3.4.52 \(\int \frac {(A+B x) (a+c x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=149 \[ \frac {5 A c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]

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Rubi [A]  time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {835, 807, 266, 47, 63, 208} \begin {gather*} \frac {5 A c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(5/2))/x^9,x]

[Out]

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

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]

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 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 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])

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 53, normalized size = 0.36 \begin {gather*} -\frac {\left (a+c x^2\right )^{7/2} \left (a^4 B+A c^4 x^7 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {c x^2}{a}+1\right )\right )}{7 a^5 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^9,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.01, size = 139, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-336 a^3 A-384 a^3 B x-952 a^2 A c x^2-1152 a^2 B c x^3-826 a A c^2 x^4-1152 a B c^2 x^5-105 A c^3 x^6-384 B c^3 x^7\right )}{2688 a x^8}-\frac {5 A c^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{64 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^(5/2))/x^9,x]

[Out]

(Sqrt[a + c*x^2]*(-336*a^3*A - 384*a^3*B*x - 952*a^2*A*c*x^2 - 1152*a^2*B*c*x^3 - 826*a*A*c^2*x^4 - 1152*a*B*c
^2*x^5 - 105*A*c^3*x^6 - 384*B*c^3*x^7))/(2688*a*x^8) - (5*A*c^4*ArcTanh[(Sqrt[c]*x)/Sqrt[a] - Sqrt[a + c*x^2]
/Sqrt[a]])/(64*a^(3/2))

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fricas [A]  time = 0.53, size = 267, normalized size = 1.79 \begin {gather*} \left [\frac {105 \, A \sqrt {a} c^{4} x^{8} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{5376 \, a^{2} x^{8}}, -\frac {105 \, A \sqrt {-a} c^{4} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{2688 \, a^{2} x^{8}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="fricas")

[Out]

[1/5376*(105*A*sqrt(a)*c^4*x^8*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(384*B*a*c^3*x^7 + 105*
A*a*c^3*x^6 + 1152*B*a^2*c^2*x^5 + 826*A*a^2*c^2*x^4 + 1152*B*a^3*c*x^3 + 952*A*a^3*c*x^2 + 384*B*a^4*x + 336*
A*a^4)*sqrt(c*x^2 + a))/(a^2*x^8), -1/2688*(105*A*sqrt(-a)*c^4*x^8*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (384*B*a
*c^3*x^7 + 105*A*a*c^3*x^6 + 1152*B*a^2*c^2*x^5 + 826*A*a^2*c^2*x^4 + 1152*B*a^3*c*x^3 + 952*A*a^3*c*x^2 + 384
*B*a^4*x + 336*A*a^4)*sqrt(c*x^2 + a))/(a^2*x^8)]

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giac [B]  time = 0.24, size = 491, normalized size = 3.30 \begin {gather*} -\frac {5 \, A c^{4} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a} + \frac {105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{15} A c^{4} + 2688 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{14} B a c^{\frac {7}{2}} + 2779 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} A a c^{4} - 2688 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} B a^{2} c^{\frac {7}{2}} + 6265 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} A a^{2} c^{4} + 13440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} B a^{3} c^{\frac {7}{2}} + 12355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} A a^{3} c^{4} - 13440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} B a^{4} c^{\frac {7}{2}} + 12355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A a^{4} c^{4} + 8064 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a^{5} c^{\frac {7}{2}} + 6265 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a^{5} c^{4} - 8064 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{6} c^{\frac {7}{2}} + 2779 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{6} c^{4} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{7} c^{\frac {7}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{7} c^{4} - 384 \, B a^{8} c^{\frac {7}{2}}}{1344 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{8} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="giac")

[Out]

-5/64*A*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/1344*(105*(sqrt(c)*x - sqrt(c*x^2
+ a))^15*A*c^4 + 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^14*B*a*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^13*A*
a*c^4 - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^12*B*a^2*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^11*A*a^2*c^4
+ 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*B*a^3*c^(7/2) + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^9*A*a^3*c^4 - 134
40*(sqrt(c)*x - sqrt(c*x^2 + a))^8*B*a^4*c^(7/2) + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a^4*c^4 + 8064*(sqr
t(c)*x - sqrt(c*x^2 + a))^6*B*a^5*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^5*c^4 - 8064*(sqrt(c)*x -
sqrt(c*x^2 + a))^4*B*a^6*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a^6*c^4 + 384*(sqrt(c)*x - sqrt(c*x
^2 + a))^2*B*a^7*c^(7/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^7*c^4 - 384*B*a^8*c^(7/2))/(((sqrt(c)*x - sqr
t(c*x^2 + a))^2 - a)^8*a)

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maple [A]  time = 0.13, size = 185, normalized size = 1.24 \begin {gather*} \frac {5 A \,c^{4} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+a}\, A \,c^{4}}{128 a^{2}}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{4}}{384 a^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{4}}{128 a^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,c^{3}}{128 a^{4} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,c^{2}}{192 a^{3} x^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{48 a^{2} x^{6}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{7 a \,x^{7}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{8 a \,x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(5/2)/x^9,x)

[Out]

