3.7.25 \(\int x^3 (a+b x^2)^2 (c+d x^2)^{5/2} \, dx\) [625]

Optimal. Leaf size=114 \[ -\frac {c (b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{11/2}}{11 d^4}+\frac {b^2 \left (c+d x^2\right )^{13/2}}{13 d^4} \]

[Out]

-1/7*c*(-a*d+b*c)^2*(d*x^2+c)^(7/2)/d^4+1/9*(-a*d+b*c)*(-a*d+3*b*c)*(d*x^2+c)^(9/2)/d^4-1/11*b*(-2*a*d+3*b*c)*
(d*x^2+c)^(11/2)/d^4+1/13*b^2*(d*x^2+c)^(13/2)/d^4

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Rubi [A]
time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, 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 = {457, 78} \begin {gather*} -\frac {b \left (c+d x^2\right )^{11/2} (3 b c-2 a d)}{11 d^4}+\frac {\left (c+d x^2\right )^{9/2} (b c-a d) (3 b c-a d)}{9 d^4}-\frac {c \left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^4}+\frac {b^2 \left (c+d x^2\right )^{13/2}}{13 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(c*(b*c - a*d)^2*(c + d*x^2)^(7/2))/d^4 + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^4) - (b*(3*b
*c - 2*a*d)*(c + d*x^2)^(11/2))/(11*d^4) + (b^2*(c + d*x^2)^(13/2))/(13*d^4)

Rule 78

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]))))

Rule 457

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]]

Rubi steps

\begin {align*} \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^2 (c+d x)^{5/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {c (b c-a d)^2 (c+d x)^{5/2}}{d^3}+\frac {(b c-a d) (3 b c-a d) (c+d x)^{7/2}}{d^3}-\frac {b (3 b c-2 a d) (c+d x)^{9/2}}{d^3}+\frac {b^2 (c+d x)^{11/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {c (b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{11/2}}{11 d^4}+\frac {b^2 \left (c+d x^2\right )^{13/2}}{13 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 99, normalized size = 0.87 \begin {gather*} \frac {\left (c+d x^2\right )^{7/2} \left (143 a^2 d^2 \left (-2 c+7 d x^2\right )+26 a b d \left (8 c^2-28 c d x^2+63 d^2 x^4\right )+b^2 \left (-48 c^3+168 c^2 d x^2-378 c d^2 x^4+693 d^3 x^6\right )\right )}{9009 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x^2)^(7/2)*(143*a^2*d^2*(-2*c + 7*d*x^2) + 26*a*b*d*(8*c^2 - 28*c*d*x^2 + 63*d^2*x^4) + b^2*(-48*c^3 +
168*c^2*d*x^2 - 378*c*d^2*x^4 + 693*d^3*x^6)))/(9009*d^4)

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Maple [A]
time = 0.09, size = 185, normalized size = 1.62

