3.349 \(\int \frac{(A+B x) (a+c x^2)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=140 \[ -\frac{c^2 \sqrt{a+c x^2} (8 A-15 B x)}{8 x}-\frac{c \left (a+c x^2\right )^{3/2} (8 A+15 B x)}{24 x^3}-\frac{\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{15}{8} \sqrt{a} B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right ) \]

[Out]

-(c^2*(8*A - 15*B*x)*Sqrt[a + c*x^2])/(8*x) - (c*(8*A + 15*B*x)*(a + c*x^2)^(3/2))/(24*x^3) - ((4*A + 5*B*x)*(
a + c*x^2)^(5/2))/(20*x^5) + A*c^(5/2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - (15*Sqrt[a]*B*c^2*ArcTanh[Sqrt[a
+ c*x^2]/Sqrt[a]])/8

________________________________________________________________________________________

Rubi [A]  time = 0.118306, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {811, 813, 844, 217, 206, 266, 63, 208} \[ -\frac{c^2 \sqrt{a+c x^2} (8 A-15 B x)}{8 x}-\frac{c \left (a+c x^2\right )^{3/2} (8 A+15 B x)}{24 x^3}-\frac{\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{15}{8} \sqrt{a} B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(5/2))/x^6,x]

[Out]

-(c^2*(8*A - 15*B*x)*Sqrt[a + c*x^2])/(8*x) - (c*(8*A + 15*B*x)*(a + c*x^2)^(3/2))/(24*x^3) - ((4*A + 5*B*x)*(
a + c*x^2)^(5/2))/(20*x^5) + A*c^(5/2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - (15*Sqrt[a]*B*c^2*ArcTanh[Sqrt[a
+ c*x^2]/Sqrt[a]])/8

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx &=-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}-\frac{\int \frac{(-8 a A c-10 a B c x) \left (a+c x^2\right )^{3/2}}{x^4} \, dx}{8 a}\\ &=-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+\frac{\int \frac{\left (32 a^2 A c^2+60 a^2 B c^2 x\right ) \sqrt{a+c x^2}}{x^2} \, dx}{32 a^2}\\ &=-\frac{c^2 (8 A-15 B x) \sqrt{a+c x^2}}{8 x}-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}-\frac{\int \frac{-120 a^3 B c^2-64 a^2 A c^3 x}{x \sqrt{a+c x^2}} \, dx}{64 a^2}\\ &=-\frac{c^2 (8 A-15 B x) \sqrt{a+c x^2}}{8 x}-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+\frac{1}{8} \left (15 a B c^2\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\left (A c^3\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=-\frac{c^2 (8 A-15 B x) \sqrt{a+c x^2}}{8 x}-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+\frac{1}{16} \left (15 a B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\left (A c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=-\frac{c^2 (8 A-15 B x) \sqrt{a+c x^2}}{8 x}-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{8} (15 a B c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )\\ &=-\frac{c^2 (8 A-15 B x) \sqrt{a+c x^2}}{8 x}-\frac{c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac{(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{15}{8} \sqrt{a} B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0279559, size = 96, normalized size = 0.69 \[ -\frac{a^2 A \sqrt{a+c x^2} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{c x^2}{a}\right )}{5 x^5 \sqrt{\frac{c x^2}{a}+1}}-\frac{B c^2 \left (a+c x^2\right )^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{c x^2}{a}+1\right )}{7 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^6,x]

[Out]

-(a^2*A*Sqrt[a + c*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, -((c*x^2)/a)])/(5*x^5*Sqrt[1 + (c*x^2)/a]) - (B*c^
2*(a + c*x^2)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, 1 + (c*x^2)/a])/(7*a^3)

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 257, normalized size = 1.8 \begin{align*} -{\frac{B}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,B{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{c}^{2}}{8\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,B{c}^{2}}{8}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{15\,B{c}^{2}}{8}\sqrt{c{x}^{2}+a}}-{\frac{A}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Ac}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,A{c}^{3}x}{15\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,A{c}^{3}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{c}^{3}x}{a}\sqrt{c{x}^{2}+a}}+A{c}^{{\frac{5}{2}}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(5/2)/x^6,x)

[Out]

