3.351 \(\int x^{7/2} (a+b x^2)^2 (A+B x^2) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(17/2))/17 + (2*b^2*B*x^(21/2))/2
1

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Rubi [A]  time = 0.0299327, antiderivative size = 63, 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} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(17/2))/17 + (2*b^2*B*x^(21/2))/2
1

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

Mathematica [A]  time = 0.0310451, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 a^2 A+819 b x^4 (2 a B+A b)+1071 a x^2 (a B+2 A b)+663 b^2 B x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(1547*a^2*A + 1071*a*(2*A*b + a*B)*x^2 + 819*b*(A*b + 2*a*B)*x^4 + 663*b^2*B*x^6))/13923

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{1326\,B{b}^{2}{x}^{6}+1638\,A{b}^{2}{x}^{4}+3276\,B{x}^{4}ab+4284\,aAb{x}^{2}+2142\,B{x}^{2}{a}^{2}+3094\,{a}^{2}A}{13923}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13923*x^(9/2)*(663*B*b^2*x^6+819*A*b^2*x^4+1638*B*a*b*x^4+2142*A*a*b*x^2+1071*B*a^2*x^2+1547*A*a^2)

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Maxima [A]  time = 1.05195, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.797779, size = 146, normalized size = 2.32 \begin{align*} \frac{2}{13923} \,{\left (663 \, B b^{2} x^{10} + 819 \,{\left (2 \, B a b + A b^{2}\right )} x^{8} + 1547 \, A a^{2} x^{4} + 1071 \,{\left (B a^{2} + 2 \, A a b\right )} x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13923*(663*B*b^2*x^10 + 819*(2*B*a*b + A*b^2)*x^8 + 1547*A*a^2*x^4 + 1071*(B*a^2 + 2*A*a*b)*x^6)*sqrt(x)

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Sympy [A]  time = 20.9895, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{13}{2}}}{13} + \frac{2 A b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{13}{2}}}{13} + \frac{4 B a b x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(13/2)/13 + 2*A*b**2*x**(17/2)/17 + 2*B*a**2*x**(13/2)/13 + 4*B*a*b*x**(17/2)
/17 + 2*B*b**2*x**(21/2)/21

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Giac [A]  time = 1.12682, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{4}{17} \, B a b x^{\frac{17}{2}} + \frac{2}{17} \, A b^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B a^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A a b x^{\frac{13}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*B*b^2*x^(21/2) + 4/17*B*a*b*x^(17/2) + 2/17*A*b^2*x^(17/2) + 2/13*B*a^2*x^(13/2) + 4/13*A*a*b*x^(13/2) +
2/9*A*a^2*x^(9/2)