3.9.59 \(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=303 \[ -\frac {\left (b^2-4 a c\right )^2 \left (2 a B \left (7 b^2-4 a c\right )-A \left (9 b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2048 a^{11/2}}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{840 a^3 x^5}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2} \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right )}{1024 a^5 x^2}+\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right )}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7} \]

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Rubi [A]  time = 0.39, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{840 a^3 x^5}+\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right )}{384 a^4 x^4}-\frac {\left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2} \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right )}{1024 a^5 x^2}-\frac {\left (b^2-4 a c\right )^2 \left (2 a B \left (7 b^2-4 a c\right )-A \left (9 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 )}{2048 a^{11/2}}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x]

[Out]

-((b^2 - 4*a*c)*(9*A*b^3 - 14*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(1024*a^5*x
^2) + ((9*A*b^3 - 14*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(384*a^4*x^4) - (A
*(a + b*x + c*x^2)^(5/2))/(7*a*x^7) + ((9*A*b - 14*a*B)*(a + b*x + c*x^2)^(5/2))/(84*a^2*x^6) - ((63*A*b^2 - 9
8*a*b*B - 48*a*A*c)*(a + b*x + c*x^2)^(5/2))/(840*a^3*x^5) - ((b^2 - 4*a*c)^2*(2*a*B*(7*b^2 - 4*a*c) - A*(9*b^
3 - 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2048*a^(11/2))

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 )^{3/2}}{x^8} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}-\frac {\int \frac {\left (\frac {1}{2} (9 A b-14 a B)+2 A c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx}{7 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}+\frac {\int \frac {\left (\frac {1}{4} \left (63 A b^2-98 a b B-48 a A c\right )+\frac {1}{2} (9 A b-14 a B) c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx}{42 a^2}\\ &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}-\frac {\left (9 A b^3-14 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 )^{3/2}}{x^5} \, dx}{48 a^3}\\ &=\frac {\left (9 A b^3-14 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 )^{3/2}}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}+\frac {\left (\left (b^2-4 a c\right ) \left (9 A b^3-14 a b^2 B-12 a A b c+8 a^2 B c\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{256 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a B \left (7 b^2-4 a c\right )-3 A \left (3 b^3-4 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{1024 a^5 x^2}+\frac {\left (9 A b^3-14 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 )^{3/2}}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (9 A b^3-14 a b^2 B-12 a A b c+8 a^2 B c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2048 a^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a B \left (7 b^2-4 a c\right )-3 A \left (3 b^3-4 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{1024 a^5 x^2}+\frac {\left (9 A b^3-14 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 )^{3/2}}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}+\frac {\left (\left (b^2-4 a c\right )^2 \left (9 A b^3-14 a b^2 B-12 a A b c+8 a^2 B c\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 )}{1024 a^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a B \left (7 b^2-4 a c\right )-3 A \left (3 b^3-4 a b c\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{1024 a^5 x^2}+\frac {\left (9 A b^3-14 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 )^{3/2}}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}-\frac {\left (b^2-4 a c\right )^2 \left (2 a B \left (7 b^2-4 a c\right )-3 A \left (3 b^3-4 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2048 a^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 234, normalized size = 0.77 \begin {gather*} \frac {\frac {7 \left (3 A \left (3 b^3-4 a b c\right )+2 a B \left (4 a c-7 b^2\right )\right ) \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)} \left (8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right )+3 x^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )}{512 a^{7/2} x^4}+\frac {(a+x (b+c x))^{5/2} \left (48 a A c+98 a b B-63 A b^2\right )}{10 a x^5}+\frac {(9 A b-14 a B) (a+x (b+c x))^{5/2}}{x^6}-\frac {12 a A (a+x (b+c x))^{5/2}}{x^7}}{84 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x]

[Out]

