3.599 \(\int x^3 (a+b x^2)^2 \sqrt{c+d x^2} \, dx\)

Optimal. Leaf size=114 \[ -\frac{b \left (c+d x^2\right )^{7/2} (3 b c-2 a d)}{7 d^4}+\frac{\left (c+d x^2\right )^{5/2} (b c-a d) (3 b c-a d)}{5 d^4}-\frac{c \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^4}+\frac{b^2 \left (c+d x^2\right )^{9/2}}{9 d^4} \]

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^4) - (b*(3*b
*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^4) + (b^2*(c + d*x^2)^(9/2))/(9*d^4)

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Rubi [A]  time = 0.0959832, antiderivative size = 114, 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} \[ -\frac{b \left (c+d x^2\right )^{7/2} (3 b c-2 a d)}{7 d^4}+\frac{\left (c+d x^2\right )^{5/2} (b c-a d) (3 b c-a d)}{5 d^4}-\frac{c \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^4}+\frac{b^2 \left (c+d x^2\right )^{9/2}}{9 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^4) - (b*(3*b
*c - 2*a*d)*(c + d*x^2)^(7/2))/(7*d^4) + (b^2*(c + d*x^2)^(9/2))/(9*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 x^3 \left (a+b x^2\right )^2 \sqrt{c+d x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^2 \sqrt{c+d x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2 \sqrt{c+d x}}{d^3}+\frac{(b c-a d) (3 b c-a d) (c+d x)^{3/2}}{d^3}-\frac{b (3 b c-2 a d) (c+d x)^{5/2}}{d^3}+\frac{b^2 (c+d x)^{7/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^4}+\frac{(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^4}-\frac{b (3 b c-2 a d) \left (c+d x^2\right )^{7/2}}{7 d^4}+\frac{b^2 \left (c+d x^2\right )^{9/2}}{9 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0764489, size = 99, normalized size = 0.87 \[ \frac{\left (c+d x^2\right )^{3/2} \left (21 a^2 d^2 \left (3 d x^2-2 c\right )+6 a b d \left (8 c^2-12 c d x^2+15 d^2 x^4\right )+b^2 \left (24 c^2 d x^2-16 c^3-30 c d^2 x^4+35 d^3 x^6\right )\right )}{315 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x^2)^(3/2)*(21*a^2*d^2*(-2*c + 3*d*x^2) + 6*a*b*d*(8*c^2 - 12*c*d*x^2 + 15*d^2*x^4) + b^2*(-16*c^3 + 2
4*c^2*d*x^2 - 30*c*d^2*x^4 + 35*d^3*x^6)))/(315*d^4)

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Maple [A]  time = 0.007, size = 108, normalized size = 1. \begin{align*} -{\frac{-35\,{b}^{2}{x}^{6}{d}^{3}-90\,ab{d}^{3}{x}^{4}+30\,{b}^{2}c{d}^{2}{x}^{4}-63\,{a}^{2}{d}^{3}{x}^{2}+72\,abc{d}^{2}{x}^{2}-24\,{b}^{2}{c}^{2}d{x}^{2}+42\,{a}^{2}c{d}^{2}-48\,ab{c}^{2}d+16\,{b}^{2}{c}^{3}}{315\,{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)^(1/2),x)

[Out]

-1/315*(d*x^2+c)^(3/2)*(-35*b^2*d^3*x^6-90*a*b*d^3*x^4+30*b^2*c*d^2*x^4-63*a^2*d^3*x^2+72*a*b*c*d^2*x^2-24*b^2
*c^2*d*x^2+42*a^2*c*d^2-48*a*b*c^2*d+16*b^2*c^3)/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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.63593, size = 304, normalized size = 2.67 \begin{align*} \frac{{\left (35 \, b^{2} d^{4} x^{8} + 5 \,{\left (b^{2} c d^{3} + 18 \, a b d^{4}\right )} x^{6} - 16 \, b^{2} c^{4} + 48 \, a b c^{3} d - 42 \, a^{2} c^{2} d^{2} - 3 \,{\left (2 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 21 \, a^{2} d^{4}\right )} x^{4} +{\left (8 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 21 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{315 \, 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)^(1/2),x, algorithm="fricas")

