Integrand size = 23, antiderivative size = 303 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=-\frac {\left (b^2-4 a c\right ) \left (9 A b^3-14 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{1024 a^5 x^2}+\frac {\left (9 A b^3-14 a b^2 B-12 a A b c+8 a^2 B c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{384 a^4 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}+\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{84 a^2 x^6}-\frac {\left (63 A b^2-98 a b B-48 a A c\right ) \left (a+b x+c x^2\right )^{5/2}}{840 a^3 x^5}-\frac {\left (b^2-4 a c\right )^2 \left (2 a B \left (7 b^2-4 a c\right )-A \left (9 b^3-12 a b c\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2048 a^{11/2}} \] Output:
-1/1024*(-4*a*c+b^2)*(-12*A*a*b*c+9*A*b^3+8*B*a^2*c-14*B*a*b^2)*(b*x+2*a)* (c*x^2+b*x+a)^(1/2)/a^5/x^2+1/384*(-12*A*a*b*c+9*A*b^3+8*B*a^2*c-14*B*a*b^ 2)*(b*x+2*a)*(c*x^2+b*x+a)^(3/2)/a^4/x^4-1/7*A*(c*x^2+b*x+a)^(5/2)/a/x^7+1 /84*(9*A*b-14*B*a)*(c*x^2+b*x+a)^(5/2)/a^2/x^6-1/840*(-48*A*a*c+63*A*b^2-9 8*B*a*b)*(c*x^2+b*x+a)^(5/2)/a^3/x^5-1/2048*(-4*a*c+b^2)^2*(2*a*B*(-4*a*c+ 7*b^2)-A*(-12*a*b*c+9*b^3))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1 /2))/a^(11/2)
Time = 5.18 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\frac {-\sqrt {a} \sqrt {a+x (b+c x)} \left (945 A b^6 x^6+2560 a^6 (6 A+7 B x)-210 a b^4 x^5 (7 b B x+3 A (b+12 c x))+128 a^5 x (6 A (25 b+32 c x)+7 B x (26 b+35 c x))+96 a^4 x^2 \left (7 B x \left (b^2+6 b c x+10 c^2 x^2\right )+A \left (4 b^2+22 b c x+32 c^2 x^2\right )\right )+28 a^2 b^2 x^4 \left (5 b B x (7 b+76 c x)+6 A \left (3 b^2+26 b c x+98 c^2 x^2\right )\right )-16 a^3 x^3 \left (7 b B x \left (7 b^2+54 b c x+162 c^2 x^2\right )+3 A \left (9 b^3+62 b^2 c x+146 b c^2 x^2+128 c^3 x^3\right )\right )\right )-105 \left (9 A b^7+128 a^4 B c^3\right ) x^7 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-210 a b \left (7 b^5 B+42 A b^4 c-60 a b^3 B c-120 a A b^2 c^2+144 a^2 b B c^2+96 a^2 A c^3\right ) x^7 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{107520 a^{11/2} x^7} \] Input:
Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x]
Output:
(-(Sqrt[a]*Sqrt[a + x*(b + c*x)]*(945*A*b^6*x^6 + 2560*a^6*(6*A + 7*B*x) - 210*a*b^4*x^5*(7*b*B*x + 3*A*(b + 12*c*x)) + 128*a^5*x*(6*A*(25*b + 32*c* x) + 7*B*x*(26*b + 35*c*x)) + 96*a^4*x^2*(7*B*x*(b^2 + 6*b*c*x + 10*c^2*x^ 2) + A*(4*b^2 + 22*b*c*x + 32*c^2*x^2)) + 28*a^2*b^2*x^4*(5*b*B*x*(7*b + 7 6*c*x) + 6*A*(3*b^2 + 26*b*c*x + 98*c^2*x^2)) - 16*a^3*x^3*(7*b*B*x*(7*b^2 + 54*b*c*x + 162*c^2*x^2) + 3*A*(9*b^3 + 62*b^2*c*x + 146*b*c^2*x^2 + 128 *c^3*x^3)))) - 105*(9*A*b^7 + 128*a^4*B*c^3)*x^7*ArcTanh[(Sqrt[c]*x - Sqrt [a + x*(b + c*x)])/Sqrt[a]] - 210*a*b*(7*b^5*B + 42*A*b^4*c - 60*a*b^3*B*c - 120*a*A*b^2*c^2 + 144*a^2*b*B*c^2 + 96*a^2*A*c^3)*x^7*ArcTanh[(-(Sqrt[c ]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(107520*a^(11/2)*x^7)
Time = 0.51 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1237, 27, 1237, 27, 1228, 1152, 1152, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {\int \frac {(9 A b-14 a B+4 A c x) \left (c x^2+b x+a\right )^{3/2}}{2 x^7}dx}{7 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(9 A b-14 a B+4 A c x) \left (c x^2+b x+a\right )^{3/2}}{x^7}dx}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (63 A b^2-98 a B b-48 a A c+2 (9 A b-14 a B) c x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 x^6}dx}{6 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (63 A b^2-98 a B b-48 a A c+2 (9 A b-14 a B) c x\right ) \left (c x^2+b x+a\right )^{3/2}}{x^6}dx}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {-\frac {-\frac {7 \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{x^5}dx}{2 a}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{5 a x^5}}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle -\frac {-\frac {-\frac {7 \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{x^3}dx}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{5 a x^5}}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle -\frac {-\frac {-\frac {7 \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (-\frac {\left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{5 a x^5}}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {-\frac {-\frac {7 \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{4 a}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{5 a x^5}}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {-\frac {7 \left (8 a^2 B c-12 a A b c-14 a b^2 B+9 A b^3\right ) \left (-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2}}{4 a x^2}\right )}{16 a}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{8 a x^4}\right )}{2 a}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a A c-98 a b B+63 A b^2\right )}{5 a x^5}}{12 a}-\frac {(9 A b-14 a B) \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}}{14 a}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{7 a x^7}\) |
Input:
Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x]
Output:
-1/7*(A*(a + b*x + c*x^2)^(5/2))/(a*x^7) - (-1/6*((9*A*b - 14*a*B)*(a + b* x + c*x^2)^(5/2))/(a*x^6) - (-1/5*((63*A*b^2 - 98*a*b*B - 48*a*A*c)*(a + b *x + c*x^2)^(5/2))/(a*x^5) - (7*(9*A*b^3 - 14*a*b^2*B - 12*a*A*b*c + 8*a^2 *B*c)*(-1/8*((2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(a*x^4) - (3*(b^2 - 4*a* c)*(-1/4*((2*a + b*x)*Sqrt[a + b*x + c*x^2])/(a*x^2) + ((b^2 - 4*a*c)*ArcT anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(3/2))))/(16*a))) /(2*a))/(12*a))/(14*a)
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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> 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] + Simp[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] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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 = 1.65 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-6144 A \,a^{3} c^{3} x^{6}+16464 A \,a^{2} b^{2} c^{2} x^{6}-7560 A a \,b^{4} c \,x^{6}+945 A \,b^{6} x^{6}-18144 B \,a^{3} b \,c^{2} x^{6}+10640 B \,a^{2} b^{3} c \,x^{6}-1470 B a \,b^{5} x^{6}-7008 A \,a^{3} b \,c^{2} x^{5}+4368 A \,a^{2} b^{3} c \,x^{5}-630 A a \,b^{5} x^{5}+6720 B \,a^{4} c^{2} x^{5}-6048 B \,a^{3} b^{2} c \,x^{5}+980 B \,a^{2} b^{4} x^{5}+3072 A \,a^{4} c^{2} x^{4}-2976 A \,a^{3} b^{2} c \,x^{4}+504 A \,a^{2} b^{4} x^{4}+4032 B \,a^{4} b c \,x^{4}-784 B \,a^{3} b^{3} x^{4}+2112 A \,a^{4} b c \,x^{3}-432 A \,a^{3} b^{3} x^{3}+31360 B \,a^{5} c \,x^{3}+672 B \,a^{4} b^{2} x^{3}+24576 A \,a^{5} c \,x^{2}+384 A \,a^{4} b^{2} x^{2}+23296 B \,a^{5} b \,x^{2}+19200 A \,a^{5} b x +17920 B \,a^{6} x +15360 A \,a^{6}\right )}{107520 x^{7} a^{5}}-\frac {\left (192 A \,a^{3} b \,c^{3}-240 A \,a^{2} b^{3} c^{2}+84 A a \,b^{5} c -9 A \,b^{7}-128 B \,a^{4} c^{3}+288 B \,a^{3} b^{2} c^{2}-120 B \,a^{2} b^{4} c +14 B a \,b^{6}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2048 a^{\frac {11}{2}}}\) | \(446\) |
| default | \(\text {Expression too large to display}\) | \(6936\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/107520*(c*x^2+b*x+a)^(1/2)*(-6144*A*a^3*c^3*x^6+16464*A*a^2*b^2*c^2*x^6 -7560*A*a*b^4*c*x^6+945*A*b^6*x^6-18144*B*a^3*b*c^2*x^6+10640*B*a^2*b^3*c* x^6-1470*B*a*b^5*x^6-7008*A*a^3*b*c^2*x^5+4368*A*a^2*b^3*c*x^5-630*A*a*b^5 *x^5+6720*B*a^4*c^2*x^5-6048*B*a^3*b^2*c*x^5+980*B*a^2*b^4*x^5+3072*A*a^4* c^2*x^4-2976*A*a^3*b^2*c*x^4+504*A*a^2*b^4*x^4+4032*B*a^4*b*c*x^4-784*B*a^ 3*b^3*x^4+2112*A*a^4*b*c*x^3-432*A*a^3*b^3*x^3+31360*B*a^5*c*x^3+672*B*a^4 *b^2*x^3+24576*A*a^5*c*x^2+384*A*a^4*b^2*x^2+23296*B*a^5*b*x^2+19200*A*a^5 *b*x+17920*B*a^6*x+15360*A*a^6)/x^7/a^5-1/2048*(192*A*a^3*b*c^3-240*A*a^2* b^3*c^2+84*A*a*b^5*c-9*A*b^7-128*B*a^4*c^3+288*B*a^3*b^2*c^2-120*B*a^2*b^4 *c+14*B*a*b^6)/a^(11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
Time = 2.33 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x, algorithm="fricas")
Output:
[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 + 1 28*(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 - (14 70*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 - 31 5*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...
