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

Optimal. Leaf size=114 \[ -\frac{b \left (c+d x^2\right )^{9/2} (3 b c-2 a d)}{9 d^4}+\frac{\left (c+d x^2\right )^{7/2} (b c-a d) (3 b c-a d)}{7 d^4}-\frac{c \left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^4}+\frac{b^2 \left (c+d x^2\right )^{11/2}}{11 d^4} \]

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(5/2))/(5*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(7/2))/(7*d^4) - (b*(3*b
*c - 2*a*d)*(c + d*x^2)^(9/2))/(9*d^4) + (b^2*(c + d*x^2)^(11/2))/(11*d^4)

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Rubi [A]  time = 0.0896447, 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} \[ -\frac{b \left (c+d x^2\right )^{9/2} (3 b c-2 a d)}{9 d^4}+\frac{\left (c+d x^2\right )^{7/2} (b c-a d) (3 b c-a d)}{7 d^4}-\frac{c \left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^4}+\frac{b^2 \left (c+d x^2\right )^{11/2}}{11 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(5/2))/(5*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(7/2))/(7*d^4) - (b*(3*b
*c - 2*a*d)*(c + d*x^2)^(9/2))/(9*d^4) + (b^2*(c + d*x^2)^(11/2))/(11*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 \left (c+d x^2\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^2 (c+d x)^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2 (c+d x)^{3/2}}{d^3}+\frac{(b c-a d) (3 b c-a d) (c+d x)^{5/2}}{d^3}-\frac{b (3 b c-2 a d) (c+d x)^{7/2}}{d^3}+\frac{b^2 (c+d x)^{9/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{c (b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^4}+\frac{(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^4}-\frac{b (3 b c-2 a d) \left (c+d x^2\right )^{9/2}}{9 d^4}+\frac{b^2 \left (c+d x^2\right )^{11/2}}{11 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0885513, size = 100, normalized size = 0.88 \[ \frac{\left (c+d x^2\right )^{5/2} \left (99 a^2 d^2 \left (5 d x^2-2 c\right )+22 a b d \left (8 c^2-20 c d x^2+35 d^2 x^4\right )-3 b^2 \left (-40 c^2 d x^2+16 c^3+70 c d^2 x^4-105 d^3 x^6\right )\right )}{3465 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x^2)^(5/2)*(99*a^2*d^2*(-2*c + 5*d*x^2) + 22*a*b*d*(8*c^2 - 20*c*d*x^2 + 35*d^2*x^4) - 3*b^2*(16*c^3 -
40*c^2*d*x^2 + 70*c*d^2*x^4 - 105*d^3*x^6)))/(3465*d^4)

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Maple [A]  time = 0.006, size = 108, normalized size = 1. \begin{align*} -{\frac{-315\,{b}^{2}{x}^{6}{d}^{3}-770\,ab{d}^{3}{x}^{4}+210\,{b}^{2}c{d}^{2}{x}^{4}-495\,{a}^{2}{d}^{3}{x}^{2}+440\,abc{d}^{2}{x}^{2}-120\,{b}^{2}{c}^{2}d{x}^{2}+198\,{a}^{2}c{d}^{2}-176\,ab{c}^{2}d+48\,{b}^{2}{c}^{3}}{3465\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{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)^(3/2),x)

[Out]

-1/3465*(d*x^2+c)^(5/2)*(-315*b^2*d^3*x^6-770*a*b*d^3*x^4+210*b^2*c*d^2*x^4-495*a^2*d^3*x^2+440*a*b*c*d^2*x^2-
120*b^2*c^2*d*x^2+198*a^2*c*d^2-176*a*b*c^2*d+48*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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5101, size = 398, normalized size = 3.49 \begin{align*} \frac{{\left (315 \, b^{2} d^{5} x^{10} + 70 \,{\left (6 \, b^{2} c d^{4} + 11 \, a b d^{5}\right )} x^{8} - 48 \, b^{2} c^{5} + 176 \, a b c^{4} d - 198 \, a^{2} c^{3} d^{2} + 5 \,{\left (3 \, b^{2} c^{2} d^{3} + 220 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{6} - 6 \,{\left (3 \, b^{2} c^{3} d^{2} - 11 \, a b c^{2} d^{3} - 132 \, a^{2} c d^{4}\right )} x^{4} +{\left (24 \, b^{2} c^{4} d - 88 \, a b c^{3} d^{2} + 99 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3465 \, 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)^(3/2),x, algorithm="fricas")

[Out]

