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

Optimal. Leaf size=375 \[ \frac {5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{65536 a^{13/2}}-\frac {5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt {a+b x+c x^2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{12288 a^5 x^4}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{2016 a^3 x^7}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}+\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right )}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9} \]

[Out]

5/12288*(-4*a*c+b^2)*(2*a*B*(-4*a*c+9*b^2)-A*(-12*a*b*c+11*b^3))*(b*x+2*a)*(c*x^2+b*x+a)^(3/2)/a^5/x^4+1/768*(
-12*A*a*b*c+11*A*b^3+8*B*a^2*c-18*B*a*b^2)*(b*x+2*a)*(c*x^2+b*x+a)^(5/2)/a^4/x^6-1/9*A*(c*x^2+b*x+a)^(7/2)/a/x
^9+1/144*(11*A*b-18*B*a)*(c*x^2+b*x+a)^(7/2)/a^2/x^8-1/2016*(-64*A*a*c+99*A*b^2-162*B*a*b)*(c*x^2+b*x+a)^(7/2)
/a^3/x^7+5/65536*(-4*a*c+b^2)^3*(2*a*B*(-4*a*c+9*b^2)-A*(-12*a*b*c+11*b^3))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x
^2+b*x+a)^(1/2))/a^(13/2)-5/32768*(-4*a*c+b^2)^2*(2*a*B*(-4*a*c+9*b^2)-A*(-12*a*b*c+11*b^3))*(b*x+2*a)*(c*x^2+
b*x+a)^(1/2)/a^6/x^2

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Rubi [A]  time = 0.48, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {834, 806, 720, 724, 206} \[ -\frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{2016 a^3 x^7}+\frac {(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right )}{768 a^4 x^6}+\frac {5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{12288 a^5 x^4}-\frac {5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt {a+b x+c x^2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{65536 a^{13/2}}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b^2 - 4*a*c)^2*(2*a*B*(9*b^2 - 4*a*c) - A*(11*b^3 - 12*a*b*c))*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(32768*
a^6*x^2) + (5*(b^2 - 4*a*c)*(2*a*B*(9*b^2 - 4*a*c) - A*(11*b^3 - 12*a*b*c))*(2*a + b*x)*(a + b*x + c*x^2)^(3/2
))/(12288*a^5*x^4) + ((11*A*b^3 - 18*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*(a + b*x + c*x^2)^(5/2))/(7
68*a^4*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(9*a*x^9) + ((11*A*b - 18*a*B)*(a + b*x + c*x^2)^(7/2))/(144*a^2*x^8
) - ((99*A*b^2 - 162*a*b*B - 64*a*A*c)*(a + b*x + c*x^2)^(7/2))/(2016*a^3*x^7) + (5*(b^2 - 4*a*c)^3*(2*a*B*(9*
b^2 - 4*a*c) - A*(11*b^3 - 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(65536*a^(13/2))

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])

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}-\frac {\int \frac {\left (\frac {1}{2} (11 A b-18 a B)+2 A c x\right ) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx}{9 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}+\frac {\int \frac {\left (\frac {1}{4} \left (99 A b^2-162 a b B-64 a A c\right )+\frac {1}{2} (11 A b-18 a B) c x\right ) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx}{72 a^2}\\ &=-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}-\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) \int \frac {\left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx}{64 a^3}\\ &=\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}+\frac {\left (5 \left (b^2-4 a c\right ) \left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx}{1536 a^4}\\ &=\frac {5 \left (b^2-4 a c\right ) \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{12288 a^5 x^4}+\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{8192 