3.353 \(\int \frac{(A+B x) (a+c x^2)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=172 \[ \frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac{5 B c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8} \]

[Out]

(5*B*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*B*c^2*(a + c*x^2)^(3/2))/(192*a*x^4) + (B*c*(a + c*x^2)^(5/2))/(48*
a*x^6) - (A*(a + c*x^2)^(7/2))/(9*a*x^9) - (B*(a + c*x^2)^(7/2))/(8*a*x^8) + (2*A*c*(a + c*x^2)^(7/2))/(63*a^2
*x^7) + (5*B*c^4*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(128*a^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.123307, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {835, 807, 266, 47, 63, 208} \[ \frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac{5 B c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(5/2))/x^10,x]

[Out]

(5*B*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*B*c^2*(a + c*x^2)^(3/2))/(192*a*x^4) + (B*c*(a + c*x^2)^(5/2))/(48*
a*x^6) - (A*(a + c*x^2)^(7/2))/(9*a*x^9) - (B*(a + c*x^2)^(7/2))/(8*a*x^8) + (2*A*c*(a + c*x^2)^(7/2))/(63*a^2
*x^7) + (5*B*c^4*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(128*a^(3/2))

Rule 835

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

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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^{10}} \, dx &=-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{\int \frac{(-9 a B+2 A c x) \left (a+c x^2\right )^{5/2}}{x^9} \, dx}{9 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{\int \frac{(-16 a A c-9 a B c x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx}{72 a^2}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{(B c) \int \frac{\left (a+c x^2\right )^{5/2}}{x^7} \, dx}{8 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{(B c) \operatorname{Subst}\left (\int \frac{(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )}{16 a}\\ &=\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (5 B c^2\right ) \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a}\\ &=\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (5 B c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{128 a}\\ &=\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (5 B c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (5 B c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{128 a}\\ &=\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac{5 B c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0255611, size = 64, normalized size = 0.37 \[ -\frac{\left (a+c x^2\right )^{7/2} \left (a^3 A \left (7 a-2 c x^2\right )+9 B c^4 x^9 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{c x^2}{a}+1\right )\right )}{63 a^5 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^10,x]

[Out]

-((a + c*x^2)^(7/2)*(a^3*A*(7*a - 2*c*x^2) + 9*B*c^4*x^9*Hypergeometric2F1[7/2, 5, 9/2, 1 + (c*x^2)/a]))/(63*a
^5*x^9)

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 204, normalized size = 1.2 \begin{align*} -{\frac{A}{9\,a{x}^{9}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{2\,Ac}{63\,{a}^{2}{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B}{8\,a{x}^{8}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bc}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{2}}{192\,{a}^{3}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{c}^{4}}{128\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,B{c}^{4}}{384\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{c}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,B{c}^{4}}{128\,{a}^{2}}\sqrt{c{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(5/2)/x^10,x)

[Out]

-1/9*A*(c*x^2+a)^(7/2)/a/x^9+2/63*A*c*(c*x^2+a)^(7/2)/a^2/x^7-1/8*B*(c*x^2+a)^(7/2)/a/x^8+1/48*B/a^2*c/x^6*(c*
x^2+a)^(7/2)+1/192*B/a^3*c^2/x^4*(c*x^2+a)^(7/2)+1/128*B/a^4*c^3/x^2*(c*x^2+a)^(7/2)-1/128*B/a^4*c^4*(c*x^2+a)
^(5/2)-5/384*B/a^3*c^4*(c*x^2+a)^(3/2)+5/128*B/a^(3/2)*c^4*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-5/128*B/a^2*c
^4*(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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.09374, size = 718, normalized size = 4.17 \begin{align*} \left [\frac{315 \, B \sqrt{a} c^{4} x^{9} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{16128 \, a^{2} x^{9}}, -\frac{315 \, B \sqrt{-a} c^{4} x^{9} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{8064 \, a^{2} x^{9}}\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^10,x, algorithm="fricas")

[Out]

[1/16128*(315*B*sqrt(a)*c^4*x^9*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(256*A*c^4*x^8 - 315*B
*a*c^3*x^7 - 128*A*a*c^3*x^6 - 2478*B*a^2*c^2*x^5 - 1920*A*a^2*c^2*x^4 - 2856*B*a^3*c*x^3 - 2432*A*a^3*c*x^2 -
1008*B*a^4*x - 896*A*a^4)*sqrt(c*x^2 + a))/(a^2*x^9), -1/8064*(315*B*sqrt(-a)*c^4*x^9*arctan(sqrt(-a)/sqrt(c*
x^2 + a)) - (256*A*c^4*x^8 - 315*B*a*c^3*x^7 - 128*A*a*c^3*x^6 - 2478*B*a^2*c^2*x^5 - 1920*A*a^2*c^2*x^4 - 285
6*B*a^3*c*x^3 - 2432*A*a^3*c*x^2 - 1008*B*a^4*x - 896*A*a^4)*sqrt(c*x^2 + a))/(a^2*x^9)]