((-12*a*A*(a + x*(b + c*x))^(5/2))/x^7 + ((9*A*b - 14*a*B)*(a + x*(b + c*x))^(5/2))/x^6 + ((-63*A*b^2 + 98*a*b
*B + 48*a*A*c)*(a + x*(b + c*x))^(5/2))/(10*a*x^5) + (7*(2*a*B*(-7*b^2 + 4*a*c) + 3*A*(3*b^3 - 4*a*b*c))*(2*Sq
rt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^2 + 4*a*x*(2*b + 5*c*x)) + 3*(b^2 - 4*a*c)^2*x^4*ArcT
anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(512*a^(7/2)*x^4))/(84*a^2)

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IntegrateAlgebraic [A]  time = 4.66, size = 489, normalized size = 1.61 \begin {gather*} \frac {\left (-128 a^4 B c^3-9 A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{1024 a^{11/2}}+\frac {\left (-96 a^2 A b c^3-144 a^2 b^2 B c^2+120 a A b^3 c^2+60 a b^4 B c-42 A b^5 c-7 b^6 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{512 a^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (-15360 a^6 A-17920 a^6 B x-19200 a^5 A b x-24576 a^5 A c x^2-23296 a^5 b B x^2-31360 a^5 B c x^3-384 a^4 A b^2 x^2-2112 a^4 A b c x^3-3072 a^4 A c^2 x^4-672 a^4 b^2 B x^3-4032 a^4 b B c x^4-6720 a^4 B c^2 x^5+432 a^3 A b^3 x^3+2976 a^3 A b^2 c x^4+7008 a^3 A b c^2 x^5+6144 a^3 A c^3 x^6+784 a^3 b^3 B x^4+6048 a^3 b^2 B c x^5+18144 a^3 b B c^2 x^6-504 a^2 A b^4 x^4-4368 a^2 A b^3 c x^5-16464 a^2 A b^2 c^2 x^6-980 a^2 b^4 B x^5-10640 a^2 b^3 B c x^6+630 a A b^5 x^5+7560 a A b^4 c x^6+1470 a b^5 B x^6-945 A b^6 x^6\right )}{107520 a^5 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-15360*a^6*A - 19200*a^5*A*b*x - 17920*a^6*B*x - 384*a^4*A*b^2*x^2 - 23296*a^5*b*B*x^2
- 24576*a^5*A*c*x^2 + 432*a^3*A*b^3*x^3 - 672*a^4*b^2*B*x^3 - 2112*a^4*A*b*c*x^3 - 31360*a^5*B*c*x^3 - 504*a^
2*A*b^4*x^4 + 784*a^3*b^3*B*x^4 + 2976*a^3*A*b^2*c*x^4 - 4032*a^4*b*B*c*x^4 - 3072*a^4*A*c^2*x^4 + 630*a*A*b^5
*x^5 - 980*a^2*b^4*B*x^5 - 4368*a^2*A*b^3*c*x^5 + 6048*a^3*b^2*B*c*x^5 + 7008*a^3*A*b*c^2*x^5 - 6720*a^4*B*c^2
*x^5 - 945*A*b^6*x^6 + 1470*a*b^5*B*x^6 + 7560*a*A*b^4*c*x^6 - 10640*a^2*b^3*B*c*x^6 - 16464*a^2*A*b^2*c^2*x^6
+ 18144*a^3*b*B*c^2*x^6 + 6144*a^3*A*c^3*x^6))/(107520*a^5*x^7) + ((-9*A*b^7 - 128*a^4*B*c^3)*ArcTanh[(Sqrt[c
]*x - Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(1024*a^(11/2)) + ((-7*b^6*B - 42*A*b^5*c + 60*a*b^4*B*c + 120*a*A*b^3*
c^2 - 144*a^2*b^2*B*c^2 - 96*a^2*A*b*c^3)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(512*a^(9/2
))