method	result	size
gosper	\(-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}} \left (-693 b^{2} x^{6} d^{3}-1638 a b \,d^{3} x^{4}+378 b^{2} c \,d^{2} x^{4}-1001 a^{2} d^{3} x^{2}+728 a b c \,d^{2} x^{2}-168 b^{2} c^{2} d \,x^{2}+286 a^{2} c \,d^{2}-208 a b \,c^{2} d +48 b^{2} c^{3}\right )}{9009 d^{4}}\)	\(108\)
default	\(b^{2} \left (\frac {x^{6} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{13 d}-\frac {6 c \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{11 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{63 d^{2}}\right )}{11 d}\right )}{13 d}\right )+2 a b \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{11 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{63 d^{2}}\right )}{11 d}\right )+a^{2} \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{63 d^{2}}\right )\)	\(185\)
trager	\(-\frac {\left (-693 b^{2} d^{6} x^{12}-1638 a b \,d^{6} x^{10}-1701 b^{2} c \,d^{5} x^{10}-1001 a^{2} d^{6} x^{8}-4186 a b c \,d^{5} x^{8}-1113 b^{2} c^{2} d^{4} x^{8}-2717 a^{2} c \,d^{5} x^{6}-2938 a b \,c^{2} d^{4} x^{6}-15 b^{2} c^{3} d^{3} x^{6}-2145 a^{2} c^{2} d^{4} x^{4}-78 a b \,c^{3} d^{3} x^{4}+18 b^{2} c^{4} d^{2} x^{4}-143 a^{2} c^{3} d^{3} x^{2}+104 a b \,c^{4} d^{2} x^{2}-24 b^{2} c^{5} d \,x^{2}+286 a^{2} c^{4} d^{2}-208 a b \,c^{5} d +48 b^{2} c^{6}\right ) \sqrt {d \,x^{2}+c}}{9009 d^{4}}\)	\(231\)
risch	\(-\frac {\left (-693 b^{2} d^{6} x^{12}-1638 a b \,d^{6} x^{10}-1701 b^{2} c \,d^{5} x^{10}-1001 a^{2} d^{6} x^{8}-4186 a b c \,d^{5} x^{8}-1113 b^{2} c^{2} d^{4} x^{8}-2717 a^{2} c \,d^{5} x^{6}-2938 a b \,c^{2} d^{4} x^{6}-15 b^{2} c^{3} d^{3} x^{6}-2145 a^{2} c^{2} d^{4} x^{4}-78 a b \,c^{3} d^{3} x^{4}+18 b^{2} c^{4} d^{2} x^{4}-143 a^{2} c^{3} d^{3} x^{2}+104 a b \,c^{4} d^{2} x^{2}-24 b^{2} c^{5} d \,x^{2}+286 a^{2} c^{4} d^{2}-208 a b \,c^{5} d +48 b^{2} c^{6}\right ) \sqrt {d \,x^{2}+c}}{9009 d^{4}}\)	\(231\)

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,method=_RETURNVERBOSE)

[Out]

b^2*(1/13*x^6*(d*x^2+c)^(7/2)/d-6/13*c/d*(1/11*x^4*(d*x^2+c)^(7/2)/d-4/11*c/d*(1/9*x^2*(d*x^2+c)^(7/2)/d-2/63*
c/d^2*(d*x^2+c)^(7/2))))+2*a*b*(1/11*x^4*(d*x^2+c)^(7/2)/d-4/11*c/d*(1/9*x^2*(d*x^2+c)^(7/2)/d-2/63*c/d^2*(d*x
^2+c)^(7/2)))+a^2*(1/9*x^2*(d*x^2+c)^(7/2)/d-2/63*c/d^2*(d*x^2+c)^(7/2))

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Maxima [A]
time = 0.30, size = 181, normalized size = 1.59 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{6}}{13 \, d} - \frac {6 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{4}}{143 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{4}}{11 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} x^{2}}{429 \, d^{3}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c x^{2}}{99 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} x^{2}}{9 \, d} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3}}{3003 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2}}{693 \, d^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c}{63 \, d^{2}} \end {gather*}

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]

1/13*(d*x^2 + c)^(7/2)*b^2*x^6/d - 6/143*(d*x^2 + c)^(7/2)*b^2*c*x^4/d^2 + 2/11*(d*x^2 + c)^(7/2)*a*b*x^4/d +
8/429*(d*x^2 + c)^(7/2)*b^2*c^2*x^2/d^3 - 8/99*(d*x^2 + c)^(7/2)*a*b*c*x^2/d^2 + 1/9*(d*x^2 + c)^(7/2)*a^2*x^2
/d - 16/3003*(d*x^2 + c)^(7/2)*b^2*c^3/d^4 + 16/693*(d*x^2 + c)^(7/2)*a*b*c^2/d^3 - 2/63*(d*x^2 + c)^(7/2)*a^2
*c/d^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (98) = 196\).
time = 1.19, size = 216, normalized size = 1.89 \begin {gather*} \frac {{\left (693 \, b^{2} d^{6} x^{12} + 63 \, {\left (27 \, b^{2} c d^{5} + 26 \, a b d^{6}\right )} x^{10} + 7 \, {\left (159 \, b^{2} c^{2} d^{4} + 598 \, a b c d^{5} + 143 \, a^{2} d^{6}\right )} x^{8} - 48 \, b^{2} c^{6} + 208 \, a b c^{5} d - 286 \, a^{2} c^{4} d^{2} + {\left (15 \, b^{2} c^{3} d^{3} + 2938 \, a b c^{2} d^{4} + 2717 \, a^{2} c d^{5}\right )} x^{6} - 3 \, {\left (6 \, b^{2} c^{4} d^{2} - 26 \, a b c^{3} d^{3} - 715 \, a^{2} c^{2} d^{4}\right )} x^{4} + {\left (24 \, b^{2} c^{5} d - 104 \, a b c^{4} d^{2} + 143 \, a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{9009 \, d^{4}} \end {gather*}

