3.2.6 \(\int x^5 (A+B x^2) (b x^2+c x^4)^{3/2} \, dx\) [106]

Optimal. Leaf size=223 \[ \frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}} \]

[Out]

-1/768*b^2*(-14*A*c+9*B*b)*(2*c*x^2+b)*(c*x^4+b*x^2)^(3/2)/c^4+1/240*b*(-14*A*c+9*B*b)*(c*x^4+b*x^2)^(5/2)/c^3
-1/168*(-14*A*c+9*B*b)*x^2*(c*x^4+b*x^2)^(5/2)/c^2+1/14*B*x^4*(c*x^4+b*x^2)^(5/2)/c-1/2048*b^6*(-14*A*c+9*B*b)
*arctanh(x^2*c^(1/2)/(c*x^4+b*x^2)^(1/2))/c^(11/2)+1/2048*b^4*(-14*A*c+9*B*b)*(2*c*x^2+b)*(c*x^4+b*x^2)^(1/2)/
c^5

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Rubi [A]
time = 0.26, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2059, 808, 684, 654, 626, 634, 212} \begin {gather*} -\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}}+\frac {b^4 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (9 b B-14 A c)}{2048 c^5}-\frac {b^2 \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (9 b B-14 A c)}{768 c^4}+\frac {b \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{240 c^3}-\frac {x^2 \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(b^4*(9*b*B - 14*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(2048*c^5) - (b^2*(9*b*B - 14*A*c)*(b + 2*c*x^2)*(b*x
^2 + c*x^4)^(3/2))/(768*c^4) + (b*(9*b*B - 14*A*c)*(b*x^2 + c*x^4)^(5/2))/(240*c^3) - ((9*b*B - 14*A*c)*x^2*(b
*x^2 + c*x^4)^(5/2))/(168*c^2) + (B*x^4*(b*x^2 + c*x^4)^(5/2))/(14*c) - (b^6*(9*b*B - 14*A*c)*ArcTanh[(Sqrt[c]
*x^2)/Sqrt[b*x^2 + c*x^4]])/(2048*c^(11/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 808

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 2059

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rubi steps

\begin {align*} \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^2 (A+B x) \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (2 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \text {Subst}\left (\int x^2 \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{14 c}\\ &=-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {(b (9 b B-14 A c)) \text {Subst}\left (\int x \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{48 c^2}\\ &=\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^2 (9 b B-14 A c)\right ) \text {Subst}\left (\int \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{96 c^3}\\ &=-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (b^4 (9 b B-14 A c)\right ) \text {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{512 c^4}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^6 (9 b B-14 A c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{4096 c^5}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^6 (9 b B-14 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^5}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 209, normalized size = 0.94 \begin {gather*} \frac {x \left (\sqrt {c} x \left (b+c x^2\right ) \left (945 b^6 B-210 b^5 c \left (7 A+3 B x^2\right )+96 b^2 c^4 x^6 \left (7 A+4 B x^2\right )+2560 c^6 x^{10} \left (7 A+6 B x^2\right )+28 b^4 c^2 x^2 \left (35 A+18 B x^2\right )-16 b^3 c^3 x^4 \left (49 A+27 B x^2\right )+256 b c^5 x^8 \left (91 A+75 B x^2\right )\right )+105 b^6 (9 b B-14 A c) \sqrt {b+c x^2} \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{215040 c^{11/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(Sqrt[c]*x*(b + c*x^2)*(945*b^6*B - 210*b^5*c*(7*A + 3*B*x^2) + 96*b^2*c^4*x^6*(7*A + 4*B*x^2) + 2560*c^6*x
^10*(7*A + 6*B*x^2) + 28*b^4*c^2*x^2*(35*A + 18*B*x^2) - 16*b^3*c^3*x^4*(49*A + 27*B*x^2) + 256*b*c^5*x^8*(91*
A + 75*B*x^2)) + 105*b^6*(9*b*B - 14*A*c)*Sqrt[b + c*x^2]*Log[-(Sqrt[c]*x) + Sqrt[b + c*x^2]]))/(215040*c^(11/
2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]
time = 0.44, size = 328, normalized size = 1.47

