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

Optimal. Leaf size=74 \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]

[Out]

((b*c - a*d)^2*Sqrt[c + d*x^2])/d^3 - (2*b*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*d^3) + (b^2*(c + d*x^2)^(5/2))/(5
*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0535002, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ -\frac{2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*Sqrt[c + d*x^2])/d^3 - (2*b*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*d^3) + (b^2*(c + d*x^2)^(5/2))/(5
*d^3)

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{\sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^2}{d^2 \sqrt{c+d x}}-\frac{2 b (b c-a d) \sqrt{c+d x}}{d^2}+\frac{b^2 (c+d x)^{3/2}}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d)^2 \sqrt{c+d x^2}}{d^3}-\frac{2 b (b c-a d) \left (c+d x^2\right )^{3/2}}{3 d^3}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0406491, size = 66, normalized size = 0.89 \[ \frac{\sqrt{c+d x^2} \left (15 a^2 d^2+10 a b d \left (d x^2-2 c\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x^2]*(15*a^2*d^2 + 10*a*b*d*(-2*c + d*x^2) + b^2*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4)))/(15*d^3)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 69, normalized size = 0.9 \begin{align*}{\frac{3\,{b}^{2}{d}^{2}{x}^{4}+10\,ab{d}^{2}{x}^{2}-4\,{b}^{2}cd{x}^{2}+15\,{a}^{2}{d}^{2}-20\,cabd+8\,{b}^{2}{c}^{2}}{15\,{d}^{3}}\sqrt{d{x}^{2}+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(d*x^2+c)^(1/2)*(3*b^2*d^2*x^4+10*a*b*d^2*x^2-4*b^2*c*d*x^2+15*a^2*d^2-20*a*b*c*d+8*b^2*c^2)/d^3

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.35014, size = 151, normalized size = 2.04 \begin{align*} \frac{{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*b^2*d^2*x^4 + 8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 - 2*(2*b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2 + c)/d^3

________________________________________________________________________________________

Sympy [A]  time = 0.895434, size = 158, normalized size = 2.14 \begin{align*} \begin{cases} \frac{a^{2} \sqrt{c + d x^{2}}}{d} - \frac{4 a b c \sqrt{c + d x^{2}}}{3 d^{2}} + \frac{2 a b x^{2} \sqrt{c + d x^{2}}}{3 d} + \frac{8 b^{2} c^{2} \sqrt{c + d x^{2}}}{15 d^{3}} - \frac{4 b^{2} c x^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{b^{2} x^{4} \sqrt{c + d x^{2}}}{5 d} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{\sqrt{c}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sqrt(c + d*x**2)/d - 4*a*b*c*sqrt(c + d*x**2)/(3*d**2) + 2*a*b*x**2*sqrt(c + d*x**2)/(3*d) + 8
*b**2*c**2*sqrt(c + d*x**2)/(15*d**3) - 4*b**2*c*x**2*sqrt(c + d*x**2)/(15*d**2) + b**2*x**4*sqrt(c + d*x**2)/
(5*d), Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)/sqrt(c), True))

________________________________________________________________________________________

Giac [A]  time = 1.12304, size = 132, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} - 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c + 15 \, \sqrt{d x^{2} + c} b^{2} c^{2} + 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d - 30 \, \sqrt{d x^{2} + c} a b c d + 15 \, \sqrt{d x^{2} + c} a^{2} d^{2}}{15 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*(d*x^2 + c)^(5/2)*b^2 - 10*(d*x^2 + c)^(3/2)*b^2*c + 15*sqrt(d*x^2 + c)*b^2*c^2 + 10*(d*x^2 + c)^(3/2)
*a*b*d - 30*sqrt(d*x^2 + c)*a*b*c*d + 15*sqrt(d*x^2 + c)*a^2*d^2)/d^3