-1/8*A*(c*x^2+a)^(7/2)/a/x^8+1/48*A*c/a^2/x^6*(c*x^2+a)^(7/2)+1/192*A*c^2/a^3/x^4*(c*x^2+a)^(7/2)+1/128*A*c^3/
a^4/x^2*(c*x^2+a)^(7/2)-1/128*A*c^4/a^4*(c*x^2+a)^(5/2)-5/384*A*c^4/a^3*(c*x^2+a)^(3/2)+5/128*A*c^4/a^(3/2)*ln
((2*a+2*(c*x^2+a)^(1/2)*a^(1/2))/x)-5/128*A*c^4/a^2*(c*x^2+a)^(1/2)-1/7*B*(c*x^2+a)^(7/2)/a/x^7

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maxima [A]  time = 0.51, size = 173, normalized size = 1.16 \begin {gather*} \frac {5 \, A c^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{128 \, a^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A c^{4}}{128 \, a^{4}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {c x^{2} + a} A c^{4}}{128 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c^{2}}{192 \, a^{3} x^{4}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{48 \, a^{2} x^{6}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, a x^{7}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{8 \, a x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="maxima")

[Out]

5/128*A*c^4*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(3/2) - 1/128*(c*x^2 + a)^(5/2)*A*c^4/a^4 - 5/384*(c*x^2 + a)^(3/2
)*A*c^4/a^3 - 5/128*sqrt(c*x^2 + a)*A*c^4/a^2 + 1/128*(c*x^2 + a)^(7/2)*A*c^3/(a^4*x^2) + 1/192*(c*x^2 + a)^(7
/2)*A*c^2/(a^3*x^4) + 1/48*(c*x^2 + a)^(7/2)*A*c/(a^2*x^6) - 1/7*(c*x^2 + a)^(7/2)*B/(a*x^7) - 1/8*(c*x^2 + a)
^(7/2)*A/(a*x^8)

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mupad [B]  time = 5.35, size = 168, normalized size = 1.13 \begin {gather*} \frac {55\,A\,a\,{\left (c\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {73\,A\,{\left (c\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {5\,A\,a^2\,\sqrt {c\,x^2+a}}{128\,x^8}-\frac {5\,A\,{\left (c\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {B\,a^2\,\sqrt {c\,x^2+a}}{7\,x^7}-\frac {3\,B\,c^2\,\sqrt {c\,x^2+a}}{7\,x^3}-\frac {B\,c^3\,\sqrt {c\,x^2+a}}{7\,a\,x}-\frac {3\,B\,a\,c\,\sqrt {c\,x^2+a}}{7\,x^5}-\frac {A\,c^4\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{128\,a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(5/2)*(A + B*x))/x^9,x)

[Out]

(55*A*a*(a + c*x^2)^(3/2))/(384*x^8) - (A*c^4*atan(((a + c*x^2)^(1/2)*1i)/a^(1/2))*5i)/(128*a^(3/2)) - (73*A*(
a + c*x^2)^(5/2))/(384*x^8) - (5*A*a^2*(a + c*x^2)^(1/2))/(128*x^8) - (5*A*(a + c*x^2)^(7/2))/(128*a*x^8) - (B
*a^2*(a + c*x^2)^(1/2))/(7*x^7) - (3*B*c^2*(a + c*x^2)^(1/2))/(7*x^3) - (B*c^3*(a + c*x^2)^(1/2))/(7*a*x) - (3
*B*a*c*(a + c*x^2)^(1/2))/(7*x^5)

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sympy [B]  time = 25.77, size = 609, normalized size = 4.09 \begin {gather*} - \frac {A a^{3}}{8 \sqrt {c} x^{9} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {23 A a^{2} \sqrt {c}}{48 x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {127 A a c^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {133 A c^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 A c^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {5 A c^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{128 a^{\frac {3}{2}}} - \frac {15 B a^{7} c^{\frac {9}{2}} \sqrt {\frac {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}} - \frac {33 B a^{6} c^{\frac {11}{2}} x^{2} \sqrt {\frac {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}} - \frac {17 B a^{5} c^{\frac {13}{2}} x^{4} \sqrt {\frac {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}} - \frac {3 B a^{4} c^{\frac {15}{2}} x^{6} \sqrt {\frac {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}} - \frac {12 B a^{3} c^{\frac {17}{2}} x^{8} \sqrt {\frac {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}} - \frac {8 B a^{2} c^{\frac {19}{2}} x^{10} \sqrt {\frac {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}} - \frac {2 B a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {7 B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {B c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(5/2)/x**9,x)

[Out]

-A*a**3/(8*sqrt(c)*x**9*sqrt(a/(c*x**2) + 1)) - 23*A*a**2*sqrt(c)/(48*x**7*sqrt(a/(c*x**2) + 1)) - 127*A*a*c**
(3/2)/(192*x**5*sqrt(a/(c*x**2) + 1)) - 133*A*c**(5/2)/(384*x**3*sqrt(a/(c*x**2) + 1)) - 5*A*c**(7/2)/(128*a*x
*sqrt(a/(c*x**2) + 1)) + 5*A*c**4*asinh(sqrt(a)/(sqrt(c)*x))/(128*a**(3/2)) - 15*B*a**7*c**(9/2)*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) - 33*B*a**6*c**(11/2)*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) - 17*B*a**5*c**(13/2)*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) - 3*B*a**4*c**(15/2)*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) - 12*B*a**3*c**(17/2)*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) - 8*B*a**2*c**(19/2)*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) - 2*B*a*c**(3/2)*sqrt(a/(c*x*
*2) + 1)/(5*x**4) - 7*B*c**(5/2)*sqrt(a/(c*x**2) + 1)/(15*x**2) - B*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a)

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