method	result	size
risch	\(-\frac {\left (-15360 B \,c^{6} x^{12}-17920 A \,c^{6} x^{10}-19200 B b \,c^{5} x^{10}-23296 A b \,c^{5} x^{8}-384 B \,b^{2} c^{4} x^{8}-672 A \,b^{2} c^{4} x^{6}+432 B \,b^{3} c^{3} x^{6}+784 A \,b^{3} c^{3} x^{4}-504 B \,b^{4} c^{2} x^{4}-980 A \,b^{4} c^{2} x^{2}+630 B \,b^{5} c \,x^{2}+1470 A \,b^{5} c -945 B \,b^{6}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{215040 c^{5}}+\frac {\left (\frac {7 b^{6} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) A}{1024 c^{\frac {9}{2}}}-\frac {9 b^{7} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) B}{2048 c^{\frac {11}{2}}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\)	\(231\)
default	\(\frac {\left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} \left (15360 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {9}{2}} x^{9}+17920 A \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {9}{2}} x^{7}-11520 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {7}{2}} b \,x^{7}-12544 A \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {7}{2}} b \,x^{5}+8064 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {5}{2}} b^{2} x^{5}+7840 A \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {5}{2}} b^{2} x^{3}-5040 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {3}{2}} b^{3} x^{3}-3920 A \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {3}{2}} b^{3} x +980 A \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} b^{4} x +2520 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}\, b^{4} x +1470 A \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} b^{5} x -630 B \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, b^{5} x -945 B \sqrt {c \,x^{2}+b}\, \sqrt {c}\, b^{6} x +1470 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{6} c -945 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{7}\right )}{215040 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {11}{2}}}\)	\(328\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/215040*(c*x^4+b*x^2)^(3/2)*(15360*B*(c*x^2+b)^(5/2)*c^(9/2)*x^9+17920*A*(c*x^2+b)^(5/2)*c^(9/2)*x^7-11520*B*
(c*x^2+b)^(5/2)*c^(7/2)*b*x^7-12544*A*(c*x^2+b)^(5/2)*c^(7/2)*b*x^5+8064*B*(c*x^2+b)^(5/2)*c^(5/2)*b^2*x^5+784
0*A*(c*x^2+b)^(5/2)*c^(5/2)*b^2*x^3-5040*B*(c*x^2+b)^(5/2)*c^(3/2)*b^3*x^3-3920*A*(c*x^2+b)^(5/2)*c^(3/2)*b^3*
x+980*A*(c*x^2+b)^(3/2)*c^(3/2)*b^4*x+2520*B*(c*x^2+b)^(5/2)*c^(1/2)*b^4*x+1470*A*(c*x^2+b)^(1/2)*c^(3/2)*b^5*
x-630*B*(c*x^2+b)^(3/2)*c^(1/2)*b^5*x-945*B*(c*x^2+b)^(1/2)*c^(1/2)*b^6*x+1470*A*ln(c^(1/2)*x+(c*x^2+b)^(1/2))
*b^6*c-945*B*ln(c^(1/2)*x+(c*x^2+b)^(1/2))*b^7)/x^3/(c*x^2+b)^(3/2)/c^(11/2)

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Maxima [A]
time = 0.29, size = 363, normalized size = 1.63 \begin {gather*} -\frac {1}{30720} \, {\left (\frac {420 \, \sqrt {c x^{4} + b x^{2}} b^{4} x^{2}}{c^{3}} - \frac {1120 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2} x^{2}}{c^{2}} - \frac {2560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{2}}{c} - \frac {105 \, b^{6} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {9}{2}}} + \frac {210 \, \sqrt {c x^{4} + b x^{2}} b^{5}}{c^{4}} - \frac {560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3}}{c^{3}} + \frac {1792 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b}{c^{2}}\right )} A + \frac {1}{143360} \, {\left (\frac {10240 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{4}}{c} + \frac {1260 \, \sqrt {c x^{4} + b x^{2}} b^{5} x^{2}}{c^{4}} - \frac {3360 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3} x^{2}}{c^{3}} - \frac {7680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b x^{2}}{c^{2}} - \frac {315 \, b^{7} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {11}{2}}} + \frac {630 \, \sqrt {c x^{4} + b x^{2}} b^{6}}{c^{5}} - \frac {1680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{4}}{c^{4}} + \frac {5376 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b^{2}}{c^{3}}\right )} B \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

