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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.351677, antiderivative size = 207, normalized size of antiderivative = 1., 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} \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{45045 b^6 x^{10}}+\frac{64 c^3 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{9009 b^5 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{1287 b^4 x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{429 b^3 x^{16}}-\frac{\left (b x^2+c x^4\right )^{5/2} (3 b B-2 A c)}{39 b^2 x^{18}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{15 b x^{20}} \]

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 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]

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 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 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]

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 = 0.0670279, size = 89, normalized size = 0.43 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (-x^2 \left (560 b^2 c^2 x^4-840 b^3 c x^2+1155 b^4-320 b c^3 x^6+128 c^4 x^8\right ) (3 b B-2 A c)-3003 A b^5\right )}{45045 b^6 x^{20}} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 142, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -256\,A{c}^{5}{x}^{10}+384\,Bb{c}^{4}{x}^{10}+640\,Ab{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 ) }{45045\,{x}^{18}{b}^{6}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.55657, size = 409, normalized size = 1.98 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Giac [B]  time = 6.31061, size = 786, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

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