3.2.18 \(\int \frac {(A+B x^2) (b x^2+c x^4)^{3/2}}{x^{19}} \, dx\)

Optimal. Leaf size=207 \[ -\frac {128 c^4 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{45045 b^6 x^{10}}+\frac {64 c^3 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{9009 b^5 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{1287 b^4 x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{429 b^3 x^{16}}-\frac {\left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{39 b^2 x^{18}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.35, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \begin {gather*} -\frac {128 c^4 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{45045 b^6 x^{10}}+\frac {64 c^3 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{9009 b^5 x^{12}}-\frac {16 c^2 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{1287 b^4 x^{14}}+\frac {8 c \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{429 b^3 x^{16}}-\frac {\left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{39 b^2 x^{18}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19,x]

[Out]

-(A*(b*x^2 + c*x^4)^(5/2))/(15*b*x^20) - ((3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(39*b^2*x^18) + (8*c*(3*b*B -
2*A*c)*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^16) - (16*c^2*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(1287*b^4*x^14)
+ (64*c^3*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(9009*b^5*x^12) - (128*c^4*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(
5/2))/(45045*b^6*x^10)

Rule 650

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

Rule 658

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

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 2034

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 \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{19}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{10}} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}+\frac {\left (-10 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^9} \, dx,x,x^2\right )}{15 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{39 b^2 x^{18}}-\frac {(4 c (3 b B-2 A c)) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^8} \, dx,x,x^2\right )}{39 b^2}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{39 b^2 x^{18}}+\frac {8 c (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{16}}+\frac {\left (8 c^2 (3 b B-2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )}{143 b^3}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{39 b^2 x^{18}}+\frac {8 c (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{16}}-\frac {16 c^2 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{1287 b^4 x^{14}}-\frac {\left (32 c^3 (3 b B-2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{1287 b^4}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{39 b^2 x^{18}}+\frac {8 c (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{16}}-\frac {16 c^2 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{1287 b^4 x^{14}}+\frac {64 c^3 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{9009 b^5 x^{12}}+\frac {\left (64 c^4 (3 b B-2 A c)\right ) \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{9009 b^5}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{39 b^2 x^{18}}+\frac {8 c (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{16}}-\frac {16 c^2 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{1287 b^4 x^{14}}+\frac {64 c^3 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{9009 b^5 x^{12}}-\frac {128 c^4 (3 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{45045 b^6 x^{10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 89, normalized size = 0.43 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (-3003 A b^5-x^2 \left (1155 b^4-840 b^3 c x^2+560 b^2 c^2 x^4-320 b c^3 x^6+128 c^4 x^8\right ) (3 b B-2 A c)\right )}{45045 b^6 x^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19,x]

[Out]

((x^2*(b + c*x^2))^(5/2)*(-3003*A*b^5 - (3*b*B - 2*A*c)*x^2*(1155*b^4 - 840*b^3*c*x^2 + 560*b^2*c^2*x^4 - 320*
b*c^3*x^6 + 128*c^4*x^8)))/(45045*b^6*x^20)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.56, size = 186, normalized size = 0.90 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-3003 A b^7-3696 A b^6 c x^2-63 A b^5 c^2 x^4+70 A b^4 c^3 x^6-80 A b^3 c^4 x^8+96 A b^2 c^5 x^{10}-128 A b c^6 x^{12}+256 A c^7 x^{14}-3465 b^7 B x^2-4410 b^6 B c x^4-105 b^5 B c^2 x^6+120 b^4 B c^3 x^8-144 b^3 B c^4 x^{10}+192 b^2 B c^5 x^{12}-384 b B c^6 x^{14}\right )}{45045 b^6 x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19,x]

[Out]

(Sqrt[b*x^2 + c*x^4]*(-3003*A*b^7 - 3465*b^7*B*x^2 - 3696*A*b^6*c*x^2 - 4410*b^6*B*c*x^4 - 63*A*b^5*c^2*x^4 -
105*b^5*B*c^2*x^6 + 70*A*b^4*c^3*x^6 + 120*b^4*B*c^3*x^8 - 80*A*b^3*c^4*x^8 - 144*b^3*B*c^4*x^10 + 96*A*b^2*c^
5*x^10 + 192*b^2*B*c^5*x^12 - 128*A*b*c^6*x^12 - 384*b*B*c^6*x^14 + 256*A*c^7*x^14))/(45045*b^6*x^16)

