Integrand size = 23, antiderivative size = 310 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=-\frac {\left (3 b^4 B-10 A b^3 c-28 a b^2 B c-440 a A b c^2-128 a^2 B c^2+2 c \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2}+\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}-\frac {1}{2} a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\left (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}} \] Output:
-1/128*(3*B*b^4-10*A*b^3*c-28*B*a*b^2*c-440*A*a*b*c^2-128*B*a^2*c^2+2*c*(- 120*A*a*c^2-10*A*b^2*c-28*B*a*b*c+3*B*b^3)*x)*(c*x^2+b*x+a)^(1/2)/c^2+1/48 *(3*B*b^2+70*A*b*c+16*B*a*c+6*c*(10*A*c+B*b)*x)*(c*x^2+b*x+a)^(3/2)/c-1/5* (-B*x+5*A)*(c*x^2+b*x+a)^(5/2)/x-1/2*a^(3/2)*(5*A*b+2*B*a)*arctanh(1/2*(b* x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))+1/256*(480*A*a^2*c^3+240*A*a*b^2*c^2-1 0*A*b^4*c+240*B*a^2*b*c^2-40*B*a*b^3*c+3*B*b^5)*arctanh(1/2*(2*c*x+b)/c^(1 /2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)
Time = 2.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\frac {\sqrt {a+x (b+c x)} \left (-128 a^2 c^2 (15 A-23 B x)+x \left (-45 b^4 B+30 b^3 c (5 A+B x)+96 c^4 x^3 (5 A+4 B x)+16 b c^3 x^2 (85 A+63 B x)+4 b^2 c^2 x (295 A+186 B x)\right )+4 a c x \left (135 b^2 B+4 c^2 x (135 A+88 B x)+2 b c (695 A+311 B x)\right )\right )}{1920 c^2 x}+a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-\frac {\left (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\right ) \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{256 c^{5/2}} \] Input:
Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2,x]
Output:
(Sqrt[a + x*(b + c*x)]*(-128*a^2*c^2*(15*A - 23*B*x) + x*(-45*b^4*B + 30*b ^3*c*(5*A + B*x) + 96*c^4*x^3*(5*A + 4*B*x) + 16*b*c^3*x^2*(85*A + 63*B*x) + 4*b^2*c^2*x*(295*A + 186*B*x)) + 4*a*c*x*(135*b^2*B + 4*c^2*x*(135*A + 88*B*x) + 2*b*c*(695*A + 311*B*x))))/(1920*c^2*x) + a^(3/2)*(5*A*b + 2*a*B )*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - ((3*b^5*B - 10*A* b^4*c - 40*a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*c^2 + 480*a^2*A*c^3)* Log[c^2*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(5/2))
Time = 0.66 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1230, 25, 1231, 27, 1231, 27, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {1}{2} \int -\frac {(5 A b+2 a B+(b B+10 A c) x) \left (c x^2+b x+a\right )^{3/2}}{x}dx-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \int \frac {(5 A b+2 a B+(b B+10 A c) x) \left (c x^2+b x+a\right )^{3/2}}{x}dx-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}-\frac {\int -\frac {\left (16 a (5 A b+2 a B) c-\left (3 B b^3-10 A c b^2-28 a B c b-120 a A c^2\right ) x\right ) \sqrt {c x^2+b x+a}}{2 x}dx}{8 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\left (16 a (5 A b+2 a B) c-\left (3 B b^3-10 A c b^2-28 a B c b-120 a A c^2\right ) x\right ) \sqrt {c x^2+b x+a}}{x}dx}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int -\frac {128 a^2 (5 A b+2 a B) c^2+\left (3 B b^5-10 A c b^4-40 a B c b^3+240 a A c^2 b^2+240 a^2 B c^2 b+480 a^2 A c^3\right ) x}{2 x \sqrt {c x^2+b x+a}}dx}{4 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {128 a^2 (5 A b+2 a B) c^2+\left (3 B b^5-10 A c b^4-40 a B c b^3+240 a A c^2 