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

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

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Rubi [A]  time = 0.12, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {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} \end {gather*}

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 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^{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*}

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Mathematica [C]  time = 0.02, size = 64, normalized size = 0.37 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/63*((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]))/
(a^5*x^9)

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IntegrateAlgebraic [A]  time = 1.08, size = 154, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-896 a^4 A-1008 a^4 B x-2432 a^3 A c x^2-2856 a^3 B c x^3-1920 a^2 A c^2 x^4-2478 a^2 B c^2 x^5-128 a A c^3 x^6-315 a B c^3 x^7+256 A c^4 x^8\right )}{8064 a^2 x^9}-\frac {5 B 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^10,x]

[Out]

(Sqrt[a + c*x^2]*(-896*a^4*A - 1008*a^4*B*x - 2432*a^3*A*c*x^2 - 2856*a^3*B*c*x^3 - 1920*a^2*A*c^2*x^4 - 2478*
a^2*B*c^2*x^5 - 128*a*A*c^3*x^6 - 315*a*B*c^3*x^7 + 256*A*c^4*x^8))/(8064*a^2*x^9) - (5*B*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 = 0.56, size = 286, normalized size = 1.66 \begin {gather*} \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 {gather*}

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)]

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giac [B]  time = 0.25, size = 491, normalized size = 2.85 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.09, size = 204, normalized size = 1.19 \begin {gather*} \frac {5 B \,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}\, B \,c^{4}}{128 a^{2}}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{4}}{384 a^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,c^{4}}{128 a^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,c^{3}}{128 a^{4} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,c^{2}}{192 a^{3} x^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B c}{48 a^{2} x^{6}}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{63 a^{2} x^{7}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{8 a \,x^{8}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{9 a \,x^{9}} \end {gather*}

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]

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

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maxima [A]  time = 0.59, size = 192, normalized size = 1.12 \begin {gather*} \frac {5 \, B 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}} B c^{4}}{128 \, a^{4}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {c x^{2} + a} B c^{4}}{128 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{192 \, a^{3} x^{4}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{48 \, a^{2} x^{6}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{63 \, a^{2} x^{7}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{8 \, a x^{8}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{9 \, a x^{9}} \end {gather*}

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]

5/128*B*c^4*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(3/2) - 1/128*(c*x^2 + a)^(5/2)*B*c^4/a^4 - 5/384*(c*x^2 + a)^(3/2
)*B*c^4/a^3 - 5/128*sqrt(c*x^2 + a)*B*c^4/a^2 + 1/128*(c*x^2 + a)^(7/2)*B*c^3/(a^4*x^2) + 1/192*(c*x^2 + a)^(7
/2)*B*c^2/(a^3*x^4) + 1/48*(c*x^2 + a)^(7/2)*B*c/(a^2*x^6) + 2/63*(c*x^2 + a)^(7/2)*A*c/(a^2*x^7) - 1/8*(c*x^2
+ a)^(7/2)*B/(a*x^8) - 1/9*(c*x^2 + a)^(7/2)*A/(a*x^9)

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mupad [B]  time = 6.34, size = 189, normalized size = 1.10 \begin {gather*} \frac {55\,B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {73\,B\,{\left (c\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {A\,a^2\,\sqrt {c\,x^2+a}}{9\,x^9}-\frac {5\,B\,a^2\,\sqrt {c\,x^2+a}}{128\,x^8}-\frac {5\,B\,{\left (c\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {5\,A\,c^2\,\sqrt {c\,x^2+a}}{21\,x^5}-\frac {A\,c^3\,\sqrt {c\,x^2+a}}{63\,a\,x^3}+\frac {2\,A\,c^4\,\sqrt {c\,x^2+a}}{63\,a^2\,x}-\frac {19\,A\,a\,c\,\sqrt {c\,x^2+a}}{63\,x^7}-\frac {B\,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^10,x)

[Out]

(55*B*a*(a + c*x^2)^(3/2))/(384*x^8) - (B*c^4*atan(((a + c*x^2)^(1/2)*1i)/a^(1/2))*5i)/(128*a^(3/2)) - (73*B*(
a + c*x^2)^(5/2))/(384*x^8) - (A*a^2*(a + c*x^2)^(1/2))/(9*x^9) - (5*B*a^2*(a + c*x^2)^(1/2))/(128*x^8) - (5*B
*(a + c*x^2)^(7/2))/(128*a*x^8) - (5*A*c^2*(a + c*x^2)^(1/2))/(21*x^5) - (A*c^3*(a + c*x^2)^(1/2))/(63*a*x^3)
+ (2*A*c^4*(a + c*x^2)^(1/2))/(63*a^2*x) - (19*A*a*c*(a + c*x^2)^(1/2))/(63*x^7)

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sympy [B]  time = 25.67, size = 1202, normalized size = 6.99 \begin {gather*} - \frac {35 A a^{9} c^{\frac {19}{2}} \sqrt {\frac {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}} - \frac {110 A a^{8} c^{\frac {21}{2}} x^{2} \sqrt {\frac {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}} - \frac {114 A a^{7} c^{\frac {23}{2}} x^{4} \sqrt {\frac {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}} - \frac {40 A a^{6} c^{\frac {25}{2}} x^{6} \sqrt {\frac {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}} - \frac {30 A a^{6} c^{\frac {11}{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 {5 A a^{5} c^{\frac {27}{2}} x^{8} \sqrt {\frac {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}} - \frac {66 A a^{5} c^{\frac {13}{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 {30 A a^{4} c^{\frac {29}{2}} x^{10} \sqrt {\frac {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}} - \frac {34 A a^{4} c^{\frac {15}{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 {40 A a^{3} c^{\frac {31}{2}} x^{12} \sqrt {\frac {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}} - \frac {6 A a^{3} c^{\frac {17}{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 {16 A a^{2} c^{\frac {33}{2}} x^{14} \sqrt {\frac {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}} - \frac {24 A a^{2} c^{\frac {19}{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 {16 A a c^{\frac {21}{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 {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a x^{2}} + \frac {2 A c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{2}} - \frac {B a^{3}}{8 \sqrt {c} x^{9} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {23 B a^{2} \sqrt {c}}{48 x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {127 B a c^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {133 B c^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 B c^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {5 B c^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{128 a^{\frac {3}{2}}} \end {gather*}

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))

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