3.357 \(\int \frac{(a+b x^2)^2 (A+B x^2)}{x^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

[Out]

(-2*a^2*A)/(3*x^(3/2)) + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(5/2))/5 + (2*b^2*B*x^(9/2))/9

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Rubi [A]  time = 0.0288711, antiderivative size = 61, 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} \[ -\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(A + B*x^2))/x^(5/2),x]

[Out]

(-2*a^2*A)/(3*x^(3/2)) + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(5/2))/5 + (2*b^2*B*x^(9/2))/9

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 \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{a^2 A}{x^{5/2}}+\frac{a (2 A b+a B)}{\sqrt{x}}+b (A b+2 a B) x^{3/2}+b^2 B x^{7/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{3 x^{3/2}}+2 a (2 A b+a B) \sqrt{x}+\frac{2}{5} b (A b+2 a B) x^{5/2}+\frac{2}{9} b^2 B x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0178349, size = 57, normalized size = 0.93 \[ \frac{-30 a^2 \left (A-3 B x^2\right )+36 a b x^2 \left (5 A+B x^2\right )+2 b^2 x^4 \left (9 A+5 B x^2\right )}{45 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(5/2),x]

[Out]

(-30*a^2*(A - 3*B*x^2) + 36*a*b*x^2*(5*A + B*x^2) + 2*b^2*x^4*(9*A + 5*B*x^2))/(45*x^(3/2))

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-10\,B{b}^{2}{x}^{6}-18\,A{b}^{2}{x}^{4}-36\,B{x}^{4}ab-180\,aAb{x}^{2}-90\,B{x}^{2}{a}^{2}+30\,{a}^{2}A}{45}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^(5/2),x)

[Out]

-2/45*(-5*B*b^2*x^6-9*A*b^2*x^4-18*B*a*b*x^4-90*A*a*b*x^2-45*B*a^2*x^2+15*A*a^2)/x^(3/2)

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Maxima [A]  time = 1.07084, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{3 \, x^{\frac{3}{2}}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(5/2),x, algorithm="maxima")

[Out]

2/9*B*b^2*x^(9/2) + 2/5*(2*B*a*b + A*b^2)*x^(5/2) - 2/3*A*a^2/x^(3/2) + 2*(B*a^2 + 2*A*a*b)*sqrt(x)

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Fricas [A]  time = 0.724228, size = 124, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (5 \, B b^{2} x^{6} + 9 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 45 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{45 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(5/2),x, algorithm="fricas")

[Out]

2/45*(5*B*b^2*x^6 + 9*(2*B*a*b + A*b^2)*x^4 - 15*A*a^2 + 45*(B*a^2 + 2*A*a*b)*x^2)/x^(3/2)

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Sympy [A]  time = 3.01178, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**(5/2),x)

[Out]

-2*A*a**2/(3*x**(3/2)) + 4*A*a*b*sqrt(x) + 2*A*b**2*x**(5/2)/5 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(5/2)/5 + 2*B*b
**2*x**(9/2)/9

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Giac [A]  time = 1.13106, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(5/2),x, algorithm="giac")

[Out]

2/9*B*b^2*x^(9/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2/3*A*a^2/x^(3
/2)