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 181, normalized size = 0.87 \begin {gather*} -\frac {{\left (128 \, {\left (3 \, B b c^{6} - 2 \, A c^{7}\right )} x^{14} - 64 \, {\left (3 \, B b^{2} c^{5} - 2 \, A b c^{6}\right )} x^{12} + 48 \, {\left (3 \, B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} x^{10} - 40 \, {\left (3 \, B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} x^{8} + 3003 \, A b^{7} + 35 \, {\left (3 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} x^{6} + 63 \, {\left (70 \, B b^{6} c + A b^{5} c^{2}\right )} x^{4} + 231 \, {\left (15 \, B b^{7} + 16 \, A b^{6} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{45045 \, b^{6} x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^19,x, algorithm="fricas")

[Out]

-1/45045*(128*(3*B*b*c^6 - 2*A*c^7)*x^14 - 64*(3*B*b^2*c^5 - 2*A*b*c^6)*x^12 + 48*(3*B*b^3*c^4 - 2*A*b^2*c^5)*
x^10 - 40*(3*B*b^4*c^3 - 2*A*b^3*c^4)*x^8 + 3003*A*b^7 + 35*(3*B*b^5*c^2 - 2*A*b^4*c^3)*x^6 + 63*(70*B*b^6*c +
A*b^5*c^2)*x^4 + 231*(15*B*b^7 + 16*A*b^6*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^6*x^16)

________________________________________________________________________________________

giac [B]  time = 5.43, size = 582, normalized size = 2.81 \begin {gather*} \frac {256 \, {\left (18018 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{20} B c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 60060 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{18} A c^{\frac {15}{2}} \mathrm {sgn}\relax (x) - 12870 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{16} B b^{2} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 128700 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{16} A b c^{\frac {15}{2}} \mathrm {sgn}\relax (x) - 32175 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} B b^{3} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 141570 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} A b^{2} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b^{4} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 50050 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A b^{3} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{5} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 6006 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b^{4} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 4095 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{6} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 2730 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{5} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) - 1365 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{7} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 910 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{6} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{8} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{7} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{9} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{8} c^{\frac {15}{2}} \mathrm {sgn}\relax (x) + 3 \, B b^{10} c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 2 \, A b^{9} c^{\frac {15}{2}} \mathrm {sgn}\relax (x)\right )}}{45045 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^19,x, algorithm="giac")

[Out]

256/45045*(18018*(sqrt(c)*x - sqrt(c*x^2 + b))^20*B*c^(13/2)*sgn(x) + 60060*(sqrt(c)*x - sqrt(c*x^2 + b))^18*A
*c^(15/2)*sgn(x) - 12870*(sqrt(c)*x - sqrt(c*x^2 + b))^16*B*b^2*c^(13/2)*sgn(x) + 128700*(sqrt(c)*x - sqrt(c*x
^2 + b))^16*A*b*c^(15/2)*sgn(x) - 32175*(sqrt(c)*x - sqrt(c*x^2 + b))^14*B*b^3*c^(13/2)*sgn(x) + 141570*(sqrt(
c)*x - sqrt(c*x^2 + b))^14*A*b^2*c^(15/2)*sgn(x) + 15015*(sqrt(c)*x - sqrt(c*x^2 + b))^12*B*b^4*c^(13/2)*sgn(x
) + 50050*(sqrt(c)*x - sqrt(c*x^2 + b))^12*A*b^3*c^(15/2)*sgn(x) + 9009*(sqrt(c)*x - sqrt(c*x^2 + b))^10*B*b^5
*c^(13/2)*sgn(x) + 6006*(sqrt(c)*x - sqrt(c*x^2 + b))^10*A*b^4*c^(15/2)*sgn(x) + 4095*(sqrt(c)*x - sqrt(c*x^2
+ b))^8*B*b^6*c^(13/2)*sgn(x) - 2730*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A*b^5*c^(15/2)*sgn(x) - 1365*(sqrt(c)*x -
sqrt(c*x^2 + b))^6*B*b^7*c^(13/2)*sgn(x) + 910*(sqrt(c)*x - sqrt(c*x^2 + b))^6*A*b^6*c^(15/2)*sgn(x) + 315*(s
qrt(c)*x - sqrt(c*x^2 + b))^4*B*b^8*c^(13/2)*sgn(x) - 210*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^7*c^(15/2)*sgn(x
) - 45*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^9*c^(13/2)*sgn(x) + 30*(sqrt(c)*x - sqrt(c*x^2 + b))^2*A*b^8*c^(15/
2)*sgn(x) + 3*B*b^10*c^(13/2)*sgn(x) - 2*A*b^9*c^(15/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^15

