Integrand size = 22, antiderivative size = 73 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=-\frac {2 d \sqrt {a x+b x^2}}{x}-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}+2 \sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \] Output:
-2*d*(b*x^2+a*x)^(1/2)/x-2/3*c*(b*x^2+a*x)^(3/2)/a/x^3+2*b^(1/2)*d*arctanh (b^(1/2)*x/(b*x^2+a*x)^(1/2))
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=-\frac {2 \sqrt {x (a+b x)} (a c+b c x+3 a d x)}{3 a x^2}-\frac {2 \sqrt {b} d \sqrt {x (a+b x)} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {x} \sqrt {a+b x}} \] Input:
Integrate[((c + d*x)*Sqrt[a*x + b*x^2])/x^3,x]
Output:
(-2*Sqrt[x*(a + b*x)]*(a*c + b*c*x + 3*a*d*x))/(3*a*x^2) - (2*Sqrt[b]*d*Sq rt[x*(a + b*x)]*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(Sqrt[x]*Sqrt[a + b*x])
Time = 0.39 (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.
\(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)}{x^3} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle d \int \frac {\sqrt {b x^2+a x}}{x^2}dx-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
\(\Big \downarrow \) 1125 |
\(\displaystyle d \left (-\int -\frac {b}{\sqrt {b x^2+a x}}dx-\frac {2 \sqrt {a x+b x^2}}{x}\right )-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle d \left (\int \frac {b}{\sqrt {b x^2+a x}}dx-\frac {2 \sqrt {a x+b x^2}}{x}\right )-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d \left (b \int \frac {1}{\sqrt {b x^2+a x}}dx-\frac {2 \sqrt {a x+b x^2}}{x}\right )-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle d \left (2 b \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}-\frac {2 \sqrt {a x+b x^2}}{x}\right )-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle d \left (2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )-\frac {2 \sqrt {a x+b x^2}}{x}\right )-\frac {2 c \left (a x+b x^2\right )^{3/2}}{3 a x^3}\) |
Input:
Int[((c + d*x)*Sqrt[a*x + b*x^2])/x^3,x]
Output:
(-2*c*(a*x + b*x^2)^(3/2))/(3*a*x^3) + d*((-2*Sqrt[a*x + b*x^2])/x + 2*Sqr t[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*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 = 0.44 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {6 d a \sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) x^{2}-2 \sqrt {x \left (b x +a \right )}\, \left (\left (3 d x +c \right ) a +c b x \right )}{3 a \,x^{2}}\) | \(61\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (3 a d x +c b x +a c \right )}{3 x \sqrt {x \left (b x +a \right )}\, a}+\sqrt {b}\, d \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )\) | \(66\) |
default | \(-\frac {2 c \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 a \,x^{3}}+d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{a \,x^{2}}+\frac {2 b \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )}{a}\right )\) | \(92\) |
Input:
int((d*x+c)*(b*x^2+a*x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/3*(6*d*a*b^(1/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))*x^2-2*(x*(b*x+a))^ (1/2)*((3*d*x+c)*a+c*b*x))/a/x^2
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.95 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=\left [\frac {3 \, a \sqrt {b} d x^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, \sqrt {b x^{2} + a x} {\left (a c + {\left (b c + 3 \, a d\right )} x\right )}}{3 \, a x^{2}}, -\frac {2 \, {\left (3 \, a \sqrt {-b} d x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + \sqrt {b x^{2} + a x} {\left (a c + {\left (b c + 3 \, a d\right )} x\right )}\right )}}{3 \, a x^{2}}\right ] \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x^3,x, algorithm="fricas")
Output:
[1/3*(3*a*sqrt(b)*d*x^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*s qrt(b*x^2 + a*x)*(a*c + (b*c + 3*a*d)*x))/(a*x^2), -2/3*(3*a*sqrt(-b)*d*x^ 2*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + sqrt(b*x^2 + a*x)*(a*c + (b*c + 3*a*d)*x))/(a*x^2)]
\[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (c + d x\right )}{x^{3}}\, dx \] Input:
integrate((d*x+c)*(b*x**2+a*x)**(1/2)/x**3,x)
Output:
Integral(sqrt(x*(a + b*x))*(c + d*x)/x**3, x)
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx={\left (\sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {2 \, \sqrt {b x^{2} + a x}}{x}\right )} d - \frac {2}{3} \, c {\left (\frac {\sqrt {b x^{2} + a x} b}{a x} + \frac {\sqrt {b x^{2} + a x}}{x^{2}}\right )} \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x^3,x, algorithm="maxima")
Output:
(sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*sqrt(b*x^2 + a*x )/x)*d - 2/3*c*(sqrt(b*x^2 + a*x)*b/(a*x) + sqrt(b*x^2 + a*x)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (61) = 122\).
Time = 0.34 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.92 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=-\sqrt {b} d \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b c + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a \sqrt {b} c + a^{2} c\right )}}{3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3}} \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x^3,x, algorithm="giac")
Output:
-sqrt(b)*d*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a)) + 2/3*( 3*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*b*c + 3*(sqrt(b)*x - sqrt(b*x^2 + a*x) )^2*a*d + 3*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a*sqrt(b)*c + a^2*c)/(sqrt(b)* x - sqrt(b*x^2 + a*x))^3
Timed out. \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=\int \frac {\sqrt {b\,x^2+a\,x}\,\left (c+d\,x\right )}{x^3} \,d x \] Input:
int(((a*x + b*x^2)^(1/2)*(c + d*x))/x^3,x)
Output:
int(((a*x + b*x^2)^(1/2)*(c + d*x))/x^3, x)
Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x^3} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a c}{3}-2 \sqrt {x}\, \sqrt {b x +a}\, a d x -\frac {2 \sqrt {x}\, \sqrt {b x +a}\, b c x}{3}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a d \,x^{2}+\frac {2 \sqrt {b}\, a d \,x^{2}}{3}-\frac {2 \sqrt {b}\, b c \,x^{2}}{3}}{a \,x^{2}} \] Input:
int((d*x+c)*(b*x^2+a*x)^(1/2)/x^3,x)
Output:
(2*( - sqrt(x)*sqrt(a + b*x)*a*c - 3*sqrt(x)*sqrt(a + b*x)*a*d*x - sqrt(x) *sqrt(a + b*x)*b*c*x + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqr t(a))*a*d*x**2 + sqrt(b)*a*d*x**2 - sqrt(b)*b*c*x**2))/(3*a*x**2)