Integrand size = 23, antiderivative size = 281 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \]
-1/256*(-96*A*a^2*c^3+240*A*a*b^2*c^2-70*A*b^4*c+240*B*a^2*b*c^2-280*B*a*b ^3*c+63*B*b^5)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2) +1/240*(-70*A*b*c-64*B*a*c+63*B*b^2)*x^2*(c*x^2+b*x+a)^(1/2)/c^3-1/40*(-10 *A*c+9*B*b)*x^3*(c*x^2+b*x+a)^(1/2)/c^2+1/5*B*x^4*(c*x^2+b*x+a)^(1/2)/c+1/ 1920*(945*b^4*B-1050*A*b^3*c-2940*a*b^2*B*c+2200*a*A*b*c^2+1024*a^2*B*c^2- 2*c*(360*A*a*c^2-350*A*b^2*c-644*B*a*b*c+315*B*b^3)*x)*(c*x^2+b*x+a)^(1/2) /c^5
Time = 0.68 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.80 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^4 B-210 b^3 c (5 A+3 B x)+28 b^2 c (-105 a B+c x (25 A+18 B x))+16 c^2 \left (64 a^2 B+6 c^2 x^3 (5 A+4 B x)-a c x (45 A+32 B x)\right )+8 b c^2 \left (-2 c x^2 (35 A+27 B x)+a (275 A+161 B x)\right )\right )+15 \left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3840 c^{11/2}} \]
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4*B - 210*b^3*c*(5*A + 3*B*x) + 28 *b^2*c*(-105*a*B + c*x*(25*A + 18*B*x)) + 16*c^2*(64*a^2*B + 6*c^2*x^3*(5* A + 4*B*x) - a*c*x*(45*A + 32*B*x)) + 8*b*c^2*(-2*c*x^2*(35*A + 27*B*x) + a*(275*A + 161*B*x))) + 15*(63*b^5*B - 70*A*b^4*c - 280*a*b^3*B*c + 240*a* A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt [a + x*(b + c*x)]])/(3840*c^(11/2))
Time = 0.59 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1236, 27, 1236, 27, 1236, 27, 1225, 1092, 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.
| \(\displaystyle \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {x^3 (8 a B+(9 b B-10 A c) x)}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\int \frac {x^3 (8 a B+(9 b B-10 A c) x)}{\sqrt {c x^2+b x+a}}dx}{10 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {\int -\frac {x^2 \left (6 a (9 b B-10 A c)+\left (63 B b^2-70 A c b-64 a B c\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{4 c}+\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}}{10 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\int \frac {x^2 \left (6 a (9 b B-10 A c)+\left (63 B b^2-70 A c b-64 a B c\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{8 c}}{10 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\frac {\int -\frac {x \left (4 a \left (63 B b^2-70 A c b-64 a B c\right )+\left (315 B b^3-350 A c b^2-644 a B c b+360 a A c^2\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{3 c}}{8 c}}{10 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{3 c}-\frac {\int \frac {x \left (4 a \left (63 B b^2-70 A c b-64 a B c\right )+\left (315 B b^3-350 A c b^2-644 a B c b+360 a A c^2\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}}{8 c}}{10 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{3 c}-\frac {\frac {15 \left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}-\frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{4 c^2}}{6 c}}{8 c}}{10 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{3 c}-\frac {\frac {15 \left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}-\frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{4 c^2}}{6 c}}{8 c}}{10 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}-\frac {\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{4 c}-\frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{3 c}-\frac {\frac {15 \left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{4 c^2}}{6 c}}{8 c}}{10 c}\) |
