Integrand size = 23, antiderivative size = 133 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=-\frac {(2 a A+(A b+4 a B) x) \sqrt {a+b x+c x^2}}{4 a x^2}+\frac {\left (A b^2-4 a b B-4 a A c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}+B \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:
-1/4*(2*a*A+(A*b+4*B*a)*x)*(c*x^2+b*x+a)^(1/2)/a/x^2+1/8*(-4*A*a*c+A*b^2-4 *B*a*b)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(3/2)+B*c^(1/ 2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))
Time = 1.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.20 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=-\frac {(A b x+2 a (A+2 B x)) \sqrt {a+x (b+c x)}}{4 a x^2}-\frac {A b^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {(b B+A c) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}-B \sqrt {c} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \] Input:
Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^3,x]
Output:
-1/4*((A*b*x + 2*a*(A + 2*B*x))*Sqrt[a + x*(b + c*x)])/(a*x^2) - (A*b^2*Ar cTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(4*a^(3/2)) - ((b*B + A*c)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/Sqrt[a] - B* Sqrt[c]*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]
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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{2 x \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {4 a b B+8 a c x B-A \left (b^2-4 a c\right )}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {A b^2-4 a B b-4 a A c-8 a B c x}{x \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}\) |
Input:
Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^3,x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Time = 1.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (A b x +4 B a x +2 A a \right )}{4 x^{2} a}+\frac {-\frac {\left (4 A a c -b^{2} A +4 a b B \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}+8 a B \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 a}\) | \(125\) |
| default | \(A \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{2 a \,x^{2}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{4 a}+\frac {c \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}\right )+B \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )\) | \(467\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(c*x^2+b*x+a)^(1/2)*(A*b*x+4*B*a*x+2*A*a)/x^2/a+1/8/a*(-(4*A*a*c-A*b^ 2+4*B*a*b)/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+8*a*B*c^( 1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
Time = 0.39 (sec) , antiderivative size = 699, normalized size of antiderivative = 5.26 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,x, algorithm="fricas")
Output:
[1/16*(8*B*a^2*sqrt(c)*x^2*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*B*a*b - A*b^2 + 4*A*a*c)*sqrt( a)*x^2*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*(2*A*a^2 + (4*B*a^2 + A*a*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^2), -1/16*(16*B*a^2*sqrt(-c)*x^2*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)) - (4*B*a*b - A*b^ 2 + 4*A*a*c)*sqrt(a)*x^2*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*(2*A*a^2 + (4*B*a^2 + A*a *b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^2), 1/8*(4*B*a^2*sqrt(c)*x^2*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*B*a*b - A*b^2 + 4*A*a*c)*sqrt(-a)*x^2*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*(2*A*a^2 + (4*B*a^ 2 + A*a*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^2), -1/8*(8*B*a^2*sqrt(-c)*x^2 *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)) - (4*B*a*b - A*b^2 + 4*A*a*c)*sqrt(-a)*x^2*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*(2*A*a^2 + (4*B* a^2 + A*a*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^2)]
\[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{3}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**3,x)
Output:
Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**3, x)
Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,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. 356 vs. \(2 (112) = 224\).
Time = 0.27 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.68 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=-B \sqrt {c} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right ) + \frac {{\left (4 \, B a b - A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,x, algorithm="giac")
Output:
-B*sqrt(c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b)) + 1 /4*(4*B*a*b - A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) /sqrt(-a))/(sqrt(-a)*a) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a *b + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2 + 4*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^3*A*a*c + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*sqr t(c) + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b*sqrt(c) - 4*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))*B*a^2*b + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A *a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c - 8*B*a^3*sqrt(c))/ (((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2*a)
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^3} \,d x \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^3,x)
Output:
int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^3, x)
Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx=\frac {-4 \sqrt {c \,x^{2}+b x +a}\, a^{2}-10 \sqrt {c \,x^{2}+b x +a}\, a b x +4 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a c \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{2} x^{2}-4 \sqrt {a}\, \mathrm {log}\left (x \right ) a c \,x^{2}-3 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} x^{2}+8 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a b \,x^{2}}{8 a \,x^{2}} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,x)
Output:
( - 4*sqrt(a + b*x + c*x**2)*a**2 - 10*sqrt(a + b*x + c*x**2)*a*b*x + 4*sq rt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*c*x**2 + 3*sqrt( a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**2*x**2 - 4*sqrt(a) *log(x)*a*c*x**2 - 3*sqrt(a)*log(x)*b**2*x**2 + 8*sqrt(c)*log( - 2*sqrt(c) *sqrt(a + b*x + c*x**2) - b - 2*c*x)*a*b*x**2)/(8*a*x**2)