a^5}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right ) \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{12288 a^5 x^4}+\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{65536 a^6}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right ) \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{12288 a^5 x^4}+\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}+\frac {\left (5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{32768 a^6}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{32768 a^6 x^2}+\frac {5 \left (b^2-4 a c\right ) \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{12288 a^5 x^4}+\frac {\left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{768 a^4 x^6}-\frac {A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9}+\frac {(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac {\left (99 A b^2-162 a b B-64 a A c\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 a^3 x^7}-\frac {5 \left (b^2-4 a c\right )^3 \left (11 A b^3-18 a b^2 B-12 a A b c+8 a^2 B c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{65536 a^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 0.73, size = 292, normalized size = 0.78 \[ \frac {\frac {3 \left (A \left (11 b^3-12 a b c\right )+2 a B \left (4 a c-9 b^2\right )\right ) \left (256 a^{5/2} (2 a+b x) (a+x (b+c x))^{5/2}-5 x^2 \left (b^2-4 a c\right ) \left (16 a^{3/2} (2 a+b x) (a+x (b+c x))^{3/2}-3 x^2 \left (b^2-4 a c\right ) \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}-x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{65536 a^{11/2} x^6}+\frac {(a+x (b+c x))^{7/2} \left (64 a A c+162 a b B-99 A b^2\right )}{224 a^2 x^7}+\frac {(11 A b-18 a B) (a+x (b+c x))^{7/2}}{16 a x^8}-\frac {A (a+x (b+c x))^{7/2}}{x^9}}{9 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((A*(a + x*(b + c*x))^(7/2))/x^9) + ((11*A*b - 18*a*B)*(a + x*(b + c*x))^(7/2))/(16*a*x^8) + ((-99*A*b^2 + 1
62*a*b*B + 64*a*A*c)*(a + x*(b + c*x))^(7/2))/(224*a^2*x^7) + (3*(2*a*B*(-9*b^2 + 4*a*c) + A*(11*b^3 - 12*a*b*
c))*(256*a^(5/2)*(2*a + b*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*x^2*(16*a^(3/2)*(2*a + b*x)*(a + x*(b +
c*x))^(3/2) - 3*(b^2 - 4*a*c)*x^2*(2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*x^2*ArcTanh[(2
*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))))/(65536*a^(11/2)*x^6))/(9*a)

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fricas [A]  time = 23.88, size = 1315, normalized size = 3.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8257536*(315*(18*B*a*b^8 - 11*A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 256*(6*B*a^4*b^2 - 5*A*a^3*b^3)*c^3 +
96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*c^2 - 16*(14*B*a^2*b^6 - 9*A*a*b^7)*c)*sqrt(a)*x^9*log(-(8*a*b*x + (b^2 + 4*a
*c)*x^2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(229376*A*a^9 + (5670*B*a^2*b^7 - 3465
*A*a*b^8 - 65536*A*a^5*c^4 - 576*(442*B*a^5*b - 407*A*a^4*b^2)*c^3 + 336*(674*B*a^4*b^3 - 483*A*a^3*b^4)*c^2 -
420*(150*B*a^3*b^5 - 97*A*a^2*b^6)*c)*x^8 - 2*(1890*B*a^3*b^6 - 1155*A*a^2*b^7 - 64*(630*B*a^6 - 689*A*a^5*b)
*c^3 + 144*(398*B*a^5*b^2 - 293*A*a^4*b^3)*c^2 - 84*(226*B*a^4*b^4 - 147*A*a^3*b^5)*c)*x^7 + 8*(378*B*a^4*b^5
- 231*A*a^3*b^6 + 4096*A*a^6*c^3 + 48*(174*B*a^6*b - 133*A*a^5*b^2)*c^2 - 24*(142*B*a^5*b^3 - 93*A*a^4*b^4)*c)
*x^6 - 16*(162*B*a^5*b^4 - 99*A*a^4*b^5 - 48*(826*B*a^7 + 41*A*a^6*b)*c^2 - 8*(162*B*a^6*b^2 - 107*A*a^5*b^3)*