1/3465*(315*b^2*d^5*x^10 + 70*(6*b^2*c*d^4 + 11*a*b*d^5)*x^8 - 48*b^2*c^5 + 176*a*b*c^4*d - 198*a^2*c^3*d^2 +
5*(3*b^2*c^2*d^3 + 220*a*b*c*d^4 + 99*a^2*d^5)*x^6 - 6*(3*b^2*c^3*d^2 - 11*a*b*c^2*d^3 - 132*a^2*c*d^4)*x^4 +
(24*b^2*c^4*d - 88*a*b*c^3*d^2 + 99*a^2*c^2*d^3)*x^2)*sqrt(d*x^2 + c)/d^4

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Sympy [A]  time = 4.1899, size = 384, normalized size = 3.37 \begin{align*} \begin{cases} - \frac{2 a^{2} c^{3} \sqrt{c + d x^{2}}}{35 d^{2}} + \frac{a^{2} c^{2} x^{2} \sqrt{c + d x^{2}}}{35 d} + \frac{8 a^{2} c x^{4} \sqrt{c + d x^{2}}}{35} + \frac{a^{2} d x^{6} \sqrt{c + d x^{2}}}{7} + \frac{16 a b c^{4} \sqrt{c + d x^{2}}}{315 d^{3}} - \frac{8 a b c^{3} x^{2} \sqrt{c + d x^{2}}}{315 d^{2}} + \frac{2 a b c^{2} x^{4} \sqrt{c + d x^{2}}}{105 d} + \frac{20 a b c x^{6} \sqrt{c + d x^{2}}}{63} + \frac{2 a b d x^{8} \sqrt{c + d x^{2}}}{9} - \frac{16 b^{2} c^{5} \sqrt{c + d x^{2}}}{1155 d^{4}} + \frac{8 b^{2} c^{4} x^{2} \sqrt{c + d x^{2}}}{1155 d^{3}} - \frac{2 b^{2} c^{3} x^{4} \sqrt{c + d x^{2}}}{385 d^{2}} + \frac{b^{2} c^{2} x^{6} \sqrt{c + d x^{2}}}{231 d} + \frac{4 b^{2} c x^{8} \sqrt{c + d x^{2}}}{33} + \frac{b^{2} d x^{10} \sqrt{c + d x^{2}}}{11} & \text{for}\: d \neq 0 \\c^{\frac{3}{2}} \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)**(3/2),x)

[Out]

Piecewise((-2*a**2*c**3*sqrt(c + d*x**2)/(35*d**2) + a**2*c**2*x**2*sqrt(c + d*x**2)/(35*d) + 8*a**2*c*x**4*sq
rt(c + d*x**2)/35 + a**2*d*x**6*sqrt(c + d*x**2)/7 + 16*a*b*c**4*sqrt(c + d*x**2)/(315*d**3) - 8*a*b*c**3*x**2
*sqrt(c + d*x**2)/(315*d**2) + 2*a*b*c**2*x**4*sqrt(c + d*x**2)/(105*d) + 20*a*b*c*x**6*sqrt(c + d*x**2)/63 +
2*a*b*d*x**8*sqrt(c + d*x**2)/9 - 16*b**2*c**5*sqrt(c + d*x**2)/(1155*d**4) + 8*b**2*c**4*x**2*sqrt(c + d*x**2
)/(1155*d**3) - 2*b**2*c**3*x**4*sqrt(c + d*x**2)/(385*d**2) + b**2*c**2*x**6*sqrt(c + d*x**2)/(231*d) + 4*b**
2*c*x**8*sqrt(c + d*x**2)/33 + b**2*d*x**10*sqrt(c + d*x**2)/11, Ne(d, 0)), (c**(3/2)*(a**2*x**4/4 + a*b*x**6/
3 + b**2*x**8/8), True))

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Giac [B]  time = 1.12617, size = 437, normalized size = 3.83 \begin{align*} \frac{\frac{231 \,{\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} c}{d} + \frac{66 \,{\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 c}{d^{2}} + \frac{33 \,{\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^{2}}{d} + \frac{11 \,{\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} c}{d^{3}} + \frac{22 \,{\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 )} a b}{d^{2}} + \frac{{\left (315 \,{\left (d x^{2} + c\right )}^{\frac{11}{2}} - 1540 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} c + 2970 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c^{2} - 2772 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{3} + 1155 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{4}\right )} b^{2}}{d^{3}}}{3465 \, 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)^(3/2),x, algorithm="giac")

[Out]

1/3465*(231*(3*(d*x^2 + c)^(5/2) - 5*(d*x^2 + c)^(3/2)*c)*a^2*c/d + 66*(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*c/d^2 + 33*(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^2/d + 11*(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*c/d^3 + 22*(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)*a*b/d^2 + (315*(d*x^2 + c)^(11/2) - 1540*(d*x^2 + c)^(9/2)*c + 2970*(d*x^2 + c
)^(7/2)*c^2 - 2772*(d*x^2 + c)^(5/2)*c^3 + 1155*(d*x^2 + c)^(3/2)*c^4)*b^2/d^3)/d