b^2+240 a^2 B c^2 b+480 a^2 A c^3\right ) x}{x \sqrt {c x^2+b x+a}}dx}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+128 a^2 c^2 (2 a B+5 A b) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+128 a^2 c^2 (2 a B+5 A b) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {128 a^2 c^2 (2 a B+5 A b) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx+\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-256 a^2 c^2 (2 a B+5 A b) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c}}-128 a^{3/2} c^2 (2 a B+5 A b) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 c}-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{4 c}}{16 c}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{24 c}\right )-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}\) |
Input:
Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2,x]
Output:
-1/5*((5*A - B*x)*(a + b*x + c*x^2)^(5/2))/x + (((3*b^2*B + 70*A*b*c + 16* a*B*c + 6*c*(b*B + 10*A*c)*x)*(a + b*x + c*x^2)^(3/2))/(24*c) + (-1/4*((3* b^4*B - 10*A*b^3*c - 28*a*b^2*B*c - 440*a*A*b*c^2 - 128*a^2*B*c^2 + 2*c*(3 *b^3*B - 10*A*b^2*c - 28*a*b*B*c - 120*a*A*c^2)*x)*Sqrt[a + b*x + c*x^2])/ c + (-128*a^(3/2)*(5*A*b + 2*a*B)*c^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[ a + b*x + c*x^2])] + ((3*b^5*B - 10*A*b^4*c - 40*a*b^3*B*c + 240*a*A*b^2*c ^2 + 240*a^2*b*B*c^2 + 480*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[ a + b*x + c*x^2])])/Sqrt[c])/(8*c))/(16*c))/2
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[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[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(599\) vs. \(2(278)=556\).
Time = 1.28 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(-\frac {a^{2} A \sqrt {c \,x^{2}+b x +a}}{x}+\frac {B \,c^{2} x^{4} \sqrt {c \,x^{2}+b x +a}}{5}-\frac {5 b^{3} a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B}{32 c^{\frac {3}{2}}}+\frac {15 a^{2} A \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}+\frac {9 c a x \sqrt {c \,x^{2}+b x +a}\, A}{8}+\frac {311 a x \sqrt {c \,x^{2}+b x +a}\, B b}{240}-\frac {5 A \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b}{2}-B \,a^{\frac {5}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )+\frac {15 A a \,b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {15 B \,a^{2} b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {5 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) A}{128 c^{\frac {3}{2}}}+\frac {3 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) B}{256 c^{\frac {5}{2}}}+\frac {9 b^{2} a \sqrt {c \,x^{2}+b x +a}\, B}{32 c}+\frac {c^{2} x^{3} \sqrt {c \,x^{2}+b x +a}\, A}{4}+\frac {31 b^{2} x^{2} \sqrt {c \,x^{2}+b x +a}\, B}{80}+\frac {59 b^{2} x \sqrt {c \,x^{2}+b x +a}\, A}{96}+\frac {5 b^{3} \sqrt {c \,x^{2}+b x +a}\, A}{64 c}-\frac {3 b^{4} \sqrt {c \,x^{2}+b x +a}\, B}{128 c^{2}}+\frac {11 c \,x^{2} \sqrt {c \,x^{2}+b x +a}\, a B}{15}+\frac {139 b a \sqrt {c \,x^{2}+b x +a}\, A}{48}+\frac {21 c \,x^{3} \sqrt {c \,x^{2}+b x +a}\, B b}{40}+\frac {17 c b \,x^{2} \sqrt {c \,x^{2}+b x +a}\, A}{24}+\frac {b^{3} x \sqrt {c \,x^{2}+b x +a}\, B}{64 c}+\frac {23 a^{2} \sqrt {c \,x^{2}+b x +a}\, B}{15}\) | \(600\) |
