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

Optimal. Leaf size=63 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(15/2))/15

________________________________________________________________________________________

Rubi [A]  time = 0.0293406, 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} \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(15/2))/15

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 \sqrt{x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A \sqrt{x}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{13/2}\right ) \, dx\\ &=\frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} a (2 A b+a B) x^{7/2}+\frac{2}{11} b (A b+2 a B) x^{11/2}+\frac{2}{15} b^2 B x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0287531, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 a^2 A+105 b x^4 (2 a B+A b)+165 a x^2 (a B+2 A b)+77 b^2 B x^6\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(385*a^2*A + 165*a*(2*A*b + a*B)*x^2 + 105*b*(A*b + 2*a*B)*x^4 + 77*b^2*B*x^6))/1155

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*}{\frac{154\,B{b}^{2}{x}^{6}+210\,A{b}^{2}{x}^{4}+420\,B{x}^{4}ab+660\,aAb{x}^{2}+330\,B{x}^{2}{a}^{2}+770\,{a}^{2}A}{1155}{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^(1/2),x)

[Out]

2/1155*x^(3/2)*(77*B*b^2*x^6+105*A*b^2*x^4+210*B*a*b*x^4+330*A*a*b*x^2+165*B*a^2*x^2+385*A*a^2)

________________________________________________________________________________________

Maxima [A]  time = 1.04412, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{7}{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]

2/15*B*b^2*x^(15/2) + 2/11*(2*B*a*b + A*b^2)*x^(11/2) + 2/3*A*a^2*x^(3/2) + 2/7*(B*a^2 + 2*A*a*b)*x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 0.912158, size = 136, normalized size = 2.16 \begin{align*} \frac{2}{1155} \,{\left (77 \, B b^{2} x^{7} + 105 \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + 385 \, A a^{2} x + 165 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3}\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]

2/1155*(77*B*b^2*x^7 + 105*(2*B*a*b + A*b^2)*x^5 + 385*A*a^2*x + 165*(B*a^2 + 2*A*a*b)*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 2.46814, size = 66, normalized size = 1.05 \begin{align*} \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{2 x^{\frac{11}{2}} \left (A b^{2} + 2 B a b\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (2 A a b + B a^{2}\right )}{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**(1/2),x)

[Out]

2*A*a**2*x**(3/2)/3 + 2*B*b**2*x**(15/2)/15 + 2*x**(11/2)*(A*b**2 + 2*B*a*b)/11 + 2*x**(7/2)*(2*A*a*b + B*a**2
)/7

________________________________________________________________________________________

Giac [A]  time = 1.15207, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{2}} + \frac{2}{3} \, A a^{2} 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^(1/2),x, algorithm="giac")

[Out]

2/15*B*b^2*x^(15/2) + 4/11*B*a*b*x^(11/2) + 2/11*A*b^2*x^(11/2) + 2/7*B*a^2*x^(7/2) + 4/7*A*a*b*x^(7/2) + 2/3*
A*a^2*x^(3/2)