-1/30720*(420*sqrt(c*x^4 + b*x^2)*b^4*x^2/c^3 - 1120*(c*x^4 + b*x^2)^(3/2)*b^2*x^2/c^2 - 2560*(c*x^4 + b*x^2)^
(5/2)*x^2/c - 105*b^6*log(2*c*x^2 + b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c))/c^(9/2) + 210*sqrt(c*x^4 + b*x^2)*b^5/c
^4 - 560*(c*x^4 + b*x^2)^(3/2)*b^3/c^3 + 1792*(c*x^4 + b*x^2)^(5/2)*b/c^2)*A + 1/143360*(10240*(c*x^4 + b*x^2)
^(5/2)*x^4/c + 1260*sqrt(c*x^4 + b*x^2)*b^5*x^2/c^4 - 3360*(c*x^4 + b*x^2)^(3/2)*b^3*x^2/c^3 - 7680*(c*x^4 + b
*x^2)^(5/2)*b*x^2/c^2 - 315*b^7*log(2*c*x^2 + b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c))/c^(11/2) + 630*sqrt(c*x^4 + b
*x^2)*b^6/c^5 - 1680*(c*x^4 + b*x^2)^(3/2)*b^4/c^4 + 5376*(c*x^4 + b*x^2)^(5/2)*b^2/c^3)*B

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Fricas [A]
time = 1.54, size = 418, normalized size = 1.87 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{430080 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{215040 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/430080*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(c)*log(-2*c*x^2 - b - 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) - 2*(15360*B*
c^7*x^12 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^10 + 128*(3*B*b^2*c^5 + 182*A*b*c^6)*x^8 + 945*B*b^6*c - 1470*A*b^5*
c^2 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^6 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^4 - 70*(9*B*b^5*c^2 - 14*A*b^4*c
^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6, 1/215040*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(-c)*arctan(sqrt(c*x^4 + b*x^2)*sq
rt(-c)/(c*x^2 + b)) + (15360*B*c^7*x^12 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^10 + 128*(3*B*b^2*c^5 + 182*A*b*c^6)*
x^8 + 945*B*b^6*c - 1470*A*b^5*c^2 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^6 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^4
- 70*(9*B*b^5*c^2 - 14*A*b^4*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**5*(x**2*(b + c*x**2))**(3/2)*(A + B*x**2), x)

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Giac [A]
time = 0.57, size = 280, normalized size = 1.26 \begin {gather*} \frac {1}{215040} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x^{2} \mathrm {sgn}\left (x\right ) + \frac {15 \, B b c^{12} \mathrm {sgn}\left (x\right ) + 14 \, A c^{13} \mathrm {sgn}\left (x\right )}{c^{12}}\right )} x^{2} + \frac {3 \, B b^{2} c^{11} \mathrm {sgn}\left (x\right ) + 182 \, A b c^{12} \mathrm {sgn}\left (x\right )}{c^{12}}\right )} x^{2} - \frac {3 \, {\left (9 \, B b^{3} c^{10} \mathrm {sgn}\left (x\right ) - 14 \, A b^{2} c^{11} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} + \frac {7 \, {\left (9 \, B b^{4} c^{9} \mathrm {sgn}\left (x\right ) - 14 \, A b^{3} c^{10} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} - \frac {35 \, {\left (9 \, B b^{5} c^{8} \mathrm {sgn}\left (x\right ) - 14 \, A b^{4} c^{9} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} + \frac {105 \, {\left (9 \, B b^{6} c^{7} \mathrm {sgn}\left (x\right ) - 14 \, A b^{5} c^{8} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (9 \, B b^{7} \mathrm {sgn}\left (x\right ) - 14 \, A b^{6} c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {{\left (9 \, B b^{7} \log \left ({\left | b \right |}\right ) - 14 \, A b^{6} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{4096 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

1/215040*(2*(4*(2*(8*(10*(12*B*c*x^2*sgn(x) + (15*B*b*c^12*sgn(x) + 14*A*c^13*sgn(x))/c^12)*x^2 + (3*B*b^2*c^1
1*sgn(x) + 182*A*b*c^12*sgn(x))/c^12)*x^2 - 3*(9*B*b^3*c^10*sgn(x) - 14*A*b^2*c^11*sgn(x))/c^12)*x^2 + 7*(9*B*
b^4*c^9*sgn(x) - 14*A*b^3*c^10*sgn(x))/c^12)*x^2 - 35*(9*B*b^5*c^8*sgn(x) - 14*A*b^4*c^9*sgn(x))/c^12)*x^2 + 1
05*(9*B*b^6*c^7*sgn(x) - 14*A*b^5*c^8*sgn(x))/c^12)*sqrt(c*x^2 + b)*x + 1/2048*(9*B*b^7*sgn(x) - 14*A*b^6*c*sg
n(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + b)))/c^(11/2) - 1/4096*(9*B*b^7*log(abs(b)) - 14*A*b^6*c*log(abs(b)))*
sgn(x)/c^(11/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x)

[Out]

int(x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x)

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