Integrand size = 23, antiderivative size = 306 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}-\frac {\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}} \]
-1/256*(2*a*B*(48*a^2*c^2-120*a*b^2*c+35*b^4)-A*(240*a^2*b*c^2-280*a*b^3*c +63*b^5))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(11/2)-1/5* A*(c*x^2+b*x+a)^(1/2)/a/x^5+1/40*(9*A*b-10*B*a)*(c*x^2+b*x+a)^(1/2)/a^2/x^ 4-1/240*(-64*A*a*c+63*A*b^2-70*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^3/x^3+1/960*(- 644*A*a*b*c+315*A*b^3+360*B*a^2*c-350*B*a*b^2)*(c*x^2+b*x+a)^(1/2)/a^4/x^2 +1/1920*(50*a*b*B*(-44*a*c+21*b^2)-A*(1024*a^2*c^2-2940*a*b^2*c+945*b^4))* (c*x^2+b*x+a)^(1/2)/a^5/x
Time = 1.48 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (-945 A b^4 x^4-96 a^4 (4 A+5 B x)+210 a b^2 x^3 (3 A b+5 b B x+14 A c x)+16 a^3 x (5 B x (7 b+9 c x)+A (27 b+32 c x))-4 a^2 x^2 \left (25 b B x (7 b+22 c x)+2 A \left (63 b^2+161 b c x+128 c^2 x^2\right )\right )\right )}{x^5}-45 \left (21 A b^5-32 a^3 B c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+150 a b \left (-7 b^3 B-28 A b^2 c+24 a b B c+24 a A c^2\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{1920 a^{11/2}} \]
((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-945*A*b^4*x^4 - 96*a^4*(4*A + 5*B*x) + 2 10*a*b^2*x^3*(3*A*b + 5*b*B*x + 14*A*c*x) + 16*a^3*x*(5*B*x*(7*b + 9*c*x) + A*(27*b + 32*c*x)) - 4*a^2*x^2*(25*b*B*x*(7*b + 22*c*x) + 2*A*(63*b^2 + 161*b*c*x + 128*c^2*x^2))))/x^5 - 45*(21*A*b^5 - 32*a^3*B*c^2)*ArcTanh[(Sq rt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 150*a*b*(-7*b^3*B - 28*A*b^2*c + 24*a*b*B*c + 24*a*A*c^2)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)]) /Sqrt[a]])/(1920*a^(11/2))
Time = 0.68 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {1237, 27, 1237, 27, 1237, 27, 25, 1237, 27, 1228, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {\int \frac {9 A b-10 a B+8 A c x}{2 x^5 \sqrt {c x^2+b x+a}}dx}{5 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {9 A b-10 a B+8 A c x}{x^5 \sqrt {c x^2+b x+a}}dx}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {\int \frac {63 A b^2-70 a B b-64 a A c+6 (9 A b-10 a B) c x}{2 x^4 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {63 A b^2-70 a B b-64 a A c+6 (9 A b-10 a B) c x}{x^4 \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {-\frac {\int -\frac {10 a B \left (35 b^2-36 a c\right )-7 A \left (45 b^3-92 a b c\right )-4 c \left (63 A b^2-70 a B b-64 a A c\right ) x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {315 A b^3-350 a B b^2-644 a A c b+360 a^2 B c+4 c \left (63 A b^2-70 a B b-64 a A c\right ) x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {315 A b^3-350 a B b^2-644 a A c b+360 a^2 B c+4 c \left (63 A b^2-70 a B b-64 a A c\right ) x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {\int -\frac {50 a b B \left (21 b^2-44 a c\right )-2 A \left (\frac {945 b^4}{2}-1470 a c b^2+512 a^2 c^2\right )-2 c \left (315 A b^3-350 a B b^2-644 a A c b+360 a^2 B c\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a c b^2+1024 a^2 c^2\right )-2 c \left (315 A b^3-350 a B b^2-644 a A c b+360 a^2 B c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {-\frac {15 \left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {15 \left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {15 \left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{2 a x^2}}{6 a}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{3 a x^3}}{8 a}-\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{4 a x^4}}{10 a}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}\) |
-1/5*(A*Sqrt[a + b*x + c*x^2])/(a*x^5) - (-1/4*((9*A*b - 10*a*B)*Sqrt[a + b*x + c*x^2])/(a*x^4) - (-1/3*((63*A*b^2 - 70*a*b*B - 64*a*A*c)*Sqrt[a + b *x + c*x^2])/(a*x^3) - (-1/2*((315*A*b^3 - 350*a*b^2*B - 644*a*A*b*c + 360 *a^2*B*c)*Sqrt[a + b*x + c*x^2])/(a*x^2) + (-(((50*a*b*B*(21*b^2 - 44*a*c) - A*(945*b^4 - 2940*a*b^2*c + 1024*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(a*x) ) + (15*(2*a*B*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2) - A*(63*b^5 - 280*a*b^3 *c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2]) ])/(2*a^(3/2)))/(4*a))/(6*a))/(8*a))/(10*a)
3.10.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] + Simp[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] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.83 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (1024 a^{2} A \,c^{2} x^{4}-2940 A a \,b^{2} c \,x^{4}+945 A \,b^{4} x^{4}+2200 B \,a^{2} b c \,x^{4}-1050 x^{4} B \,b^{3} a +1288 A \,a^{2} b c \,x^{3}-630 a A \,b^{3} x^{3}-720 a^{3} B c \,x^{3}+700 x^{3} B \,a^{2} b^{2}-512 a^{3} A c \,x^{2}+504 a^{2} A \,b^{2} x^{2}-560 x^{2} B \,a^{3} b -432 a^{3} A b x +480 a^{4} B x +384 A \,a^{4}\right )}{1920 a^{5} x^{5}}+\frac {\left (240 A \,a^{2} b \,c^{2}-280 A a \,b^{3} c +63 A \,b^{5}-96 B \,a^{3} c^{2}+240 B \,a^{2} b^{2} c -70 B a \,b^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{256 a^{\frac {11}{2}}}\) | \(257\) |