c)*x^5 + 128*(18*B*a^6*b^3 - 11*A*a^5*b^4 + 3840*A*a^7*c^2 + 12*(614*B*a^7*b + 7*A*a^6*b^2)*c)*x^4 + 256*(1458
*B*a^7*b^2 + 5*A*a^6*b^3 + 12*(238*B*a^8 + 251*A*a^7*b)*c)*x^3 + 1024*(594*B*a^8*b + 309*A*a^7*b^2 + 608*A*a^8
*c)*x^2 + 14336*(18*B*a^9 + 37*A*a^8*b)*x)*sqrt(c*x^2 + b*x + a))/(a^7*x^9), -1/4128768*(315*(18*B*a*b^8 - 11*
A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 256*(6*B*a^4*b^2 - 5*A*a^3*b^3)*c^3 + 96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*
c^2 - 16*(14*B*a^2*b^6 - 9*A*a*b^7)*c)*sqrt(-a)*x^9*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c
*x^2 + a*b*x + a^2)) + 2*(229376*A*a^9 + (5670*B*a^2*b^7 - 3465*A*a*b^8 - 65536*A*a^5*c^4 - 576*(442*B*a^5*b -
407*A*a^4*b^2)*c^3 + 336*(674*B*a^4*b^3 - 483*A*a^3*b^4)*c^2 - 420*(150*B*a^3*b^5 - 97*A*a^2*b^6)*c)*x^8 - 2*
(1890*B*a^3*b^6 - 1155*A*a^2*b^7 - 64*(630*B*a^6 - 689*A*a^5*b)*c^3 + 144*(398*B*a^5*b^2 - 293*A*a^4*b^3)*c^2
- 84*(226*B*a^4*b^4 - 147*A*a^3*b^5)*c)*x^7 + 8*(378*B*a^4*b^5 - 231*A*a^3*b^6 + 4096*A*a^6*c^3 + 48*(174*B*a^
6*b - 133*A*a^5*b^2)*c^2 - 24*(142*B*a^5*b^3 - 93*A*a^4*b^4)*c)*x^6 - 16*(162*B*a^5*b^4 - 99*A*a^4*b^5 - 48*(8
26*B*a^7 + 41*A*a^6*b)*c^2 - 8*(162*B*a^6*b^2 - 107*A*a^5*b^3)*c)*x^5 + 128*(18*B*a^6*b^3 - 11*A*a^5*b^4 + 384
0*A*a^7*c^2 + 12*(614*B*a^7*b + 7*A*a^6*b^2)*c)*x^4 + 256*(1458*B*a^7*b^2 + 5*A*a^6*b^3 + 12*(238*B*a^8 + 251*
A*a^7*b)*c)*x^3 + 1024*(594*B*a^8*b + 309*A*a^7*b^2 + 608*A*a^8*c)*x^2 + 14336*(18*B*a^9 + 37*A*a^8*b)*x)*sqrt
(c*x^2 + b*x + a))/(a^7*x^9)]

________________________________________________________________________________________

giac [B]  time = 0.58, size = 4427, normalized size = 11.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/32768*(18*B*a*b^8 - 11*A*b^9 - 224*B*a^2*b^6*c + 144*A*a*b^7*c + 960*B*a^3*b^4*c^2 - 672*A*a^2*b^5*c^2 - 15
36*B*a^4*b^2*c^3 + 1280*A*a^3*b^3*c^3 + 512*B*a^5*c^4 - 768*A*a^4*b*c^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))/sqrt(-a))/(sqrt(-a)*a^6) + 1/2064384*(5670*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a*b^8 - 3465*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^17*A*b^9 - 70560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a^2*b^6*c + 45360*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^17*A*a*b^7*c + 302400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a^3*b^4*c^2 - 2
11680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*A*a^2*b^5*c^2 - 483840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*B*a
^4*b^2*c^3 + 403200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*A*a^3*b^3*c^3 + 161280*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^17*B*a^5*c^4 - 241920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^17*A*a^4*b*c^4 - 49140*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^15*B*a^2*b^8 + 30030*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a*b^9 + 611520*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^15*B*a^3*b^6*c - 393120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^2*b^7*c - 2620800*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^15*B*a^4*b^4*c^2 + 1834560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^3*b^5*c^2 +
4193280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^5*b^2*c^3 - 3494400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15