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]

1/9009*(693*b^2*d^6*x^12 + 63*(27*b^2*c*d^5 + 26*a*b*d^6)*x^10 + 7*(159*b^2*c^2*d^4 + 598*a*b*c*d^5 + 143*a^2*
d^6)*x^8 - 48*b^2*c^6 + 208*a*b*c^5*d - 286*a^2*c^4*d^2 + (15*b^2*c^3*d^3 + 2938*a*b*c^2*d^4 + 2717*a^2*c*d^5)
*x^6 - 3*(6*b^2*c^4*d^2 - 26*a*b*c^3*d^3 - 715*a^2*c^2*d^4)*x^4 + (24*b^2*c^5*d - 104*a*b*c^4*d^2 + 143*a^2*c^
3*d^3)*x^2)*sqrt(d*x^2 + c)/d^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (102) = 204\).
time = 0.69, size = 468, normalized size = 4.11 \begin {gather*} \begin {cases} - \frac {2 a^{2} c^{4} \sqrt {c + d x^{2}}}{63 d^{2}} + \frac {a^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{63 d} + \frac {5 a^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{21} + \frac {19 a^{2} c d x^{6} \sqrt {c + d x^{2}}}{63} + \frac {a^{2} d^{2} x^{8} \sqrt {c + d x^{2}}}{9} + \frac {16 a b c^{5} \sqrt {c + d x^{2}}}{693 d^{3}} - \frac {8 a b c^{4} x^{2} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {2 a b c^{3} x^{4} \sqrt {c + d x^{2}}}{231 d} + \frac {226 a b c^{2} x^{6} \sqrt {c + d x^{2}}}{693} + \frac {46 a b c d x^{8} \sqrt {c + d x^{2}}}{99} + \frac {2 a b d^{2} x^{10} \sqrt {c + d x^{2}}}{11} - \frac {16 b^{2} c^{6} \sqrt {c + d x^{2}}}{3003 d^{4}} + \frac {8 b^{2} c^{5} x^{2} \sqrt {c + d x^{2}}}{3003 d^{3}} - \frac {2 b^{2} c^{4} x^{4} \sqrt {c + d x^{2}}}{1001 d^{2}} + \frac {5 b^{2} c^{3} x^{6} \sqrt {c + d x^{2}}}{3003 d} + \frac {53 b^{2} c^{2} x^{8} \sqrt {c + d x^{2}}}{429} + \frac {27 b^{2} c d x^{10} \sqrt {c + d x^{2}}}{143} + \frac {b^{2} d^{2} x^{12} \sqrt {c + d x^{2}}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{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 {gather*}

