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

Optimal. Leaf size=37 \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0153744, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0139022, size = 33, normalized size = 0.89 \[ \frac{2}{45} \sqrt{x} \left (9 x^2 (a B+A b)+45 a A+5 b B x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(45*a*A + 9*(A*b + a*B)*x^2 + 5*b*B*x^4))/45

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 32, normalized size = 0.9 \begin{align*}{\frac{10\,bB{x}^{4}+18\,A{x}^{2}b+18\,B{x}^{2}a+90\,Aa}{45}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*x^(1/2)*(5*B*b*x^4+9*A*b*x^2+9*B*a*x^2+45*A*a)

________________________________________________________________________________________

Maxima [A]  time = 1.00321, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.828137, size = 74, normalized size = 2. \begin{align*} \frac{2}{45} \,{\left (5 \, B b x^{4} + 9 \,{\left (B a + A b\right )} x^{2} + 45 \, A a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(1/2),x, algorithm="fricas")

[Out]

2/45*(5*B*b*x^4 + 9*(B*a + A*b)*x^2 + 45*A*a)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 0.76436, size = 44, normalized size = 1.19 \begin{align*} 2 A a \sqrt{x} + \frac{2 A b x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12298, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^(1/2),x, algorithm="giac")

[Out]

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