3.661 \(\int \frac{x^3 (a+b x^2)^2}{(c+d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{b \sqrt{c+d x^2} (3 b c-2 a d)}{d^4}-\frac{(b c-a d) (3 b c-a d)}{d^4 \sqrt{c+d x^2}}+\frac{c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0895646, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {446, 77} \[ -\frac{b \sqrt{c+d x^2} (3 b c-2 a d)}{d^4}-\frac{(b c-a d) (3 b c-a d)}{d^4 \sqrt{c+d x^2}}+\frac{c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (a+b x)^2}{(c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2}{d^3 (c+d x)^{5/2}}+\frac{(b c-a d) (3 b c-a d)}{d^3 (c+d x)^{3/2}}-\frac{b (3 b c-2 a d)}{d^3 \sqrt{c+d x}}+\frac{b^2 \sqrt{c+d x}}{d^3}\right ) \, dx,x,x^2\right )\\ &=\frac{c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}-\frac{(b c-a d) (3 b c-a d)}{d^4 \sqrt{c+d x^2}}-\frac{b (3 b c-2 a d) \sqrt{c+d x^2}}{d^4}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0590191, size = 98, normalized size = 0.89 \[ \frac{-a^2 d^2 \left (2 c+3 d x^2\right )+2 a b d \left (8 c^2+12 c d x^2+3 d^2 x^4\right )+b^2 \left (-24 c^2 d x^2-16 c^3-6 c d^2 x^4+d^3 x^6\right )}{3 d^4 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*d^2*(2*c + 3*d*x^2)) + 2*a*b*d*(8*c^2 + 12*c*d*x^2 + 3*d^2*x^4) + b^2*(-16*c^3 - 24*c^2*d*x^2 - 6*c*d^2
*x^4 + d^3*x^6))/(3*d^4*(c + d*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 108, normalized size = 1. \begin{align*} -{\frac{-{b}^{2}{x}^{6}{d}^{3}-6\,ab{d}^{3}{x}^{4}+6\,{b}^{2}c{d}^{2}{x}^{4}+3\,{a}^{2}{d}^{3}{x}^{2}-24\,abc{d}^{2}{x}^{2}+24\,{b}^{2}{c}^{2}d{x}^{2}+2\,{a}^{2}c{d}^{2}-16\,ab{c}^{2}d+16\,{b}^{2}{c}^{3}}{3\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

-1/3*(-b^2*d^3*x^6-6*a*b*d^3*x^4+6*b^2*c*d^2*x^4+3*a^2*d^3*x^2-24*a*b*c*d^2*x^2+24*b^2*c^2*d*x^2+2*a^2*c*d^2-1
6*a*b*c^2*d+16*b^2*c^3)/(d*x^2+c)^(3/2)/d^4

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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^3*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37507, size = 252, normalized size = 2.29 \begin{align*} \frac{{\left (b^{2} d^{3} x^{6} - 16 \, b^{2} c^{3} + 16 \, a b c^{2} d - 2 \, a^{2} c d^{2} - 6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} - 3 \,{\left (8 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

1/3*(b^2*d^3*x^6 - 16*b^2*c^3 + 16*a*b*c^2*d - 2*a^2*c*d^2 - 6*(b^2*c*d^2 - a*b*d^3)*x^4 - 3*(8*b^2*c^2*d - 8*
a*b*c*d^2 + a^2*d^3)*x^2)*sqrt(d*x^2 + c)/(d^6*x^4 + 2*c*d^5*x^2 + c^2*d^4)

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Sympy [A]  time = 1.97742, size = 454, normalized size = 4.13 \begin{align*} \begin{cases} - \frac{2 a^{2} c d^{2}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} - \frac{3 a^{2} d^{3} x^{2}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} + \frac{16 a b c^{2} d}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} + \frac{24 a b c d^{2} x^{2}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} + \frac{6 a b d^{3} x^{4}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} - \frac{16 b^{2} c^{3}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} - \frac{24 b^{2} c^{2} d x^{2}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} - \frac{6 b^{2} c d^{2} x^{4}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} + \frac{b^{2} d^{3} x^{6}}{3 c d^{4} \sqrt{c + d x^{2}} + 3 d^{5} x^{2} \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}}{c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Piecewise((-2*a**2*c*d**2/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)) - 3*a**2*d**3*x**2/(3*c*d
**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)) + 16*a*b*c**2*d/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*
sqrt(c + d*x**2)) + 24*a*b*c*d**2*x**2/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)) + 6*a*b*d**3
*x**4/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)) - 16*b**2*c**3/(3*c*d**4*sqrt(c + d*x**2) + 3
*d**5*x**2*sqrt(c + d*x**2)) - 24*b**2*c**2*d*x**2/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2))
- 6*b**2*c*d**2*x**4/(3*c*d**4*sqrt(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)) + b**2*d**3*x**6/(3*c*d**4*sqr
t(c + d*x**2) + 3*d**5*x**2*sqrt(c + d*x**2)), Ne(d, 0)), ((a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8)/c**(5/2),
True))

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Giac [A]  time = 1.15051, size = 173, normalized size = 1.57 \begin{align*} \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} - 9 \, \sqrt{d x^{2} + c} b^{2} c + 6 \, \sqrt{d x^{2} + c} a b d - \frac{9 \,{\left (d x^{2} + c\right )} b^{2} c^{2} - b^{2} c^{3} - 12 \,{\left (d x^{2} + c\right )} a b c d + 2 \, a b c^{2} d + 3 \,{\left (d x^{2} + c\right )} a^{2} d^{2} - a^{2} c d^{2}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}}{3 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/3*((d*x^2 + c)^(3/2)*b^2 - 9*sqrt(d*x^2 + c)*b^2*c + 6*sqrt(d*x^2 + c)*a*b*d - (9*(d*x^2 + c)*b^2*c^2 - b^2*
c^3 - 12*(d*x^2 + c)*a*b*c*d + 2*a*b*c^2*d + 3*(d*x^2 + c)*a^2*d^2 - a^2*c*d^2)/(d*x^2 + c)^(3/2))/d^4