________________________________________________________________________________________

Sympy [B]  time = 29.4006, size = 1202, normalized size = 6.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(5/2)/x**10,x)

[Out]

-35*A*a**9*c**(19/2)*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 +
315*a**4*c**12*x**14) - 110*A*a**8*c**(21/2)*x**2*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x*
*10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 114*A*a**7*c**(23/2)*x**4*sqrt(a/(c*x**2) + 1)/(315*a**7*
c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 40*A*a**6*c**(25/2)*x**6*sqr
t(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) -
30*A*a**6*c**(11/2)*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 5*A
*a**5*c**(27/2)*x**8*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 +
315*a**4*c**12*x**14) - 66*A*a**5*c**(13/2)*x**2*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8
+ 105*a**3*c**6*x**10) + 30*A*a**4*c**(29/2)*x**10*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*
x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 34*A*a**4*c**(15/2)*x**4*sqrt(a/(c*x**2) + 1)/(105*a**5
*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 40*A*a**3*c**(31/2)*x**12*sqrt(a/(c*x**2) + 1)/(315*a
**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 6*A*a**3*c**(17/2)*x**6*
sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 16*A*a**2*c**(33/2)*x**
14*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**
14) - 24*A*a**2*c**(19/2)*x**8*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x
**10) - 16*A*a*c**(21/2)*x**10*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x
**10) - A*c**(5/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - A*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a*x**2) + 2*A*c**(9/2)*
sqrt(a/(c*x**2) + 1)/(15*a**2) - B*a**3/(8*sqrt(c)*x**9*sqrt(a/(c*x**2) + 1)) - 23*B*a**2*sqrt(c)/(48*x**7*sqr
t(a/(c*x**2) + 1)) - 127*B*a*c**(3/2)/(192*x**5*sqrt(a/(c*x**2) + 1)) - 133*B*c**(5/2)/(384*x**3*sqrt(a/(c*x**
2) + 1)) - 5*B*c**(7/2)/(128*a*x*sqrt(a/(c*x**2) + 1)) + 5*B*c**4*asinh(sqrt(a)/(sqrt(c)*x))/(128*a**(3/2))

________________________________________________________________________________________

Giac [B]  time = 1.18294, size = 663, normalized size = 3.85 \begin{align*} -\frac{5 \, B c^{4} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a} + \frac{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{17} B c^{4} + 8022 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{15} B a c^{4} + 16128 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{14} A a c^{\frac{9}{2}} + 10458 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} B a^{2} c^{4} + 26880 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} A a^{2} c^{\frac{9}{2}} + 18270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} B a^{3} c^{4} + 80640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} A a^{3} c^{\frac{9}{2}} + 48384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a^{4} c^{\frac{9}{2}} - 18270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a^{5} c^{4} + 48384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a^{5} c^{\frac{9}{2}} - 10458 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{6} c^{4} + 6912 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{6} c^{\frac{9}{2}} - 8022 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{7} c^{4} + 2304 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{7} c^{\frac{9}{2}} - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{8} c^{4} - 256 \, A a^{8} c^{\frac{9}{2}}}{4032 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{9} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^10,x, algorithm="giac")

[Out]

-5/64*B*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/4032*(315*(sqrt(c)*x - sqrt(c*x^2
+ a))^17*B*c^4 + 8022*(sqrt(c)*x - sqrt(c*x^2 + a))^15*B*a*c^4 + 16128*(sqrt(c)*x - sqrt(c*x^2 + a))^14*A*a*c
^(9/2) + 10458*(sqrt(c)*x - sqrt(c*x^2 + a))^13*B*a^2*c^4 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^12*A*a^2*c^(9/
2) + 18270*(sqrt(c)*x - sqrt(c*x^2 + a))^11*B*a^3*c^4 + 80640*(sqrt(c)*x - sqrt(c*x^2 + a))^10*A*a^3*c^(9/2) +
48384*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a^4*c^(9/2) - 18270*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a^5*c^4 + 48384
*(sqrt(c)*x - sqrt(c*x^2 + a))^6*A*a^5*c^(9/2) - 10458*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a^6*c^4 + 6912*(sqrt(
c)*x - sqrt(c*x^2 + a))^4*A*a^6*c^(9/2) - 8022*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^7*c^4 + 2304*(sqrt(c)*x - s
qrt(c*x^2 + a))^2*A*a^7*c^(9/2) - 315*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^8*c^4 - 256*A*a^8*c^(9/2))/(((sqrt(c)*
x - sqrt(c*x^2 + a))^2 - a)^9*a)