________________________________________________________________________________________

maple [A]  time = 0.05, size = 142, normalized size = 0.69 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-256 A \,c^{5} x^{10}+384 B b \,c^{4} x^{10}+640 A b \,c^{4} x^{8}-960 B \,b^{2} c^{3} x^{8}-1120 A \,b^{2} c^{3} x^{6}+1680 B \,b^{3} c^{2} x^{6}+1680 A \,b^{3} c^{2} x^{4}-2520 B \,b^{4} c \,x^{4}-2310 A \,b^{4} c \,x^{2}+3465 B \,b^{5} x^{2}+3003 A \,b^{5}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{45045 b^{6} x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^19,x)

[Out]

-1/45045*(c*x^2+b)*(-256*A*c^5*x^10+384*B*b*c^4*x^10+640*A*b*c^4*x^8-960*B*b^2*c^3*x^8-1120*A*b^2*c^3*x^6+1680
*B*b^3*c^2*x^6+1680*A*b^3*c^2*x^4-2520*B*b^4*c*x^4-2310*A*b^4*c*x^2+3465*B*b^5*x^2+3003*A*b^5)*(c*x^4+b*x^2)^(
3/2)/x^18/b^6

________________________________________________________________________________________

maxima [B]  time = 1.65, size = 385, normalized size = 1.86 \begin {gather*} -\frac {1}{30030} \, B {\left (\frac {256 \, \sqrt {c x^{4} + b x^{2}} c^{6}}{b^{5} x^{2}} - \frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{4} x^{4}} + \frac {96 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{3} x^{6}} - \frac {80 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{8}} + \frac {70 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{10}} - \frac {63 \, \sqrt {c x^{4} + b x^{2}} c}{x^{12}} - \frac {693 \, \sqrt {c x^{4} + b x^{2}} b}{x^{14}} + \frac {3003 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{16}}\right )} + \frac {1}{180180} \, A {\left (\frac {1024 \, \sqrt {c x^{4} + b x^{2}} c^{7}}{b^{6} x^{2}} - \frac {512 \, \sqrt {c x^{4} + b x^{2}} c^{6}}{b^{5} x^{4}} + \frac {384 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{4} x^{6}} - \frac {320 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{3} x^{8}} + \frac {280 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{10}} - \frac {252 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{12}} + \frac {231 \, \sqrt {c x^{4} + b x^{2}} c}{x^{14}} + \frac {3003 \, \sqrt {c x^{4} + b x^{2}} b}{x^{16}} - \frac {15015 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{18}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^19,x, algorithm="maxima")

[Out]