(B*x^4*Sqrt[a + b*x + c*x^2])/(5*c) - (((9*b*B - 10*A*c)*x^3*Sqrt[a + b*x + c*x^2])/(4*c) - (((63*b^2*B - 70*A*b*c - 64*a*B*c)*x^2*Sqrt[a + b*x + c* x^2])/(3*c) - (-1/4*((945*b^4*B - 1050*A*b^3*c - 2940*a*b^2*B*c + 2200*a*A *b*c^2 + 1024*a^2*B*c^2 - 2*c*(315*b^3*B - 350*A*b^2*c - 644*a*b*B*c + 360 *a*A*c^2)*x)*Sqrt[a + b*x + c*x^2])/c^2 + (15*(63*b^5*B - 70*A*b^4*c - 280 *a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*c^2 - 96*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c))/(8*c))/(1 0*c)
3.10.50.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(\frac {\left (384 B \,x^{4} c^{4}+480 A \,c^{4} x^{3}-432 B b \,c^{3} x^{3}-560 A b \,c^{3} x^{2}-512 B a \,c^{3} x^{2}+504 B \,b^{2} c^{2} x^{2}-720 A a \,c^{3} x +700 A \,b^{2} c^{2} x +1288 B a b \,c^{2} x -630 B \,b^{3} c x +2200 a A b \,c^{2}-1050 A \,b^{3} c +1024 a^{2} B \,c^{2}-2940 a \,b^{2} B c +945 b^{4} B \right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{5}}+\frac {\left (96 A \,a^{2} c^{3}-240 A a \,b^{2} c^{2}+70 A \,b^{4} c -240 B \,a^{2} b \,c^{2}+280 B a \,b^{3} c -63 B \,b^{5}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}\) | \(234\) |
| default | \(B \left (\frac {x^{4} \sqrt {c \,x^{2}+b x +a}}{5 c}-\frac {9 b \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {4 a \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{5 c}\right )+A \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )\) | \(868\) |
1/1920*(384*B*c^4*x^4+480*A*c^4*x^3-432*B*b*c^3*x^3-560*A*b*c^3*x^2-512*B* a*c^3*x^2+504*B*b^2*c^2*x^2-720*A*a*c^3*x+700*A*b^2*c^2*x+1288*B*a*b*c^2*x -630*B*b^3*c*x+2200*A*a*b*c^2-1050*A*b^3*c+1024*B*a^2*c^2-2940*B*a*b^2*c+9 45*B*b^4)*(c*x^2+b*x+a)^(1/2)/c^5+1/256*(96*A*a^2*c^3-240*A*a*b^2*c^2+70*A *b^4*c-240*B*a^2*b*c^2+280*B*a*b^3*c-63*B*b^5)/c^(11/2)*ln((1/2*b+c*x)/c^( 1/2)+(c*x^2+b*x+a)^(1/2))
Time = 0.34 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.85 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\left [-\frac {15 \, {\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {15 \, {\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \]
[-1/7680*(15*(63*B*b^5 - 96*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 70*( 4*B*a*b^3 + A*b^4)*c)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^ 2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*B*c^5*x^4 + 945*B*b^4*c + 8*(128*B*a^2 + 275*A*a*b)*c^3 - 48*(9*B*b*c^4 - 10*A*c^5)*x^3 - 210*(14 *B*a*b^2 + 5*A*b^3)*c^2 + 8*(63*B*b^2*c^3 - 2*(32*B*a + 35*A*b)*c^4)*x^2 - 2*(315*B*b^3*c^2 + 360*A*a*c^4 - 14*(46*B*a*b + 25*A*b^2)*c^3)*x)*sqrt(c* x^2 + b*x + a))/c^6, 1/3840*(15*(63*B*b^5 - 96*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 70*(4*B*a*b^3 + A*b^4)*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(384*B*c^5*x^4 + 945*B*b^4*c + 8*(128*B*a^2 + 275*A*a*b)*c^3 - 48*(9*B*b*c^4 - 10*A*c^5)* x^3 - 210*(14*B*a*b^2 + 5*A*b^3)*c^2 + 8*(63*B*b^2*c^3 - 2*(32*B*a + 35*A* b)*c^4)*x^2 - 2*(315*B*b^3*c^2 + 360*A*a*c^4 - 14*(46*B*a*b + 25*A*b^2)*c^ 3)*x)*sqrt(c*x^2 + b*x + a))/c^6]
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (309) = 618\).