[Out]

1/315*(35*b^2*d^4*x^8 + 5*(b^2*c*d^3 + 18*a*b*d^4)*x^6 - 16*b^2*c^4 + 48*a*b*c^3*d - 42*a^2*c^2*d^2 - 3*(2*b^2
*c^2*d^2 - 6*a*b*c*d^3 - 21*a^2*d^4)*x^4 + (8*b^2*c^3*d - 24*a*b*c^2*d^2 + 21*a^2*c*d^3)*x^2)*sqrt(d*x^2 + c)/
d^4

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Sympy [A]  time = 1.59804, size = 308, normalized size = 2.7 \begin{align*} \begin{cases} - \frac{2 a^{2} c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{a^{2} c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{a^{2} x^{4} \sqrt{c + d x^{2}}}{5} + \frac{16 a b c^{3} \sqrt{c + d x^{2}}}{105 d^{3}} - \frac{8 a b c^{2} x^{2} \sqrt{c + d x^{2}}}{105 d^{2}} + \frac{2 a b c x^{4} \sqrt{c + d x^{2}}}{35 d} + \frac{2 a b x^{6} \sqrt{c + d x^{2}}}{7} - \frac{16 b^{2} c^{4} \sqrt{c + d x^{2}}}{315 d^{4}} + \frac{8 b^{2} c^{3} x^{2} \sqrt{c + d x^{2}}}{315 d^{3}} - \frac{2 b^{2} c^{2} x^{4} \sqrt{c + d x^{2}}}{105 d^{2}} + \frac{b^{2} c x^{6} \sqrt{c + d x^{2}}}{63 d} + \frac{b^{2} x^{8} \sqrt{c + d x^{2}}}{9} & \text{for}\: d \neq 0 \\\sqrt{c} \left (\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}\right ) & \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)**(1/2),x)

[Out]

Piecewise((-2*a**2*c**2*sqrt(c + d*x**2)/(15*d**2) + a**2*c*x**2*sqrt(c + d*x**2)/(15*d) + a**2*x**4*sqrt(c +
d*x**2)/5 + 16*a*b*c**3*sqrt(c + d*x**2)/(105*d**3) - 8*a*b*c**2*x**2*sqrt(c + d*x**2)/(105*d**2) + 2*a*b*c*x*
*4*sqrt(c + d*x**2)/(35*d) + 2*a*b*x**6*sqrt(c + d*x**2)/7 - 16*b**2*c**4*sqrt(c + d*x**2)/(315*d**4) + 8*b**2
*c**3*x**2*sqrt(c + d*x**2)/(315*d**3) - 2*b**2*c**2*x**4*sqrt(c + d*x**2)/(105*d**2) + b**2*c*x**6*sqrt(c + d
*x**2)/(63*d) + b**2*x**8*sqrt(c + d*x**2)/9, Ne(d, 0)), (sqrt(c)*(a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8), Tr
ue))

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Giac [A]  time = 1.1356, size = 192, normalized size = 1.68 \begin{align*} \frac{\frac{21 \,{\left (3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c\right )} a^{2}}{d} + \frac{6 \,{\left (15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2}\right )} a b}{d^{2}} + \frac{{\left (35 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{3}\right )} b^{2}}{d^{3}}}{315 \, d} \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)^(1/2),x, algorithm="giac")

[Out]

1/315*(21*(3*(d*x^2 + c)^(5/2) - 5*(d*x^2 + c)^(3/2)*c)*a^2/d + 6*(15*(d*x^2 + c)^(7/2) - 42*(d*x^2 + c)^(5/2)
*c + 35*(d*x^2 + c)^(3/2)*c^2)*a*b/d^2 + (35*(d*x^2 + c)^(9/2) - 135*(d*x^2 + c)^(7/2)*c + 189*(d*x^2 + c)^(5/
2)*c^2 - 105*(d*x^2 + c)^(3/2)*c^3)*b^2/d^3)/d