3.129 \(\int \frac{(A+B x^2) (b x^2+c x^4)^{3/2}}{x^{16}} \, dx\)

Optimal. Leaf size=251 \[ -\frac{c^4 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1024 b^4 x^3}+\frac{c^3 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1536 b^3 x^5}-\frac{c^2 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1920 b^2 x^7}+\frac{c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{1024 b^{9/2}}-\frac{c \sqrt{b x^2+c x^4} (12 b B-7 A c)}{320 b x^9}-\frac{\left (b x^2+c x^4\right )^{3/2} (12 b B-7 A c)}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}} \]

[Out]

-(c*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(320*b*x^9) - (c^2*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(1920*b^2*x
^7) + (c^3*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(1536*b^3*x^5) - (c^4*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(
1024*b^4*x^3) - ((12*b*B - 7*A*c)*(b*x^2 + c*x^4)^(3/2))/(120*b*x^13) - (A*(b*x^2 + c*x^4)^(5/2))/(12*b*x^17)
+ (c^5*(12*b*B - 7*A*c)*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(1024*b^(9/2))

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Rubi [A]  time = 0.389782, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2038, 2020, 2025, 2008, 206} \[ -\frac{c^4 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1024 b^4 x^3}+\frac{c^3 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1536 b^3 x^5}-\frac{c^2 \sqrt{b x^2+c x^4} (12 b B-7 A c)}{1920 b^2 x^7}+\frac{c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{1024 b^{9/2}}-\frac{c \sqrt{b x^2+c x^4} (12 b B-7 A c)}{320 b x^9}-\frac{\left (b x^2+c x^4\right )^{3/2} (12 b B-7 A c)}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^16,x]

[Out]

-(c*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(320*b*x^9) - (c^2*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(1920*b^2*x
^7) + (c^3*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(1536*b^3*x^5) - (c^4*(12*b*B - 7*A*c)*Sqrt[b*x^2 + c*x^4])/(
1024*b^4*x^3) - ((12*b*B - 7*A*c)*(b*x^2 + c*x^4)^(3/2))/(120*b*x^13) - (A*(b*x^2 + c*x^4)^(5/2))/(12*b*x^17)
+ (c^5*(12*b*B - 7*A*c)*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(1024*b^(9/2))

Rule 2038

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] + Dist[(a*d*(m + j*p + 1
) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; F
reeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m
+ j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, -(n*p) - 1])) && (GtQ[e, 0] || IntegersQ[j,
n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]

Rule 2020

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b*
x^n)^p)/(c*(m + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2025

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{16}} \, dx &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac{(-12 b B+7 A c) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{14}} \, dx}{12 b}\\ &=-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac{(c (12 b B-7 A c)) \int \frac{\sqrt{b x^2+c x^4}}{x^{10}} \, dx}{40 b}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac{\left (c^2 (12 b B-7 A c)\right ) \int \frac{1}{x^6 \sqrt{b x^2+c x^4}} \, dx}{320 b}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{c^2 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1920 b^2 x^7}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac{\left (c^3 (12 b B-7 A c)\right ) \int \frac{1}{x^4 \sqrt{b x^2+c x^4}} \, dx}{384 b^2}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{c^2 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1920 b^2 x^7}+\frac{c^3 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1536 b^3 x^5}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac{\left (c^4 (12 b B-7 A c)\right ) \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx}{512 b^3}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{c^2 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1920 b^2 x^7}+\frac{c^3 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1536 b^3 x^5}-\frac{c^4 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1024 b^4 x^3}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac{\left (c^5 (12 b B-7 A c)\right ) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{1024 b^4}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{c^2 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1920 b^2 x^7}+\frac{c^3 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1536 b^3 x^5}-\frac{c^4 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1024 b^4 x^3}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac{\left (c^5 (12 b B-7 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{1024 b^4}\\ &=-\frac{c (12 b B-7 A c) \sqrt{b x^2+c x^4}}{320 b x^9}-\frac{c^2 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1920 b^2 x^7}+\frac{c^3 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1536 b^3 x^5}-\frac{c^4 (12 b B-7 A c) \sqrt{b x^2+c x^4}}{1024 b^4 x^3}-\frac{(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac{c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{1024 b^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0366375, size = 66, normalized size = 0.26 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (c^5 x^{12} (12 b B-7 A c) \, _2F_1\left (\frac{5}{2},6;\frac{7}{2};\frac{c x^2}{b}+1\right )-5 A b^6\right )}{60 b^7 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^16,x]