*A*a^4*b^3*c^3 + 4107264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^6*c^4 + 2096640*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^15*A*a^5*b*c^4 + 28901376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*B*a^6*b*c^(7/2) + 8257536*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^14*A*a^6*c^(9/2) + 188244*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^3*b^8 - 115038
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^2*b^9 - 2342592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^4*b^6*c
+ 1505952*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^3*b^7*c + 10039680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^1
3*B*a^5*b^4*c^2 - 7027776*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^4*b^5*c^2 + 54125568*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^13*B*a^6*b^2*c^3 + 13386240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^5*b^3*c^3 + 5354496*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^7*c^4 + 41513472*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^6*b*c^4 +
100466688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*B*a^6*b^3*c^(5/2) + 12386304*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^12*B*a^7*b*c^(7/2) + 136249344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*A*a^6*b^2*c^(7/2) + 13762560*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^12*A*a^7*c^(9/2) - 417636*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^4*b^8 + 2552
22*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^3*b^9 + 5197248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^5*b^6
*c - 3341088*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^4*b^7*c + 69148800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^11*B*a^6*b^4*c^2 + 15591744*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^5*b^5*c^2 + 71027712*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^11*B*a^7*b^2*c^3 + 192665088*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^6*b^3*c^3 + 9354240
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^8*c^4 + 109831680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^7*b*c
^4 + 53673984*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^6*b^5*c^(3/2) + 37158912*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^10*B*a^7*b^3*c^(5/2) + 235339776*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^6*b^4*c^(5/2) + 61931520*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^8*b*c^(7/2) + 247726080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^7
*b^2*c^(7/2) + 41287680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^8*c^(9/2) + 589824*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^9*B*a^5*b^8 - 360448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*b^9 + 12386304*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^9*B*a^6*b^6*c + 4718592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^5*b^7*c + 41287680*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^9*B*a^7*b^4*c^2 + 144506880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^6*b^5*c^2 +
20643840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^8*b^2*c^3 + 405995520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
9*A*a^7*b^3*c^3 + 165150720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^8*b*c^4 + 4128768*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^8*B*a^6*b^7*sqrt(c) - 12386304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^7*b^5*c^(3/2) + 7844659
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^6*b^6*c^(3/2) + 24772608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a
