Integrand size = 22, antiderivative size = 73 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=-\frac {2 B \sqrt {b x+c x^2}}{x}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}+2 B \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \] Output:
-2*B*(c*x^2+b*x)^(1/2)/x-2/3*A*(c*x^2+b*x)^(3/2)/b/x^3+2*B*c^(1/2)*arctanh (c^(1/2)*x/(c*x^2+b*x)^(1/2))
Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=-\frac {2 \sqrt {x (b+c x)} (A b+3 b B x+A c x)}{3 b x^2}-\frac {2 B \sqrt {c} \sqrt {x (b+c x)} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}} \] Input:
Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^3,x]
Output:
(-2*Sqrt[x*(b + c*x)]*(A*b + 3*b*B*x + A*c*x))/(3*b*x^2) - (2*B*Sqrt[c]*Sq rt[x*(b + c*x)]*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])
Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1220, 1125, 25, 27, 1091, 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.
\(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle B \int \frac {\sqrt {c x^2+b x}}{x^2}dx-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
\(\Big \downarrow \) 1125 |
\(\displaystyle B \left (-\int -\frac {c}{\sqrt {c x^2+b x}}dx-\frac {2 \sqrt {b x+c x^2}}{x}\right )-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle B \left (\int \frac {c}{\sqrt {c x^2+b x}}dx-\frac {2 \sqrt {b x+c x^2}}{x}\right )-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle B \left (c \int \frac {1}{\sqrt {c x^2+b x}}dx-\frac {2 \sqrt {b x+c x^2}}{x}\right )-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle B \left (2 c \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}-\frac {2 \sqrt {b x+c x^2}}{x}\right )-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle B \left (2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )-\frac {2 \sqrt {b x+c x^2}}{x}\right )-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}\) |
Input:
Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^3,x]
Output:
(-2*A*(b*x + c*x^2)^(3/2))/(3*b*x^3) + B*((-2*Sqrt[b*x + c*x^2])/x + 2*Sqr t[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2) Int[(1/Sqrt[a + b*x + c*x^2])*Expan dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x ), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && EqQ[m + p, -3/2]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) 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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 0.89 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {6 B b \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) x^{2}-2 \sqrt {x \left (c x +b \right )}\, \left (\left (3 B x +A \right ) b +A c x \right )}{3 b \,x^{2}}\) | \(61\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (A c x +3 B b x +A b \right )}{3 x \sqrt {x \left (c x +b \right )}\, b}+B \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )\) | \(66\) |
default | \(-\frac {2 A \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 b \,x^{3}}+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{b \,x^{2}}+\frac {2 c \left (\sqrt {c \,x^{2}+b x}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 \sqrt {c}}\right )}{b}\right )\) | \(92\) |
Input:
int((B*x+A)*(c*x^2+b*x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/3*(6*B*b*c^(1/2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))*x^2-2*(x*(c*x+b))^ (1/2)*((3*B*x+A)*b+A*c*x))/b/x^2
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=\left [\frac {3 \, B b \sqrt {c} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x} {\left (A b + {\left (3 \, B b + A c\right )} x\right )}}{3 \, b x^{2}}, -\frac {2 \, {\left (3 \, B b \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + \sqrt {c x^{2} + b x} {\left (A b + {\left (3 \, B b + A c\right )} x\right )}\right )}}{3 \, b x^{2}}\right ] \] Input:
integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^3,x, algorithm="fricas")
Output:
[1/3*(3*B*b*sqrt(c)*x^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*s qrt(c*x^2 + b*x)*(A*b + (3*B*b + A*c)*x))/(b*x^2), -2/3*(3*B*b*sqrt(-c)*x^ 2*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + sqrt(c*x^2 + b*x)*(A*b + (3*B*b + A*c)*x))/(b*x^2)]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{3}}\, dx \] Input:
integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**3,x)
Output:
Integral(sqrt(x*(b + c*x))*(A + B*x)/x**3, x)
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx={\left (\sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {2 \, \sqrt {c x^{2} + b x}}{x}\right )} B - \frac {2}{3} \, A {\left (\frac {\sqrt {c x^{2} + b x} c}{b x} + \frac {\sqrt {c x^{2} + b x}}{x^{2}}\right )} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^3,x, algorithm="maxima")
Output:
(sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*sqrt(c*x^2 + b*x )/x)*B - 2/3*A*(sqrt(c*x^2 + b*x)*c/(b*x) + sqrt(c*x^2 + b*x)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (61) = 122\).
Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.92 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=-B \sqrt {c} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b \sqrt {c} + A b^{2}\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^3,x, algorithm="giac")
Output:
-B*sqrt(c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b)) + 2/3*( 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x) )^2*A*c + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*sqrt(c) + A*b^2)/(sqrt(c)* x - sqrt(c*x^2 + b*x))^3
Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{x^3} \,d x \] Input:
int(((b*x + c*x^2)^(1/2)*(A + B*x))/x^3,x)
Output:
int(((b*x + c*x^2)^(1/2)*(A + B*x))/x^3, x)
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a b}{3}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a c x}{3}-2 \sqrt {x}\, \sqrt {c x +b}\, b^{2} x +2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{2} x^{2}-\frac {2 \sqrt {c}\, a c \,x^{2}}{3}+\frac {2 \sqrt {c}\, b^{2} x^{2}}{3}}{b \,x^{2}} \] Input:
int((B*x+A)*(c*x^2+b*x)^(1/2)/x^3,x)
Output:
(2*( - sqrt(x)*sqrt(b + c*x)*a*b - sqrt(x)*sqrt(b + c*x)*a*c*x - 3*sqrt(x) *sqrt(b + c*x)*b**2*x + 3*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sq rt(b))*b**2*x**2 - sqrt(c)*a*c*x**2 + sqrt(c)*b**2*x**2))/(3*b*x**2)