(3*B*a*b^2 + A*b^3)*x^8 + 16065*(B*a^2*b + A*a*b^2)*x^6 + 13923*A*a^3*x^2 + 77
35*(B*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [A]  time = 11.3913, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(5/2)/5 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b**3*x**(17/2)/17 + 2*B*a**3*x**(9
/2)/9 + 6*B*a**2*b*x**(13/2)/13 + 6*B*a*b**2*x**(17/2)/17 + 2*B*b**3*x**(21/2)/21

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Giac [A]  time = 1.13626, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, B a b^{2} x^{\frac{17}{2}} + \frac{2}{17} \, A b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="giac")

[Out]

2/21*B*b^3*x^(21/2) + 6/17*B*a*b^2*x^(17/2) + 2/17*A*b^3*x^(17/2) + 6/13*B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13
/2) + 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) + 2/5*A*a^3*x^(5/2)