^8*b^3*c^(5/2) + 322043904*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^7*b^4*c^(5/2) - 53673984*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^8*B*a^9*b*c^(7/2) + 383975424*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^8*b^2*c^(7/2) + 24
772608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^9*c^(9/2) - 172188*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^
6*b^8 + 334602*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^5*b^9 - 5197248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7
*B*a^7*b^6*c + 19266336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^6*b^7*c - 48504960*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^7*B*a^8*b^4*c^2 + 194975424*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^7*b^5*c^2 - 54512640*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^7*B*a^9*b^2*c^3 + 389491200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^8*b^3*c^3 -
9354240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^10*c^4 + 137894400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a
^9*b*c^4 - 4128768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^7*b^7*sqrt(c) + 4128768*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*A*a^6*b^8*sqrt(c) - 20643840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^8*b^5*c^(3/2) + 55050240*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^7*b^6*c^(3/2) - 112852992*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^9
*b^3*c^(5/2) + 280756224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^8*b^4*c^(5/2) - 12386304*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^6*B*a^10*b*c^(7/2) + 198180864*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^9*b^2*c^(7/2) + 247
72608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^10*c^(9/2) - 188244*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^
7*b^8 + 115038*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^6*b^9 - 10043712*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
5*B*a^8*b^6*c + 10880352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^7*b^7*c - 51327360*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*B*a^9*b^4*c^2 + 93731904*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^8*b^5*c^2 - 70640640*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^5*B*a^10*b^2*c^3 + 193052160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^9*b^3*c^3 -
5354496*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^11*c^4 + 57576960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a
^10*b*c^4 - 20643840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^9*b^5*c^(3/2) + 20643840*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^4*A*a^8*b^6*c^(3/2) - 37158912*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^10*b^3*c^(5/2) + 743178
24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^9*b^4*c^(5/2) - 33619968*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*
a^11*b*c^(7/2) + 83165184*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^10*b^2*c^(7/2) + 3538944*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^4*A*a^11*c^(9/2) + 49140*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^8*b^8 - 30030*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*A*a^7*b^9 - 611520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^9*b^6*c + 393120*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^8*b^7*c - 18023040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^10*b^4*c^2
+ 18809280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^9*b^5*c^2 - 20708352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*a^11*b^2*c^3 + 37900800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^10*b^3*c^3 - 4107264*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*B*a^12*c^4 + 14418432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^11*b*c^4 - 12386304*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^2*B*a^11*b^3*c^(5/2) + 12386304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^10*b^4*
c^(5/2) - 2949120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^12*b*c^(7/2) + 7077888*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*A*a^11*b^2*c^(7/2) + 1179648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^12*c^(9/2) - 5670*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^9*b^8 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^8*b^9 + 70560*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*B*a^10*b^6*c - 45360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^9*b^7*c - 302400*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*B*a^11*b^4*c^2 + 211680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^10*b^5*c^2 - 364492
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^12*b^2*c^3 + 3725568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^11*b^3*
c^3 - 161280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^13*c^4 + 241920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^1
2*b*c^4 - 589824*B*a^13*b*c^(7/2) + 589824*A*a^12*b^2*c^(7/2) - 131072*A*a^13*c^(9/2))/(((sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2 - a)^9*a^6)

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maple [B]  time = 0.16, size = 2677, normalized size = 7.14 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8192*B/a^7*b^6/x^2*(c*x^2+b*x+a)^(7/2)+1/128*B*c^3/a^4/x^2*(c*x^2+b*x+a)^(7/2)+5/256*B*c^4/a^4*b*(c*x^2+b*x+
a)^(3/2)*x-45/16384*B/a^6*b^7*(c*x^2+b*x+a)^(1/2)*x*c+5/256*B*c^4/a^3*b*(c*x^2+b*x+a)^(1/2)*x+13/768*B/a^5*b^3
*c/x^3*(c*x^2+b*x+a)^(7/2)+5/256*B*c^4/a^5*b*(c*x^2+b*x+a)^(5/2)*x-145/3072*B/a^6*b^3*c^3*(c*x^2+b*x+a)^(5/2)*
x-55/1024*B/a^4*b^3*c^3*(c*x^2+b*x+a)^(1/2)*x-55/6144*B/a^6*b^4*c/x^2*(c*x^2+b*x+a)^(7/2)-155/3072*B/a^5*b^3*c
^3*(c*x^2+b*x+a)^(3/2)*x+145/3072*B/a^6*b^3*c^2/x*(c*x^2+b*x+a)^(7/2)+95/4096*B/a^5*b^5*c^2*(c*x^2+b*x+a)^(1/2
)*x-5/256*B*c^3/a^5*b/x*(c*x^2+b*x+a)^(7/2)+55/98304*A/a^8*b^8*c*(c*x^2+b*x+a)^(3/2)*x+119/24576*A/a^8*b^6*c/x
*(c*x^2+b*x+a)^(7/2)-3/256*A/a^5*b*c^3/x^2*(c*x^2+b*x+a)^(7/2)-1/128*A/a^4*b*c^2/x^4*(c*x^2+b*x+a)^(7/2)-1/32*
A/a^3*b*c/x^6*(c*x^2+b*x+a)^(7/2)+3/256*A/a^5*b^2*c^2/x^3*(c*x^2+b*x+a)^(7/2)-15/512*A/a^4*b^2*c^4*(c*x^2+b*x+
a)^(1/2)*x+15/512*A/a^6*b^2*c^3/x*(c*x^2+b*x+a)^(7/2)-15/512*A/a^6*b^2*c^4*(c*x^2+b*x+a)^(5/2)*x-15/512*A/a^5*
b^2*c^4*(c*x^2+b*x+a)^(3/2)*x+55/32768*A/a^7*b^8*(c*x^2+b*x+a)^(1/2)*x*c-5/512*A/a^6*b^4*c/x^3*(c*x^2+b*x+a)^(
7/2)+85/2048*A/a^5*b^4*c^3*(c*x^2+b*x+a)^(1/2)*x-65/2048*A/a^7*b^4*c^2/x*(c*x^2+b*x+a)^(7/2)+65/2048*A/a^7*b^4
*c^3*(c*x^2+b*x+a)^(5/2)*x+75/2048*A/a^6*b^4*c^3*(c*x^2+b*x+a)^(3/2)*x+23/4096*A/a^7*b^5*c/x^2*(c*x^2+b*x+a)^(
7/2)-125/8192*A/a^6*b^6*c^2*(c*x^2+b*x+a)^(1/2)*x+1/64*A/a^4*b^2*c/x^5*(c*x^2+b*x+a)^(7/2)+5/1024*A/a^6*b^3*c^
2/x^2*(c*x^2+b*x+a)^(7/2)+1/768*A/a^5*b^3*c/x^4*(c*x^2+b*x+a)^(7/2)-119/24576*A/a^8*b^6*c^2*(c*x^2+b*x+a)^(5/2
