3.355 \(\int \frac{(a+b x^2)^2 (A+B x^2)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(13/2))/13

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Rubi [A]  time = 0.0322777, 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} \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9 + (2*b^2*B*x^(13/2))/13

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

Mathematica [A]  time = 0.0279692, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 A+65 b x^4 (2 a B+A b)+117 a x^2 (a B+2 A b)+45 b^2 B x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(585*a^2*A + 117*a*(2*A*b + a*B)*x^2 + 65*b*(A*b + 2*a*B)*x^4 + 45*b^2*B*x^6))/585

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{90\,B{b}^{2}{x}^{6}+130\,A{b}^{2}{x}^{4}+260\,B{x}^{4}ab+468\,aAb{x}^{2}+234\,B{x}^{2}{a}^{2}+1170\,{a}^{2}A}{585}\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^(1/2),x)

[Out]

2/585*x^(1/2)*(45*B*b^2*x^6+65*A*b^2*x^4+130*B*a*b*x^4+234*A*a*b*x^2+117*B*a^2*x^2+585*A*a^2)

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Maxima [A]  time = 1.08106, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{9}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} 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^(1/2),x, algorithm="maxima")

[Out]

2/13*B*b^2*x^(13/2) + 2/9*(2*B*a*b + A*b^2)*x^(9/2) + 2*A*a^2*sqrt(x) + 2/5*(B*a^2 + 2*A*a*b)*x^(5/2)

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Fricas [A]  time = 0.903859, size = 131, normalized size = 2.15 \begin{align*} \frac{2}{585} \,{\left (45 \, B b^{2} x^{6} + 65 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 585 \, A a^{2} + 117 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\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^(1/2),x, algorithm="fricas")

[Out]

2/585*(45*B*b^2*x^6 + 65*(2*B*a*b + A*b^2)*x^4 + 585*A*a^2 + 117*(B*a^2 + 2*A*a*b)*x^2)*sqrt(x)

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Sympy [A]  time = 2.12217, size = 78, normalized size = 1.28 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{5}{2}}}{5} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(5/2)/5 + 2*A*b**2*x**(9/2)/9 + 2*B*a**2*x**(5/2)/5 + 4*B*a*b*x**(9/2)/9 + 2*B*b
**2*x**(13/2)/13

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Giac [A]  time = 1.16364, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/13*B*b^2*x^(13/2) + 4/9*B*a*b*x^(9/2) + 2/9*A*b^2*x^(9/2) + 2/5*B*a^2*x^(5/2) + 4/5*A*a*b*x^(5/2) + 2*A*a^2*
sqrt(x)