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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2} \]

[Out]

(-2*a^2*A)/(5*x^(5/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(3/2))/3 + (2*b^2*B*x^(7/2))/7

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Rubi [A]  time = 0.0287255, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*A)/(5*x^(5/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(3/2))/3 + (2*b^2*B*x^(7/2))/7

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

Mathematica [A]  time = 0.0155888, size = 57, normalized size = 0.93 \[ \frac{-42 a^2 \left (A+5 B x^2\right )+140 a b x^2 \left (B x^2-3 A\right )+10 b^2 x^4 \left (7 A+3 B x^2\right )}{105 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(140*a*b*x^2*(-3*A + B*x^2) + 10*b^2*x^4*(7*A + 3*B*x^2) - 42*a^2*(A + 5*B*x^2))/(105*x^(5/2))

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-30\,B{b}^{2}{x}^{6}-70\,A{b}^{2}{x}^{4}-140\,B{x}^{4}ab+420\,aAb{x}^{2}+210\,B{x}^{2}{a}^{2}+42\,{a}^{2}A}{105}{x}^{-{\frac{5}{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^(7/2),x)

[Out]

-2/105*(-15*B*b^2*x^6-35*A*b^2*x^4-70*B*a*b*x^4+210*A*a*b*x^2+105*B*a^2*x^2+21*A*a^2)/x^(5/2)

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Maxima [A]  time = 1.01632, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (A a^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="maxima")

[Out]

2/7*B*b^2*x^(7/2) + 2/3*(2*B*a*b + A*b^2)*x^(3/2) - 2/5*(A*a^2 + 5*(B*a^2 + 2*A*a*b)*x^2)/x^(5/2)

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Fricas [A]  time = 0.796934, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 21 \, A a^{2} - 105 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{105 \, x^{\frac{5}{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^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^2*x^6 + 35*(2*B*a*b + A*b^2)*x^4 - 21*A*a^2 - 105*(B*a^2 + 2*A*a*b)*x^2)/x^(5/2)

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Sympy [A]  time = 4.14279, size = 76, normalized size = 1.25 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{\sqrt{x}} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{\sqrt{x}} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/(5*x**(5/2)) - 4*A*a*b/sqrt(x) + 2*A*b**2*x**(3/2)/3 - 2*B*a**2/sqrt(x) + 4*B*a*b*x**(3/2)/3 + 2*B*b
**2*x**(7/2)/7

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Giac [A]  time = 1.12807, size = 74, normalized size = 1.21 \begin{align*} \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + A a^{2}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="giac")

[Out]

2/7*B*b^2*x^(7/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) - 2/5*(5*B*a^2*x^2 + 10*A*a*b*x^2 + A*a^2)/x^(5/2)