3.394 \(\int \sqrt{x} (a+b x^2)^2 (c+d x^2) \, dx\)

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

[Out]

(2*a^2*c*x^(3/2))/3 + (2*a*(2*b*c + a*d)*x^(7/2))/7 + (2*b*(b*c + 2*a*d)*x^(11/2))/11 + (2*b^2*d*x^(15/2))/15

________________________________________________________________________________________

Rubi [A]  time = 0.0285362, 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} \[ \frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0295099, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 a^2 c+105 b x^4 (2 a d+b c)+165 a x^2 (a d+2 b c)+77 b^2 d x^6\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{154\,{b}^{2}d{x}^{6}+420\,{x}^{4}abd+210\,{b}^{2}c{x}^{4}+330\,{x}^{2}{a}^{2}d+660\,abc{x}^{2}+770\,{a}^{2}c}{1155}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)*x^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0643, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*b^2*d*x^(15/2) + 2/11*(b^2*c + 2*a*b*d)*x^(11/2) + 2/3*a^2*c*x^(3/2) + 2/7*(2*a*b*c + a^2*d)*x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 0.707551, size = 136, normalized size = 2.16 \begin{align*} \frac{2}{1155} \,{\left (77 \, b^{2} d x^{7} + 105 \,{\left (b^{2} c + 2 \, a b d\right )} x^{5} + 385 \, a^{2} c x + 165 \,{\left (2 \, a b c + a^{2} d\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*(d*x^2+c)*x^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.49226, size = 66, normalized size = 1.05 \begin{align*} \frac{2 a^{2} c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d x^{\frac{15}{2}}}{15} + \frac{2 x^{\frac{11}{2}} \left (2 a b d + b^{2} c\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (a^{2} d + 2 a b c\right )}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)*x**(1/2),x)

[Out]

2*a**2*c*x**(3/2)/3 + 2*b**2*d*x**(15/2)/15 + 2*x**(11/2)*(2*a*b*d + b**2*c)/11 + 2*x**(7/2)*(a**2*d + 2*a*b*c
)/7

________________________________________________________________________________________

Giac [A]  time = 1.17171, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c x^{\frac{11}{2}} + \frac{4}{11} \, a b d x^{\frac{11}{2}} + \frac{4}{7} \, a b c x^{\frac{7}{2}} + \frac{2}{7} \, a^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*b^2*d*x^(15/2) + 2/11*b^2*c*x^(11/2) + 4/11*a*b*d*x^(11/2) + 4/7*a*b*c*x^(7/2) + 2/7*a^2*d*x^(7/2) + 2/3*
a^2*c*x^(3/2)