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fricas [A]  time = 4.54, size = 889, normalized size = 2.93 \begin {gather*} \left [\frac {105 \, {\left (14 \, B a b^{6} - 9 \, A b^{7} - 64 \, {\left (2 \, B a^{4} - 3 \, A a^{3} b\right )} c^{3} + 48 \, {\left (6 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} c^{2} - 12 \, {\left (10 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} c\right )} \sqrt {a} x^{7} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (15360 \, A a^{7} - {\left (1470 \, B a^{2} b^{5} - 945 \, A a b^{6} + 6144 \, A a^{4} c^{3} + 336 \, {\left (54 \, B a^{4} b - 49 \, A a^{3} b^{2}\right )} c^{2} - 280 \, {\left (38 \, B a^{3} b^{3} - 27 \, A a^{2} b^{4}\right )} c\right )} x^{6} + 2 \, {\left (490 \, B a^{3} b^{4} - 315 \, A a^{2} b^{5} + 48 \, {\left (70 \, B a^{5} - 73 \, A a^{4} b\right )} c^{2} - 168 \, {\left (18 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} c\right )} x^{5} - 8 \, {\left (98 \, B a^{4} b^{3} - 63 \, A a^{3} b^{4} - 384 \, A a^{5} c^{2} - 12 \, {\left (42 \, B a^{5} b - 31 \, A a^{4} b^{2}\right )} c\right )} x^{4} + 16 \, {\left (42 \, B a^{5} b^{2} - 27 \, A a^{4} b^{3} + 4 \, {\left (490 \, B a^{6} + 33 \, A a^{5} b\right )} c\right )} x^{3} + 128 \, {\left (182 \, B a^{6} b + 3 \, A a^{5} b^{2} + 192 \, A a^{6} c\right )} x^{2} + 1280 \, {\left (14 \, B a^{7} + 15 \, A a^{6} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{430080 \, a^{6} x^{7}}, \frac {105 \, {\left (14 \, B a b^{6} - 9 \, A b^{7} - 64 \, {\left (2 \, B a^{4} - 3 \, A a^{3} b\right )} c^{3} + 48 \, {\left (6 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} c^{2} - 12 \, {\left (10 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} c\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (15360 \, A a^{7} - {\left (1470 \, B a^{2} b^{5} - 945 \, A a b^{6} + 6144 \, A a^{4} c^{3} + 336 \, {\left (54 \, B a^{4} b - 49 \, A a^{3} b^{2}\right )} c^{2} - 280 \, {\left (38 \, B a^{3} b^{3} - 27 \, A a^{2} b^{4}\right )} c\right )} x^{6} + 2 \, {\left (490 \, B a^{3} b^{4} - 315 \, A a^{2} b^{5} + 48 \, {\left (70 \, B a^{5} - 73 \, A a^{4} b\right )} c^{2} - 168 \, {\left (18 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} c\right )} x^{5} - 8 \, {\left (98 \, B a^{4} b^{3} - 63 \, A a^{3} b^{4} - 384 \, A a^{5} c^{2} - 12 \, {\left (42 \, B a^{5} b - 31 \, A a^{4} b^{2}\right )} c\right )} x^{4} + 16 \, {\left (42 \, B a^{5} b^{2} - 27 \, A a^{4} b^{3} + 4 \, {\left (490 \, B a^{6} + 33 \, A a^{5} b\right )} c\right )} x^{3} + 128 \, {\left (182 \, B a^{6} b + 3 \, A a^{5} b^{2} + 192 \, A a^{6} c\right )} x^{2} + 1280 \, {\left (14 \, B a^{7} + 15 \, A a^{6} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{215040 \, a^{6} x^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x, algorithm="fricas")

[Out]

[1/430080*(105*(14*B*a*b^6 - 9*A*b^7 - 64*(2*B*a^4 - 3*A*a^3*b)*c^3 + 48*(6*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 - 12*
(10*B*a^2*b^4 - 7*A*a*b^5)*c)*sqrt(a)*x^7*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*(15360*A*a^7 - (1470*B*a^2*b^5 - 945*A*a*b^6 + 6144*A*a^4*c^3 + 336*(54*B*a^4*b
- 49*A*a^3*b^2)*c^2 - 280*(38*B*a^3*b^3 - 27*A*a^2*b^4)*c)*x^6 + 2*(490*B*a^3*b^4 - 315*A*a^2*b^5 + 48*(70*B*a
^5 - 73*A*a^4*b)*c^2 - 168*(18*B*a^4*b^2 - 13*A*a^3*b^3)*c)*x^5 - 8*(98*B*a^4*b^3 - 63*A*a^3*b^4 - 384*A*a^5*c
^2 - 12*(42*B*a^5*b - 31*A*a^4*b^2)*c)*x^4 + 16*(42*B*a^5*b^2 - 27*A*a^4*b^3 + 4*(490*B*a^6 + 33*A*a^5*b)*c)*x
^3 + 128*(182*B*a^6*b + 3*A*a^5*b^2 + 192*A*a^6*c)*x^2 + 1280*(14*B*a^7 + 15*A*a^6*b)*x)*sqrt(c*x^2 + b*x + a)
)/(a^6*x^7), 1/215040*(105*(14*B*a*b^6 - 9*A*b^7 - 64*(2*B*a^4 - 3*A*a^3*b)*c^3 + 48*(6*B*a^3*b^2 - 5*A*a^2*b^
3)*c^2 - 12*(10*B*a^2*b^4 - 7*A*a*b^5)*c)*sqrt(-a)*x^7*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*(15360*A*a^7 - (1470*B*a^2*b^5 - 945*A*a*b^6 + 6144*A*a^4*c^3 + 336*(54*B*a^4*b -
49*A*a^3*b^2)*c^2 - 280*(38*B*a^3*b^3 - 27*A*a^2*b^4)*c)*x^6 + 2*(490*B*a^3*b^4 - 315*A*a^2*b^5 + 48*(70*B*a^5
- 73*A*a^4*b)*c^2 - 168*(18*B*a^4*b^2 - 13*A*a^3*b^3)*c)*x^5 - 8*(98*B*a^4*b^3 - 63*A*a^3*b^4 - 384*A*a^5*c^2
- 12*(42*B*a^5*b - 31*A*a^4*b^2)*c)*x^4 + 16*(42*B*a^5*b^2 - 27*A*a^4*b^3 + 4*(490*B*a^6 + 33*A*a^5*b)*c)*x^3
+ 128*(182*B*a^6*b + 3*A*a^5*b^2 + 192*A*a^6*c)*x^2 + 1280*(14*B*a^7 + 15*A*a^6*b)*x)*sqrt(c*x^2 + b*x + a))/
(a^6*x^7)]

________________________________________________________________________________________

giac [B]  time = 0.36, size = 2713, normalized size = 8.95

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)^(3/2)/x^8,x, algorithm="giac")

[Out]

1/1024*(14*B*a*b^6 - 9*A*b^7 - 120*B*a^2*b^4*c + 84*A*a*b^5*c + 288*B*a^3*b^2*c^2 - 240*A*a^2*b^3*c^2 - 128*B*
a^4*c^3 + 192*A*a^3*b*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/107520*(14
70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a*b^6 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*b^7 - 12600*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^2*b^4*c + 8820*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a*b^5*c + 30
240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^3*b^2*c^2 - 25200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^2*
b^3*c^2 - 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^4*c^3 + 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^1
3*A*a^3*b*c^3 - 9800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^2*b^6 + 6300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^11*A*a*b^7 + 84000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^3*b^4*c - 58800*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^11*A*a^2*b^5*c - 201600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^4*b^2*c^2 + 168000*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^11*A*a^3*b^3*c^2 - 197120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^5*c^3 - 134400*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^11*A*a^4*b*c^3 - 1075200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^5*b*c^(5/2) -
430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^5*c^(7/2) + 27734*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a
^3*b^6 - 17829*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b^7 - 237720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*
B*a^4*b^4*c + 166404*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3*b^5*c - 1192800*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^9*B*a^5*b^2*c^2 - 475440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*b^3*c^2 - 138880*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^9*B*a^6*c^3 - 1512000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^5*b*c^3 - 1576960*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^8*B*a^5*b^3*c^(3/2) + 215040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^6*b*c^(5/2)
- 3655680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^5*b^2*c^(5/2) - 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^8*A*a^6*c^(7/2) - 43008*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*b^6 + 27648*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*A*a^3*b^7 - 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*b^4*c - 258048*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*A*a^4*b^5*c - 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^6*b^2*c^2 - 3225600*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^7*A*a^5*b^3*c^2 - 2580480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^6*b*c^3 - 215040*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^5*b^5*sqrt(c) + 716800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^6*b^
3*c^(3/2) - 2580480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^5*b^4*c^(3/2) - 430080*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*B*a^7*b*c^(5/2) - 3440640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^6*b^2*c^(5/2) - 860160*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^7*c^(7/2) + 15274*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^5*b^6 - 25179*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b^7 + 237720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^6*b^4*c - 7
68516*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^5*b^5*c + 977760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^7*b
^2*c^2 - 3610320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^6*b^3*c^2 + 138880*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*B*a^8*c^3 - 1928640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^7*b*c^3 + 215040*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*B*a^6*b^5*sqrt(c) - 215040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*b^6*sqrt(c) + 430080*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^7*b^3*c^(3/2) - 1290240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^6*b^4*c
^(3/2) + 1118208*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^8*b*c^(5/2) - 2838528*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*A*a^7*b^2*c^(5/2) - 172032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^8*c^(7/2) + 9800*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*B*a^6*b^6 - 6300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^5*b^7 + 346080*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*B*a^7*b^4*c - 371280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^6*b^5*c + 631680*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^8*b^2*c^2 - 1243200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^7*b^3*c^2 +
197120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^9*c^3 - 725760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^8*b
*c^3 + 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^8*b^3*c^(3/2) - 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*A*a^7*b^4*c^(3/2) + 129024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^9*b*c^(5/2) - 344064*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*A*a^8*b^2*c^(5/2) - 86016*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^9*c^(7/2) - 1470*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^7*b^6 + 945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b^7 + 12600*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*B*a^8*b^4*c - 8820*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^7*b^5*c + 184800*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^9*b^2*c^2 - 189840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^8*b^3*c^2 + 1
3440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^10*c^3 - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^9*b*c^3 +
43008*B*a^10*b*c^(5/2) - 43008*A*a^9*b^2*c^(5/2) + 12288*A*a^10*c^(7/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2 - a)^7*a^5)

________________________________________________________________________________________

maple [B]  time = 0.08, size = 1575, normalized size = 5.20

result 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)^(3/2)/x^8,x)

[Out]

