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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

[Out]

(-2*a^2*A)/Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(7/2))/7 + (2*b^2*B*x^(11/2))/11

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Rubi [A]  time = 0.030532, 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} \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*A)/Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*b*(A*b + 2*a*B)*x^(7/2))/7 + (2*b^2*B*x^(11/2))/11

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^{3/2}} \, dx &=\int \left (\frac{a^2 A}{x^{3/2}}+a (2 A b+a B) \sqrt{x}+b (A b+2 a B) x^{5/2}+b^2 B x^{9/2}\right ) \, dx\\ &=-\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{3} a (2 A b+a B) x^{3/2}+\frac{2}{7} b (A b+2 a B) x^{7/2}+\frac{2}{11} b^2 B x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0203783, size = 60, normalized size = 0.98 \[ \frac{-154 a^2 \left (3 A-B x^2\right )+44 a b x^2 \left (7 A+3 B x^2\right )+6 b^2 x^4 \left (11 A+7 B x^2\right )}{231 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-154*a^2*(3*A - B*x^2) + 44*a*b*x^2*(7*A + 3*B*x^2) + 6*b^2*x^4*(11*A + 7*B*x^2))/(231*Sqrt[x])

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-42\,B{b}^{2}{x}^{6}-66\,A{b}^{2}{x}^{4}-132\,B{x}^{4}ab-308\,aAb{x}^{2}-154\,B{x}^{2}{a}^{2}+462\,{a}^{2}A}{231}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/231*(-21*B*b^2*x^6-33*A*b^2*x^4-66*B*a*b*x^4-154*A*a*b*x^2-77*B*a^2*x^2+231*A*a^2)/x^(1/2)

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Maxima [A]  time = 1.04592, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} 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^(3/2),x, algorithm="maxima")

[Out]

2/11*B*b^2*x^(11/2) + 2/7*(2*B*a*b + A*b^2)*x^(7/2) - 2*A*a^2/sqrt(x) + 2/3*(B*a^2 + 2*A*a*b)*x^(3/2)

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Fricas [A]  time = 0.886639, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (21 \, B b^{2} x^{6} + 33 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*B*b^2*x^6 + 33*(2*B*a*b + A*b^2)*x^4 - 231*A*a^2 + 77*(B*a^2 + 2*A*a*b)*x^2)/sqrt(x)

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Sympy [A]  time = 2.5104, size = 78, normalized size = 1.28 \begin{align*} - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(7/2)/7 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(7/2)/7 + 2*B*
b**2*x**(11/2)/11

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Giac [A]  time = 1.16097, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} - \frac{2 \, A a^{2}}{\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^(3/2),x, algorithm="giac")

[Out]

2/11*B*b^2*x^(11/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) - 2*A*a^2/
sqrt(x)