| default | \(B \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{4 a \,x^{4}}-\frac {7 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )}{8 a}-\frac {3 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{5 a \,x^{5}}-\frac {9 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{4 a \,x^{4}}-\frac {7 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )}{8 a}-\frac {3 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{10 a}-\frac {4 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )}{5 a}\right )\) | \(962\) |
-1/1920*(c*x^2+b*x+a)^(1/2)*(1024*A*a^2*c^2*x^4-2940*A*a*b^2*c*x^4+945*A*b ^4*x^4+2200*B*a^2*b*c*x^4-1050*B*a*b^3*x^4+1288*A*a^2*b*c*x^3-630*A*a*b^3* x^3-720*B*a^3*c*x^3+700*B*a^2*b^2*x^3-512*A*a^3*c*x^2+504*A*a^2*b^2*x^2-56 0*B*a^3*b*x^2-432*A*a^3*b*x+480*B*a^4*x+384*A*a^4)/a^5/x^5+1/256*(240*A*a^ 2*b*c^2-280*A*a*b^3*c+63*A*b^5-96*B*a^3*c^2+240*B*a^2*b^2*c-70*B*a*b^4)/a^ (11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
Time = 1.15 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.82 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\left [-\frac {15 \, {\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (384 \, A a^{5} - {\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \, {\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \, {\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{6} x^{5}}, \frac {15 \, {\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (384 \, A a^{5} - {\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \, {\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \, {\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{6} x^{5}}\right ] \]
[-1/7680*(15*(70*B*a*b^4 - 63*A*b^5 + 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 40*(6 *B*a^2*b^2 - 7*A*a*b^3)*c)*sqrt(a)*x^5*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*(384*A*a^5 - (1050*B*a^2*b^3 - 945*A*a*b^4 - 1024*A*a^3*c^2 - 20*(110*B*a^3*b - 147*A *a^2*b^2)*c)*x^4 + 2*(350*B*a^3*b^2 - 315*A*a^2*b^3 - 4*(90*B*a^4 - 161*A* a^3*b)*c)*x^3 - 8*(70*B*a^4*b - 63*A*a^3*b^2 + 64*A*a^4*c)*x^2 + 48*(10*B* a^5 - 9*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^5), 1/3840*(15*(70*B*a*b ^4 - 63*A*b^5 + 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 7*A*a*b^3 )*c)*sqrt(-a)*x^5*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*(384*A*a^5 - (1050*B*a^2*b^3 - 945*A*a*b^4 - 10 24*A*a^3*c^2 - 20*(110*B*a^3*b - 147*A*a^2*b^2)*c)*x^4 + 2*(350*B*a^3*b^2 - 315*A*a^2*b^3 - 4*(90*B*a^4 - 161*A*a^3*b)*c)*x^3 - 8*(70*B*a^4*b - 63*A *a^3*b^2 + 64*A*a^4*c)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^5)]
\[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{x^{6} \sqrt {a + b x + c x^{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1266 vs. \(2 (276) = 552\).
Time = 0.29 (sec) , antiderivative size = 1266, normalized size of antiderivative = 4.14 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]
1/128*(70*B*a*b^4 - 63*A*b^5 - 240*B*a^2*b^2*c + 280*A*a*b^3*c + 96*B*a^3* c^2 - 240*A*a^2*b*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a ))/(sqrt(-a)*a^5) - 1/1920*(1050*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a *b^4 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^2*b^2*c + 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^3*c + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^2 - 4900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b^4 + 4410*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^5 + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^2*c - 19600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^3*c - 6720*(sqrt(c) *x - sqrt(c*x^2 + b*x + a))^7*B*a^4*c^2 + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b*c^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3 *b^4 - 8064*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^5 - 30720*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^4*b^2*c + 35840*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))^5*A*a^3*b^3*c - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5* A*a^4*b*c^2 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b*c^(3/2) - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*c^(5/2) - 7900*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b^4 + 7110*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^5 + 13920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^5* b^2*c - 31600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*b^3*c + 6720*...
Timed out. \[ \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{x^6\,\sqrt {c\,x^2+b\,x+a}} \,d x \]