-5/128*A/a^6*b^4*c/x*(c*x^2+b*x+a)^(5/2)-1/16*B/a^4*b^3*c^2*(c*x^2+b*x+a)^(1/2)*x+11/192*B/a^5*b^3*c/x*(c*x^2+
b*x+a)^(5/2)+1/64*A/a^5*b^3*c/x^2*(c*x^2+b*x+a)^(5/2)-1/32*A/a^4*b*c^2/x^2*(c*x^2+b*x+a)^(5/2)-1/16*A/a^3*b*c/
x^4*(c*x^2+b*x+a)^(5/2)-3/64*A/a^5*b^2*c^3*(c*x^2+b*x+a)^(3/2)*x+3/64*A/a^5*b^2*c^2/x*(c*x^2+b*x+a)^(5/2)-3/64
*A/a^4*b^2*c^3*(c*x^2+b*x+a)^(1/2)*x-1/48*B*c/a^3*b/x^3*(c*x^2+b*x+a)^(5/2)+1/32*B*c^3/a^3*b*(c*x^2+b*x+a)^(1/
2)*x-1/32*B*c^2/a^4*b/x*(c*x^2+b*x+a)^(5/2)+1/32*B*c^3/a^4*b*(c*x^2+b*x+a)^(3/2)*x-1/32*B/a^4*b^2*c/x^2*(c*x^2
+b*x+a)^(5/2)-11/192*B/a^5*b^3*c^2*(c*x^2+b*x+a)^(3/2)*x+7/1536*B/a^6*b^5*c*(c*x^2+b*x+a)^(3/2)*x+7/512*B/a^5*
b^5*c*(c*x^2+b*x+a)^(1/2)*x+1/32*A/a^4*b^2*c/x^3*(c*x^2+b*x+a)^(5/2)+5/128*A/a^6*b^4*c^2*(c*x^2+b*x+a)^(3/2)*x
-3/1024*A/a^7*b^6*c*(c*x^2+b*x+a)^(3/2)*x-9/1024*A/a^6*b^6*c*(c*x^2+b*x+a)^(1/2)*x+3/64*A/a^5*b^4*c^2*(c*x^2+b
*x+a)^(1/2)*x+1/16*B*c^3/a^(3/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+7/1536*B/a^6*b^6*(c*x^2+b*x+a)^
(3/2)+7/512*B/a^5*b^6*(c*x^2+b*x+a)^(1/2)-1/48*B*c^3/a^3*(c*x^2+b*x+a)^(3/2)-1/16*B*c^3/a^2*(c*x^2+b*x+a)^(1/2
)-1/6*B/a/x^6*(c*x^2+b*x+a)^(5/2)-7/1024*B/a^(9/2)*b^6*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-3/1024*A/
a^7*b^7*(c*x^2+b*x+a)^(3/2)-9/1024*A/a^6*b^7*(c*x^2+b*x+a)^(1/2)+9/2048*A/a^(11/2)*b^7*ln((b*x+2*a+2*(c*x^2+b*
x+a)^(1/2)*a^(1/2))/x)-7/96*B/a^3*b^2/x^4*(c*x^2+b*x+a)^(5/2)+7/192*B/a^4*b^3/x^3*(c*x^2+b*x+a)^(5/2)-7/768*B/
a^5*b^4/x^2*(c*x^2+b*x+a)^(5/2)-9/64*B/a^(5/2)*b^2*c^2*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+15/256*B/
a^(7/2)*b^4*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-7/1536*B/a^6*b^5/x*(c*x^2+b*x+a)^(5/2)-37/768*B/a^
5*b^4*c*(c*x^2+b*x+a)^(3/2)-23/256*B/a^4*b^4*c*(c*x^2+b*x+a)^(1/2)+1/48*B*c^2/a^3/x^2*(c*x^2+b*x+a)^(5/2)+1/24
*B*c/a^2/x^4*(c*x^2+b*x+a)^(5/2)+3/512*A/a^6*b^5/x^2*(c*x^2+b*x+a)^(5/2)+1/32*A/a^4*b*c^3*(c*x^2+b*x+a)^(3/2)+
3/32*A/a^3*b*c^3*(c*x^2+b*x+a)^(1/2)+3/28*A/a^2*b/x^6*(c*x^2+b*x+a)^(5/2)+3/1024*A/a^7*b^6/x*(c*x^2+b*x+a)^(5/
2)+17/512*A/a^6*b^5*c*(c*x^2+b*x+a)^(3/2)+33/512*A/a^5*b^5*c*(c*x^2+b*x+a)^(1/2)-3/32*A/a^(5/2)*b*c^3*ln((b*x+
2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+15/128*A/a^(7/2)*b^3*c^2*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-2
1/512*A/a^(9/2)*b^5*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+7/60*B/a^2*b/x^5*(c*x^2+b*x+a)^(5/2)+1/16*
B/a^4*b^2*c^2*(c*x^2+b*x+a)^(3/2)+5/32*B/a^3*b^2*c^2*(c*x^2+b*x+a)^(1/2)+2/35*A*c/a^2/x^5*(c*x^2+b*x+a)^(5/2)-
3/40*A/a^3*b^2/x^5*(c*x^2+b*x+a)^(5/2)-1/16*A/a^5*b^3*c^2*(c*x^2+b*x+a)^(3/2)-9/64*A/a^4*b^3*c^2*(c*x^2+b*x+a)
^(1/2)+3/64*A/a^4*b^3/x^4*(c*x^2+b*x+a)^(5/2)-3/128*A/a^5*b^4/x^3*(c*x^2+b*x+a)^(5/2)-1/7*A*(c*x^2+b*x+a)^(5/2
)/a/x^7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,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 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**8,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**8, x)

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