\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{8}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**8,x)
Output:
Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**8, x)
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x, algorithm="maxima")
Output:
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. 2713 vs. \(2 (273) = 546\).
Time = 0.28 (sec) , antiderivative size = 2713, normalized size of antiderivative = 8.95 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x, algorithm="giac")
Output:
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(-(sq rt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/107520*(1470 *(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 + 30240* (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))^13*A*a^ 3*b*c^3 - 9800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^2*b^6 + 6300*(sq rt(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))^1 1*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 - 178 29*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b^7 - 237720*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))^9*B*a^4*b^4*c + 166404*(sqrt(c)*x - sqrt(c*x^2 + ...
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^8} \,d x \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8,x)
Output:
int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^8, x)
Time = 3.08 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.85 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^8} \, dx=\frac {-30720 \sqrt {c \,x^{2}+b x +a}\, a^{7}-74240 \sqrt {c \,x^{2}+b x +a}\, a^{6} b x -49152 \sqrt {c \,x^{2}+b x +a}\, a^{6} c \,x^{2}-47360 \sqrt {c \,x^{2}+b x +a}\, a^{5} b^{2} x^{2}-66944 \sqrt {c \,x^{2}+b x +a}\, a^{5} b c \,x^{3}-6144 \sqrt {c \,x^{2}+b x +a}\, a^{5} c^{2} x^{4}-480 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{3} x^{3}-2112 \sqrt {c \,x^{2}+b x +a}\, a^{4} b^{2} c \,x^{4}+576 \sqrt {c \,x^{2}+b x +a}\, a^{4} b \,c^{2} x^{5}+12288 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{3} x^{6}+560 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{4} x^{4}+3360 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} c \,x^{5}+3360 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c^{2} x^{6}-700 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{5} x^{5}-6160 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} c \,x^{6}+1050 \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} x^{6}+6720 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{3} b \,c^{3} x^{7}+5040 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b^{3} c^{2} x^{7}-3780 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{5} c \,x^{7}+525 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{7} x^{7}-6720 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{3} b \,c^{3} x^{7}-5040 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b^{3} c^{2} x^{7}+3780 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{5} c \,x^{7}-525 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{7} x^{7}}{215040 a^{5} x^{7}} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^8,x)
Output:
( - 30720*sqrt(a + b*x + c*x**2)*a**7 - 74240*sqrt(a + b*x + c*x**2)*a**6* b*x - 49152*sqrt(a + b*x + c*x**2)*a**6*c*x**2 - 47360*sqrt(a + b*x + c*x* *2)*a**5*b**2*x**2 - 66944*sqrt(a + b*x + c*x**2)*a**5*b*c*x**3 - 6144*sqr t(a + b*x + c*x**2)*a**5*c**2*x**4 - 480*sqrt(a + b*x + c*x**2)*a**4*b**3* x**3 - 2112*sqrt(a + b*x + c*x**2)*a**4*b**2*c*x**4 + 576*sqrt(a + b*x + c *x**2)*a**4*b*c**2*x**5 + 12288*sqrt(a + b*x + c*x**2)*a**4*c**3*x**6 + 56 0*sqrt(a + b*x + c*x**2)*a**3*b**4*x**4 + 3360*sqrt(a + b*x + c*x**2)*a**3 *b**3*c*x**5 + 3360*sqrt(a + b*x + c*x**2)*a**3*b**2*c**2*x**6 - 700*sqrt( a + b*x + c*x**2)*a**2*b**5*x**5 - 6160*sqrt(a + b*x + c*x**2)*a**2*b**4*c *x**6 + 1050*sqrt(a + b*x + c*x**2)*a*b**6*x**6 + 6720*sqrt(a)*log(2*sqrt( a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*b*c**3*x**7 + 5040*sqrt(a)*log (2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**3*c**2*x**7 - 3780* sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**5*c*x**7 + 525*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**7*x**7 - 6720*sqrt(a)*log(x)*a**3*b*c**3*x**7 - 5040*sqrt(a)*log(x)*a**2*b**3*c**2* x**7 + 3780*sqrt(a)*log(x)*a*b**5*c*x**7 - 525*sqrt(a)*log(x)*b**7*x**7)/( 215040*a**5*x**7)