| default | \(A \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{a x}+\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5}+\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2}+a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )\right )\right )}{2 a}+\frac {6 c \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{a}\right )+B \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5}+\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2}+a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2}+a \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )\right )\right )\) | \(751\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-a^2*A*(c*x^2+b*x+a)^(1/2)/x+1/5*B*c^2*x^4*(c*x^2+b*x+a)^(1/2)-5/32/c^(3/2 )*b^3*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B+15/8*a^2*A*c^(1/2)*l n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+9/8*c*a*x*(c*x^2+b*x+a)^(1/2)*A +311/240*a*x*(c*x^2+b*x+a)^(1/2)*B*b-5/2*A*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*( c*x^2+b*x+a)^(1/2))/x)*b-B*a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/ 2))/x)+15/16*A*a*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1 5/16*B*a^2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-5/128/c^( 3/2)*b^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A+3/256/c^(5/2)*b^5*l n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B+9/32/c*b^2*a*(c*x^2+b*x+a)^(1 /2)*B+1/4*c^2*x^3*(c*x^2+b*x+a)^(1/2)*A+31/80*b^2*x^2*(c*x^2+b*x+a)^(1/2)* B+59/96*b^2*x*(c*x^2+b*x+a)^(1/2)*A+5/64/c*b^3*(c*x^2+b*x+a)^(1/2)*A-3/128 /c^2*b^4*(c*x^2+b*x+a)^(1/2)*B+11/15*c*x^2*(c*x^2+b*x+a)^(1/2)*a*B+139/48* b*a*(c*x^2+b*x+a)^(1/2)*A+21/40*c*x^3*(c*x^2+b*x+a)^(1/2)*B*b+17/24*c*b*x^ 2*(c*x^2+b*x+a)^(1/2)*A+1/64/c*b^3*x*(c*x^2+b*x+a)^(1/2)*B+23/15*a^2*(c*x^ 2+b*x+a)^(1/2)*B
Time = 4.34 (sec) , antiderivative size = 1393, normalized size of antiderivative = 4.49 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,x, algorithm="fricas")
Output:
[1/7680*(1920*(2*B*a^2 + 5*A*a*b)*sqrt(a)*c^3*x*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) + 15*( 3*B*b^5 + 480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A* b^4)*c)*sqrt(c)*x*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^5 - 1920*A*a^2*c^3 + 48*(21 *B*b*c^4 + 10*A*c^5)*x^4 + 8*(93*B*b^2*c^3 + 2*(88*B*a + 85*A*b)*c^4)*x^3 + 2*(15*B*b^3*c^2 + 1080*A*a*c^4 + 2*(622*B*a*b + 295*A*b^2)*c^3)*x^2 - (4 5*B*b^4*c - 8*(368*B*a^2 + 695*A*a*b)*c^3 - 30*(18*B*a*b^2 + 5*A*b^3)*c^2) *x)*sqrt(c*x^2 + b*x + a))/(c^3*x), 1/3840*(960*(2*B*a^2 + 5*A*a*b)*sqrt(a )*c^3*x*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) - 15*(3*B*b^5 + 480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*c)*sqrt(-c)*x*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 ^5 - 1920*A*a^2*c^3 + 48*(21*B*b*c^4 + 10*A*c^5)*x^4 + 8*(93*B*b^2*c^3 + 2 *(88*B*a + 85*A*b)*c^4)*x^3 + 2*(15*B*b^3*c^2 + 1080*A*a*c^4 + 2*(622*B*a* b + 295*A*b^2)*c^3)*x^2 - (45*B*b^4*c - 8*(368*B*a^2 + 695*A*a*b)*c^3 - 30 *(18*B*a*b^2 + 5*A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(c^3*x), 1/7680*(38 40*(2*B*a^2 + 5*A*a*b)*sqrt(-a)*c^3*x*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)) + 15*(3*B*b^5 + 480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*c)*sqrt(c)*x*log(...