-1/30030*B*(256*sqrt(c*x^4 + b*x^2)*c^6/(b^5*x^2) - 128*sqrt(c*x^4 + b*x^2)*c^5/(b^4*x^4) + 96*sqrt(c*x^4 + b*
x^2)*c^4/(b^3*x^6) - 80*sqrt(c*x^4 + b*x^2)*c^3/(b^2*x^8) + 70*sqrt(c*x^4 + b*x^2)*c^2/(b*x^10) - 63*sqrt(c*x^
4 + b*x^2)*c/x^12 - 693*sqrt(c*x^4 + b*x^2)*b/x^14 + 3003*(c*x^4 + b*x^2)^(3/2)/x^16) + 1/180180*A*(1024*sqrt(
c*x^4 + b*x^2)*c^7/(b^6*x^2) - 512*sqrt(c*x^4 + b*x^2)*c^6/(b^5*x^4) + 384*sqrt(c*x^4 + b*x^2)*c^5/(b^4*x^6) -
320*sqrt(c*x^4 + b*x^2)*c^4/(b^3*x^8) + 280*sqrt(c*x^4 + b*x^2)*c^3/(b^2*x^10) - 252*sqrt(c*x^4 + b*x^2)*c^2/
(b*x^12) + 231*sqrt(c*x^4 + b*x^2)*c/x^14 + 3003*sqrt(c*x^4 + b*x^2)*b/x^16 - 15015*(c*x^4 + b*x^2)^(3/2)/x^18
)

________________________________________________________________________________________

mupad [B]  time = 3.21, size = 356, normalized size = 1.72 \begin {gather*} \frac {2\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{1287\,b^2\,x^{10}}-\frac {16\,A\,c\,\sqrt {c\,x^4+b\,x^2}}{195\,x^{14}}-\frac {B\,b\,\sqrt {c\,x^4+b\,x^2}}{13\,x^{14}}-\frac {14\,B\,c\,\sqrt {c\,x^4+b\,x^2}}{143\,x^{12}}-\frac {A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{715\,b\,x^{12}}-\frac {A\,b\,\sqrt {c\,x^4+b\,x^2}}{15\,x^{16}}-\frac {16\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{9009\,b^3\,x^8}+\frac {32\,A\,c^5\,\sqrt {c\,x^4+b\,x^2}}{15015\,b^4\,x^6}-\frac {128\,A\,c^6\,\sqrt {c\,x^4+b\,x^2}}{45045\,b^5\,x^4}+\frac {256\,A\,c^7\,\sqrt {c\,x^4+b\,x^2}}{45045\,b^6\,x^2}-\frac {B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{429\,b\,x^{10}}+\frac {8\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{3003\,b^2\,x^8}-\frac {16\,B\,c^4\,\sqrt {c\,x^4+b\,x^2}}{5005\,b^3\,x^6}+\frac {64\,B\,c^5\,\sqrt {c\,x^4+b\,x^2}}{15015\,b^4\,x^4}-\frac {128\,B\,c^6\,\sqrt {c\,x^4+b\,x^2}}{15015\,b^5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19,x)

[Out]

(2*A*c^3*(b*x^2 + c*x^4)^(1/2))/(1287*b^2*x^10) - (16*A*c*(b*x^2 + c*x^4)^(1/2))/(195*x^14) - (B*b*(b*x^2 + c*
x^4)^(1/2))/(13*x^14) - (14*B*c*(b*x^2 + c*x^4)^(1/2))/(143*x^12) - (A*c^2*(b*x^2 + c*x^4)^(1/2))/(715*b*x^12)
- (A*b*(b*x^2 + c*x^4)^(1/2))/(15*x^16) - (16*A*c^4*(b*x^2 + c*x^4)^(1/2))/(9009*b^3*x^8) + (32*A*c^5*(b*x^2
+ c*x^4)^(1/2))/(15015*b^4*x^6) - (128*A*c^6*(b*x^2 + c*x^4)^(1/2))/(45045*b^5*x^4) + (256*A*c^7*(b*x^2 + c*x^
4)^(1/2))/(45045*b^6*x^2) - (B*c^2*(b*x^2 + c*x^4)^(1/2))/(429*b*x^10) + (8*B*c^3*(b*x^2 + c*x^4)^(1/2))/(3003
*b^2*x^8) - (16*B*c^4*(b*x^2 + c*x^4)^(1/2))/(5005*b^3*x^6) + (64*B*c^5*(b*x^2 + c*x^4)^(1/2))/(15015*b^4*x^4)
- (128*B*c^6*(b*x^2 + c*x^4)^(1/2))/(15015*b^5*x^2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{19}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**19,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**19, x)

________________________________________________________________________________________