)*x-235/24576*A/a^7*b^6*c^2*(c*x^2+b*x+a)^(3/2)*x+11/32768*A/a^9*b^8*c*(c*x^2+b*x+a)^(5/2)*x-1/128*B*c^2/a^4*b
/x^3*(c*x^2+b*x+a)^(7/2)-31/4096*B/a^7*b^5*c/x*(c*x^2+b*x+a)^(7/2)-15/16384*B/a^7*b^7*c*(c*x^2+b*x+a)^(3/2)*x+
185/12288*B/a^6*b^5*c^2*(c*x^2+b*x+a)^(3/2)*x-9/16384*B/a^8*b^7*c*(c*x^2+b*x+a)^(5/2)*x+31/4096*B/a^7*b^5*c^2*
(c*x^2+b*x+a)^(5/2)*x-7/512*B/a^5*b^2*c^2/x^2*(c*x^2+b*x+a)^(7/2)-1/128*B/a^4*b^2*c/x^4*(c*x^2+b*x+a)^(7/2)-1/
96*B*c/a^3*b/x^5*(c*x^2+b*x+a)^(7/2)-15/16384*B/a^7*b^8*(c*x^2+b*x+a)^(3/2)-9/16384*B/a^8*b^8*(c*x^2+b*x+a)^(5
/2)-45/16384*B/a^6*b^8*(c*x^2+b*x+a)^(1/2)-5/384*B*c^4/a^3*(c*x^2+b*x+a)^(3/2)-1/128*B*c^4/a^4*(c*x^2+b*x+a)^(
5/2)-5/128*B*c^4/a^2*(c*x^2+b*x+a)^(1/2)-1/8*B/a/x^8*(c*x^2+b*x+a)^(7/2)+45/32768*B/a^(11/2)*b^8*ln((b*x+2*a+2
*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+5/128*B*c^4/a^(3/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+55/98304*A/
a^8*b^9*(c*x^2+b*x+a)^(3/2)+11/32768*A/a^9*b^9*(c*x^2+b*x+a)^(5/2)+55/32768*A/a^7*b^9*(c*x^2+b*x+a)^(1/2)-55/6
5536*A/a^(13/2)*b^9*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-11/224*A/a^3*b^2/x^7*(c*x^2+b*x+a)^(7/2)-145
/3072*A/a^5*b^3*c^3*(c*x^2+b*x+a)^(3/2)-35/1024*A/a^6*b^3*c^3*(c*x^2+b*x+a)^(5/2)-115/1024*A/a^4*b^3*c^3*(c*x^
2+b*x+a)^(1/2)-227/49152*A/a^8*b^7*c*(c*x^2+b*x+a)^(5/2)-11/32768*A/a^9*b^8/x*(c*x^2+b*x+a)^(7/2)+11/2048*A/a^
6*b^5/x^4*(c*x^2+b*x+a)^(7/2)+145/4096*A/a^6*b^5*c^2*(c*x^2+b*x+a)^(3/2)+107/4096*A/a^7*b^5*c^2*(c*x^2+b*x+a)^
(5/2)+295/4096*A/a^5*b^5*c^2*(c*x^2+b*x+a)^(1/2)-11/768*A/a^5*b^4/x^5*(c*x^2+b*x+a)^(7/2)+11/384*A/a^4*b^3/x^6
*(c*x^2+b*x+a)^(7/2)-11/12288*A/a^7*b^6/x^3*(c*x^2+b*x+a)^(7/2)-305/16384*A/a^6*b^7*c*(c*x^2+b*x+a)^(1/2)-415/
49152*A/a^7*b^7*c*(c*x^2+b*x+a)^(3/2)-11/49152*A/a^8*b^7/x^2*(c*x^2+b*x+a)^(7/2)+2/63*A*c/a^2/x^7*(c*x^2+b*x+a
)^(7/2)+25/256*A/a^(7/2)*b^3*c^3*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+45/4096*A/a^(11/2)*b^7*c*ln((b*
x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-105/2048*A/a^(9/2)*b^5*c^2*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/
x)-15/256*A/a^(5/2)*b*c^4*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+9/112*B/a^2*b/x^7*(c*x^2+b*x+a)^(7/2)+
25/512*B/a^4*b^2*c^3*(c*x^2+b*x+a)^(3/2)+17/512*B/a^5*b^2*c^3*(c*x^2+b*x+a)^(5/2)+1/192*B*c^2/a^3/x^4*(c*x^2+b
*x+a)^(7/2)+1/48*B*c/a^2/x^6*(c*x^2+b*x+a)^(7/2)-15/128*B/a^(5/2)*b^2*c^3*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^
(1/2))/x)-35/2048*B/a^(9/2)*b^6*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+75/1024*B/a^(7/2)*b^4*c^2*ln((
b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+65/512*B/a^3*b^2*c^3*(c*x^2+b*x+a)^(1/2)+59/8192*B/a^7*b^6*c*(c*x^2+
b*x+a)^(5/2)+9/16384*B/a^8*b^7/x*(c*x^2+b*x+a)^(7/2)-9/1024*B/a^5*b^4/x^4*(c*x^2+b*x+a)^(7/2)-305/6144*B/a^5*b
^4*c^2*(c*x^2+b*x+a)^(3/2)-235/6144*B/a^6*b^4*c^2*(c*x^2+b*x+a)^(5/2)-205/2048*B/a^4*b^4*c^2*(c*x^2+b*x+a)^(1/
2)+3/128*B/a^4*b^3/x^5*(c*x^2+b*x+a)^(7/2)-3/64*B/a^3*b^2/x^6*(c*x^2+b*x+a)^(7/2)+3/2048*B/a^6*b^5/x^3*(c*x^2+
b*x+a)^(7/2)+235/8192*B/a^5*b^6*c*(c*x^2+b*x+a)^(1/2)+325/24576*B/a^6*b^6*c*(c*x^2+b*x+a)^(3/2)+5/256*A/a^4*b*
c^4*(c*x^2+b*x+a)^(3/2)+3/256*A/a^5*b*c^4*(c*x^2+b*x+a)^(5/2)+15/256*A/a^3*b*c^4*(c*x^2+b*x+a)^(1/2)+11/144*A/
a^2*b/x^8*(c*x^2+b*x+a)^(7/2)-1/9*A*(c*x^2+b*x+a)^(7/2)/a/x^9

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^{10}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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