\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{x^{2}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**2,x)
Output:
Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**2, x)
Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,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
Time = 0.27 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, B c^{2} x + \frac {21 \, B b c^{5} + 10 \, A c^{6}}{c^{4}}\right )} x + \frac {93 \, B b^{2} c^{4} + 176 \, B a c^{5} + 170 \, A b c^{5}}{c^{4}}\right )} x + \frac {15 \, B b^{3} c^{3} + 1244 \, B a b c^{4} + 590 \, A b^{2} c^{4} + 1080 \, A a c^{5}}{c^{4}}\right )} x - \frac {45 \, B b^{4} c^{2} - 540 \, B a b^{2} c^{3} - 150 \, A b^{3} c^{3} - 2944 \, B a^{2} c^{4} - 5560 \, A a b c^{4}}{c^{4}}\right )} + \frac {{\left (2 \, B a^{3} + 5 \, A a^{2} b\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b + 2 \, A a^{3} \sqrt {c}}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a} - \frac {{\left (3 \, B b^{5} - 40 \, B a b^{3} c - 10 \, A b^{4} c + 240 \, B a^{2} b c^{2} + 240 \, A a b^{2} c^{2} + 480 \, A a^{2} c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {5}{2}}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,x, algorithm="giac")
Output:
1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*B*c^2*x + (21*B*b*c^5 + 10*A*c^6) /c^4)*x + (93*B*b^2*c^4 + 176*B*a*c^5 + 170*A*b*c^5)/c^4)*x + (15*B*b^3*c^ 3 + 1244*B*a*b*c^4 + 590*A*b^2*c^4 + 1080*A*a*c^5)/c^4)*x - (45*B*b^4*c^2 - 540*B*a*b^2*c^3 - 150*A*b^3*c^3 - 2944*B*a^2*c^4 - 5560*A*a*b*c^4)/c^4) + (2*B*a^3 + 5*A*a^2*b)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(- a))/sqrt(-a) + ((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b + 2*A*a^3*sqrt (c))/((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a) - 1/256*(3*B*b^5 - 40*B*a *b^3*c - 10*A*b^4*c + 240*B*a^2*b*c^2 + 240*A*a*b^2*c^2 + 480*A*a^2*c^3)*l og(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^2} \,d x \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2,x)
Output:
int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2, x)
Time = 0.30 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx=\frac {-3840 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{3}+17008 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{3} x +4320 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} x^{2}+1380 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{2} x +7336 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{3} x^{2}+5536 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{4} x^{3}+960 \sqrt {c \,x^{2}+b x +a}\, a \,c^{5} x^{4}-90 \sqrt {c \,x^{2}+b x +a}\, b^{5} c x +60 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{2} x^{2}+1488 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{3} x^{3}+2016 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{4} x^{4}+768 \sqrt {c \,x^{2}+b x +a}\, b \,c^{5} x^{5}+13440 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} b \,c^{3} x -13440 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{3} x +7200 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{3} c^{3} x +7200 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a^{2} b^{2} c^{2} x -750 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a \,b^{4} c x +45 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) b^{6} x}{3840 c^{3} x} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,x)
Output:
( - 3840*sqrt(a + b*x + c*x**2)*a**3*c**3 + 17008*sqrt(a + b*x + c*x**2)*a **2*b*c**3*x + 4320*sqrt(a + b*x + c*x**2)*a**2*c**4*x**2 + 1380*sqrt(a + b*x + c*x**2)*a*b**3*c**2*x + 7336*sqrt(a + b*x + c*x**2)*a*b**2*c**3*x**2 + 5536*sqrt(a + b*x + c*x**2)*a*b*c**4*x**3 + 960*sqrt(a + b*x + c*x**2)* a*c**5*x**4 - 90*sqrt(a + b*x + c*x**2)*b**5*c*x + 60*sqrt(a + b*x + c*x** 2)*b**4*c**2*x**2 + 1488*sqrt(a + b*x + c*x**2)*b**3*c**3*x**3 + 2016*sqrt (a + b*x + c*x**2)*b**2*c**4*x**4 + 768*sqrt(a + b*x + c*x**2)*b*c**5*x**5 + 13440*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b* c**3*x - 13440*sqrt(a)*log(x)*a**2*b*c**3*x + 7200*sqrt(c)*log( - 2*sqrt(c )*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a**3*c**3*x + 7200*sqrt(c)*log( - 2* sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a**2*b**2*c**2*x - 750*sqrt(c) *log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a*b**4*c*x + 45*sqrt (c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*b**6*x)/(3840*c** 3*x)