[Out]

((x^2*(b + c*x^2))^(5/2)*(-5*A*b^6 + c^5*(12*b*B - 7*A*c)*x^12*Hypergeometric2F1[5/2, 6, 7/2, 1 + (c*x^2)/b]))
/(60*b^7*x^17)

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Maple [A]  time = 0.062, size = 386, normalized size = 1.5 \begin{align*} -{\frac{1}{15360\,{x}^{15}{b}^{6}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 105\,A{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{12}{c}^{6}-35\,A \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{12}{c}^{6}-180\,B{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{12}{c}^{5}+60\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{12}b{c}^{5}+35\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{10}{c}^{5}-105\,A\sqrt{c{x}^{2}+b}{x}^{12}b{c}^{6}-60\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{10}b{c}^{4}+180\,B\sqrt{c{x}^{2}+b}{x}^{12}{b}^{2}{c}^{5}+70\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{8}b{c}^{4}-120\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{8}{b}^{2}{c}^{3}-280\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}{b}^{2}{c}^{3}+480\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}{b}^{3}{c}^{2}+560\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{3}{c}^{2}-960\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{4}c-896\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{4}c+1536\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{5}+1280\,A \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{5} \right ) \left ( c{x}^{2}+b \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^16,x)

[Out]

-1/15360*(c*x^4+b*x^2)^(3/2)*(105*A*b^(3/2)*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^12*c^6-35*A*(c*x^2+b)^(3/2)*
x^12*c^6-180*B*b^(5/2)*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^12*c^5+60*B*(c*x^2+b)^(3/2)*x^12*b*c^5+35*A*(c*x^
2+b)^(5/2)*x^10*c^5-105*A*(c*x^2+b)^(1/2)*x^12*b*c^6-60*B*(c*x^2+b)^(5/2)*x^10*b*c^4+180*B*(c*x^2+b)^(1/2)*x^1
2*b^2*c^5+70*A*(c*x^2+b)^(5/2)*x^8*b*c^4-120*B*(c*x^2+b)^(5/2)*x^8*b^2*c^3-280*A*(c*x^2+b)^(5/2)*x^6*b^2*c^3+4
80*B*(c*x^2+b)^(5/2)*x^6*b^3*c^2+560*A*(c*x^2+b)^(5/2)*x^4*b^3*c^2-960*B*(c*x^2+b)^(5/2)*x^4*b^4*c-896*A*(c*x^
2+b)^(5/2)*x^2*b^4*c+1536*B*(c*x^2+b)^(5/2)*x^2*b^5+1280*A*(c*x^2+b)^(5/2)*b^5)/x^15/(c*x^2+b)^(3/2)/b^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{16}}\,{d x} \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^16,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^16, x)

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Fricas [A]  time = 1.71437, size = 905, normalized size = 3.61 \begin{align*} \left [-\frac{15 \,{\left (12 \, B b c^{5} - 7 \, A c^{6}\right )} \sqrt{b} x^{13} \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \,{\left (15 \,{\left (12 \, B b^{2} c^{4} - 7 \, A b c^{5}\right )} x^{10} - 10 \,{\left (12 \, B b^{3} c^{3} - 7 \, A b^{2} c^{4}\right )} x^{8} + 1280 \, A b^{6} + 8 \,{\left (12 \, B b^{4} c^{2} - 7 \, A b^{3} c^{3}\right )} x^{6} + 48 \,{\left (44 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} + 128 \,{\left (12 \, B b^{6} + 13 \, A b^{5} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30720 \, b^{5} x^{13}}, -\frac{15 \,{\left (12 \, B b c^{5} - 7 \, A c^{6}\right )} \sqrt{-b} x^{13} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (15 \,{\left (12 \, B b^{2} c^{4} - 7 \, A b c^{5}\right )} x^{10} - 10 \,{\left (12 \, B b^{3} c^{3} - 7 \, A b^{2} c^{4}\right )} x^{8} + 1280 \, A b^{6} + 8 \,{\left (12 \, B b^{4} c^{2} - 7 \, A b^{3} c^{3}\right )} x^{6} + 48 \,{\left (44 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} + 128 \,{\left (12 \, B b^{6} + 13 \, A b^{5} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15360 \, b^{5} x^{13}}\right ] \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^16,x, algorithm="fricas")

[Out]

[-1/30720*(15*(12*B*b*c^5 - 7*A*c^6)*sqrt(b)*x^13*log(-(c*x^3 + 2*b*x - 2*sqrt(c*x^4 + b*x^2)*sqrt(b))/x^3) +
2*(15*(12*B*b^2*c^4 - 7*A*b*c^5)*x^10 - 10*(12*B*b^3*c^3 - 7*A*b^2*c^4)*x^8 + 1280*A*b^6 + 8*(12*B*b^4*c^2 - 7
*A*b^3*c^3)*x^6 + 48*(44*B*b^5*c + A*b^4*c^2)*x^4 + 128*(12*B*b^6 + 13*A*b^5*c)*x^2)*sqrt(c*x^4 + b*x^2))/(b^5
*x^13), -1/15360*(15*(12*B*b*c^5 - 7*A*c^6)*sqrt(-b)*x^13*arctan(sqrt(c*x^4 + b*x^2)*sqrt(-b)/(c*x^3 + b*x)) +
(15*(12*B*b^2*c^4 - 7*A*b*c^5)*x^10 - 10*(12*B*b^3*c^3 - 7*A*b^2*c^4)*x^8 + 1280*A*b^6 + 8*(12*B*b^4*c^2 - 7*
A*b^3*c^3)*x^6 + 48*(44*B*b^5*c + A*b^4*c^2)*x^4 + 128*(12*B*b^6 + 13*A*b^5*c)*x^2)*sqrt(c*x^4 + b*x^2))/(b^5*
x^13)]

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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^{16}}\, 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**16,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**16, x)

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Giac [A]  time = 1.29499, size = 397, normalized size = 1.58 \begin{align*} -\frac{\frac{15 \,{\left (12 \, B b c^{6} \mathrm{sgn}\left (x\right ) - 7 \, A c^{7} \mathrm{sgn}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{180 \,{\left (c x^{2} + b\right )}^{\frac{11}{2}} B b c^{6} \mathrm{sgn}\left (x\right ) - 1020 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} B b^{2} c^{6} \mathrm{sgn}\left (x\right ) + 2376 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} B b^{3} c^{6} \mathrm{sgn}\left (x\right ) - 696 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} B b^{4} c^{6} \mathrm{sgn}\left (x\right ) - 1020 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b^{5} c^{6} \mathrm{sgn}\left (x\right ) + 180 \, \sqrt{c x^{2} + b} B b^{6} c^{6} \mathrm{sgn}\left (x\right ) - 105 \,{\left (c x^{2} + b\right )}^{\frac{11}{2}} A c^{7} \mathrm{sgn}\left (x\right ) + 595 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} A b c^{7} \mathrm{sgn}\left (x\right ) - 1386 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} A b^{2} c^{7} \mathrm{sgn}\left (x\right ) + 1686 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} A b^{3} c^{7} \mathrm{sgn}\left (x\right ) + 595 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A b^{4} c^{7} \mathrm{sgn}\left (x\right ) - 105 \, \sqrt{c x^{2} + b} A b^{5} c^{7} \mathrm{sgn}\left (x\right )}{b^{4} c^{6} x^{12}}}{15360 \, c} \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^16,x, algorithm="giac")

[Out]

-1/15360*(15*(12*B*b*c^6*sgn(x) - 7*A*c^7*sgn(x))*arctan(sqrt(c*x^2 + b)/sqrt(-b))/(sqrt(-b)*b^4) + (180*(c*x^
2 + b)^(11/2)*B*b*c^6*sgn(x) - 1020*(c*x^2 + b)^(9/2)*B*b^2*c^6*sgn(x) + 2376*(c*x^2 + b)^(7/2)*B*b^3*c^6*sgn(
x) - 696*(c*x^2 + b)^(5/2)*B*b^4*c^6*sgn(x) - 1020*(c*x^2 + b)^(3/2)*B*b^5*c^6*sgn(x) + 180*sqrt(c*x^2 + b)*B*
b^6*c^6*sgn(x) - 105*(c*x^2 + b)^(11/2)*A*c^7*sgn(x) + 595*(c*x^2 + b)^(9/2)*A*b*c^7*sgn(x) - 1386*(c*x^2 + b)
^(7/2)*A*b^2*c^7*sgn(x) + 1686*(c*x^2 + b)^(5/2)*A*b^3*c^7*sgn(x) + 595*(c*x^2 + b)^(3/2)*A*b^4*c^7*sgn(x) - 1
05*sqrt(c*x^2 + b)*A*b^5*c^7*sgn(x))/(b^4*c^6*x^12))/c