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**4*sqrt(c + d*x**2)/(63*d**2) + a**2*c**3*x**2*sqrt(c + d*x**2)/(63*d) + 5*a**2*c**2*x**4
*sqrt(c + d*x**2)/21 + 19*a**2*c*d*x**6*sqrt(c + d*x**2)/63 + a**2*d**2*x**8*sqrt(c + d*x**2)/9 + 16*a*b*c**5*
sqrt(c + d*x**2)/(693*d**3) - 8*a*b*c**4*x**2*sqrt(c + d*x**2)/(693*d**2) + 2*a*b*c**3*x**4*sqrt(c + d*x**2)/(
231*d) + 226*a*b*c**2*x**6*sqrt(c + d*x**2)/693 + 46*a*b*c*d*x**8*sqrt(c + d*x**2)/99 + 2*a*b*d**2*x**10*sqrt(
c + d*x**2)/11 - 16*b**2*c**6*sqrt(c + d*x**2)/(3003*d**4) + 8*b**2*c**5*x**2*sqrt(c + d*x**2)/(3003*d**3) - 2
*b**2*c**4*x**4*sqrt(c + d*x**2)/(1001*d**2) + 5*b**2*c**3*x**6*sqrt(c + d*x**2)/(3003*d) + 53*b**2*c**2*x**8*
sqrt(c + d*x**2)/429 + 27*b**2*c*d*x**10*sqrt(c + d*x**2)/143 + b**2*d**2*x**12*sqrt(c + d*x**2)/13, Ne(d, 0))
, (c**(5/2)*(a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8), True))

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Giac [A]
time = 0.95, size = 150, normalized size = 1.32 \begin {gather*} \frac {693 \, {\left (d x^{2} + c\right )}^{\frac {13}{2}} b^{2} - 2457 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} c + 3003 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c^{2} - 1287 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3} + 1638 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} a b d - 4004 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b c d + 2574 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2} d + 1001 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a^{2} d^{2} - 1287 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c d^{2}}{9009 \, d^{4}} \end {gather*}

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]

1/9009*(693*(d*x^2 + c)^(13/2)*b^2 - 2457*(d*x^2 + c)^(11/2)*b^2*c + 3003*(d*x^2 + c)^(9/2)*b^2*c^2 - 1287*(d*
x^2 + c)^(7/2)*b^2*c^3 + 1638*(d*x^2 + c)^(11/2)*a*b*d - 4004*(d*x^2 + c)^(9/2)*a*b*c*d + 2574*(d*x^2 + c)^(7/
2)*a*b*c^2*d + 1001*(d*x^2 + c)^(9/2)*a^2*d^2 - 1287*(d*x^2 + c)^(7/2)*a^2*c*d^2)/d^4

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Mupad [B]
time = 0.45, size = 207, normalized size = 1.82 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {x^8\,\left (1001\,a^2\,d^6+4186\,a\,b\,c\,d^5+1113\,b^2\,c^2\,d^4\right )}{9009\,d^4}-\frac {286\,a^2\,c^4\,d^2-208\,a\,b\,c^5\,d+48\,b^2\,c^6}{9009\,d^4}+\frac {b^2\,d^2\,x^{12}}{13}+\frac {c\,x^6\,\left (2717\,a^2\,d^2+2938\,a\,b\,c\,d+15\,b^2\,c^2\right )}{9009\,d}+\frac {b\,d\,x^{10}\,\left (26\,a\,d+27\,b\,c\right )}{143}+\frac {c^3\,x^2\,\left (143\,a^2\,d^2-104\,a\,b\,c\,d+24\,b^2\,c^2\right )}{9009\,d^3}+\frac {c^2\,x^4\,\left (715\,a^2\,d^2+26\,a\,b\,c\,d-6\,b^2\,c^2\right )}{3003\,d^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c + d*x^2)^(1/2)*((x^8*(1001*a^2*d^6 + 1113*b^2*c^2*d^4 + 4186*a*b*c*d^5))/(9009*d^4) - (48*b^2*c^6 + 286*a^2
*c^4*d^2 - 208*a*b*c^5*d)/(9009*d^4) + (b^2*d^2*x^12)/13 + (c*x^6*(2717*a^2*d^2 + 15*b^2*c^2 + 2938*a*b*c*d))/
(9009*d) + (b*d*x^10*(26*a*d + 27*b*c))/143 + (c^3*x^2*(143*a^2*d^2 + 24*b^2*c^2 - 104*a*b*c*d))/(9009*d^3) +
(c^2*x^4*(715*a^2*d^2 - 6*b^2*c^2 + 26*a*b*c*d))/(3003*d^2))

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