-1/4*B/a/x^4*(c*x^2+a)^(7/2)-3/8*B/a^2*c/x^2*(c*x^2+a)^(7/2)+3/8*B/a^2*c^2*(c*x^2+a)^(5/2)+5/8*B/a*c^2*(c*x^2+
a)^(3/2)-15/8*B*a^(1/2)*c^2*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+15/8*B*c^2*(c*x^2+a)^(1/2)-1/5*A/a/x^5*(c*x^
2+a)^(7/2)-2/15*A/a^2*c/x^3*(c*x^2+a)^(7/2)-8/15*A/a^3*c^2/x*(c*x^2+a)^(7/2)+8/15*A/a^3*c^3*x*(c*x^2+a)^(5/2)+
2/3*A/a^2*c^3*x*(c*x^2+a)^(3/2)+A/a*c^3*x*(c*x^2+a)^(1/2)+A*c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

________________________________________________________________________________________

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+A)*(c*x^2+a)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.79746, size = 1351, normalized size = 9.65 \begin{align*} \left [\frac{120 \, A c^{\frac{5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 225 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x^{5}}, -\frac{240 \, A \sqrt{-c} c^{2} x^{5} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 225 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x^{5}}, \frac{225 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 60 \, A c^{\frac{5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) +{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x^{5}}, -\frac{120 \, A \sqrt{-c} c^{2} x^{5} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 225 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/240*(120*A*c^(5/2)*x^5*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 225*B*sqrt(a)*c^2*x^5*log(-(c*x^2
- 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(120*B*c^2*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30
*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5, -1/240*(240*A*sqrt(-c)*c^2*x^5*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) -
225*B*sqrt(a)*c^2*x^5*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(120*B*c^2*x^5 - 184*A*c^2*x^4
- 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5, 1/120*(225*B*sqrt(-a)*c^2*x^5*ar
ctan(sqrt(-a)/sqrt(c*x^2 + a)) + 60*A*c^(5/2)*x^5*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (120*B*c^2
*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5, -1/120*(120
*A*sqrt(-c)*c^2*x^5*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 225*B*sqrt(-a)*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^2 + a
)) - (120*B*c^2*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x
^5]

________________________________________________________________________________________

Sympy [B]  time = 15.9186, size = 294, normalized size = 2.1 \begin{align*} - \frac{A \sqrt{a} c^{2}}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{11 A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{8 A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15} + A c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{A c^{3} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{15 B \sqrt{a} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8} - \frac{B a^{3}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B a^{2} \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{x} + \frac{7 B a c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B c^{\frac{5}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(5/2)/x**6,x)

[Out]

-A*sqrt(a)*c**2/(x*sqrt(1 + c*x**2/a)) - A*a**2*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 11*A*a*c**(3/2)*sqrt(a
/(c*x**2) + 1)/(15*x**2) - 8*A*c**(5/2)*sqrt(a/(c*x**2) + 1)/15 + A*c**(5/2)*asinh(sqrt(c)*x/sqrt(a)) - A*c**3
*x/(sqrt(a)*sqrt(1 + c*x**2/a)) - 15*B*sqrt(a)*c**2*asinh(sqrt(a)/(sqrt(c)*x))/8 - B*a**3/(4*sqrt(c)*x**5*sqrt
(a/(c*x**2) + 1)) - 3*B*a**2*sqrt(c)/(8*x**3*sqrt(a/(c*x**2) + 1)) - B*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/x + 7*B
*a*c**(3/2)/(8*x*sqrt(a/(c*x**2) + 1)) + B*c**(5/2)*x/sqrt(a/(c*x**2) + 1)

________________________________________________________________________________________

Giac [B]  time = 1.21335, size = 447, normalized size = 3.19 \begin{align*} \frac{15 \, B a c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} - A c^{\frac{5}{2}} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \sqrt{c x^{2} + a} B c^{2} + \frac{135 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B a c^{2} + 360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a c^{\frac{5}{2}} - 150 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a^{2} c^{2} - 720 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a^{2} c^{\frac{5}{2}} + 1120 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{3} c^{\frac{5}{2}} + 150 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{4} c^{2} - 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{4} c^{\frac{5}{2}} - 135 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{5} c^{2} + 184 \, A a^{5} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^6,x, algorithm="giac")

[Out]

15/4*B*a*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - A*c^(5/2)*log(abs(-sqrt(c)*x + sqrt(c*
x^2 + a))) + sqrt(c*x^2 + a)*B*c^2 + 1/60*(135*(sqrt(c)*x - sqrt(c*x^2 + a))^9*B*a*c^2 + 360*(sqrt(c)*x - sqrt
(c*x^2 + a))^8*A*a*c^(5/2) - 150*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a^2*c^2 - 720*(sqrt(c)*x - sqrt(c*x^2 + a))
^6*A*a^2*c^(5/2) + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^3*c^(5/2) + 150*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*
a^4*c^2 - 560*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^4*c^(5/2) - 135*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^5*c^2 + 18
4*A*a^5*c^(5/2))/((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^5