Time = 0.66 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.21 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {a \left (- \frac {3 a \left (A - \frac {9 B b}{10 c}\right )}{4 c} - \frac {5 b \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{6 c}\right )}{2 c} - \frac {b \left (- \frac {2 a \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (A - \frac {9 B b}{10 c}\right )}{4 c} - \frac {5 b \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a + b x + c x^{2}} \left (\frac {B x^{4}}{5 c} + \frac {x^{3} \left (A - \frac {9 B b}{10 c}\right )}{4 c} + \frac {x^{2} \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{3 c} + \frac {x \left (- \frac {3 a \left (A - \frac {9 B b}{10 c}\right )}{4 c} - \frac {5 b \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{6 c}\right )}{2 c} + \frac {- \frac {2 a \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (A - \frac {9 B b}{10 c}\right )}{4 c} - \frac {5 b \left (- \frac {4 B a}{5 c} - \frac {7 b \left (A - \frac {9 B b}{10 c}\right )}{8 c}\right )}{6 c}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {B \left (a + b x\right )^{\frac {11}{2}}}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (A b - 5 B a\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{3 b} + \frac {\sqrt {a + b x} \left (A a^{4} b - B a^{5}\right )}{b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Piecewise(((-a*(-3*a*(A - 9*B*b/(10*c))/(4*c) - 5*b*(-4*B*a/(5*c) - 7*b*(A - 9*B*b/(10*c))/(8*c))/(6*c))/(2*c) - b*(-2*a*(-4*B*a/(5*c) - 7*b*(A - 9* B*b/(10*c))/(8*c))/(3*c) - 3*b*(-3*a*(A - 9*B*b/(10*c))/(4*c) - 5*b*(-4*B* a/(5*c) - 7*b*(A - 9*B*b/(10*c))/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((lo g(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c) , 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) + sqrt(a + b*x + c*x**2)*(B*x**4/(5*c) + x**3*(A - 9*B*b/(10*c))/(4*c) + x** 2*(-4*B*a/(5*c) - 7*b*(A - 9*B*b/(10*c))/(8*c))/(3*c) + x*(-3*a*(A - 9*B*b /(10*c))/(4*c) - 5*b*(-4*B*a/(5*c) - 7*b*(A - 9*B*b/(10*c))/(8*c))/(6*c))/ (2*c) + (-2*a*(-4*B*a/(5*c) - 7*b*(A - 9*B*b/(10*c))/(8*c))/(3*c) - 3*b*(- 3*a*(A - 9*B*b/(10*c))/(4*c) - 5*b*(-4*B*a/(5*c) - 7*b*(A - 9*B*b/(10*c))/ (8*c))/(6*c))/(4*c))/c), Ne(c, 0)), (2*(B*(a + b*x)**(11/2)/(11*b) + (a + b*x)**(9/2)*(A*b - 5*B*a)/(9*b) + (a + b*x)**(7/2)*(-4*A*a*b + 10*B*a**2)/ (7*b) + (a + b*x)**(5/2)*(6*A*a**2*b - 10*B*a**3)/(5*b) + (a + b*x)**(3/2) *(-4*A*a**3*b + 5*B*a**4)/(3*b) + sqrt(a + b*x)*(A*a**4*b - B*a**5)/b)/b** 5, Ne(b, 0)), ((A*x**5/5 + B*x**6/6)/sqrt(a), True))
Exception generated. \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, B x}{c} - \frac {9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac {63 \, B b^{2} c^{2} - 64 \, B a c^{3} - 70 \, A b c^{3}}{c^{5}}\right )} x - \frac {315 \, B b^{3} c - 644 \, B a b c^{2} - 350 \, A b^{2} c^{2} + 360 \, A a c^{3}}{c^{5}}\right )} x + \frac {945 \, B b^{4} - 2940 \, B a b^{2} c - 1050 \, A b^{3} c + 1024 \, B a^{2} c^{2} + 2200 \, A a b c^{2}}{c^{5}}\right )} + \frac {{\left (63 \, B b^{5} - 280 \, B a b^{3} c - 70 \, A b^{4} c + 240 \, B a^{2} b c^{2} + 240 \, A a b^{2} c^{2} - 96 \, A a^{2} c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {11}{2}}} \]
1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*B*x/c - (9*B*b*c^3 - 10*A*c^4)/c^ 5)*x + (63*B*b^2*c^2 - 64*B*a*c^3 - 70*A*b*c^3)/c^5)*x - (315*B*b^3*c - 64 4*B*a*b*c^2 - 350*A*b^2*c^2 + 360*A*a*c^3)/c^5)*x + (945*B*b^4 - 2940*B*a* b^2*c - 1050*A*b^3*c + 1024*B*a^2*c^2 + 2200*A*a*b*c^2)/c^5) + 1/256*(63*B *b^5 - 280*B*a*b^3*c - 70*A*b^4*c + 240*B*a^2*b*c^2 + 240*A*a*b^2*c^2 - 96 *A*a^2*c^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^ (11/2)
Timed out. \[ \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \]