Integrand size = 22, antiderivative size = 94 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\frac {(4 b c-a d) \sqrt {a x+b x^2}}{4 b}+\frac {d \left (a x+b x^2\right )^{3/2}}{2 b x}+\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}} \] Output:
1/4*(-a*d+4*b*c)*(b*x^2+a*x)^(1/2)/b+1/2*d*(b*x^2+a*x)^(3/2)/b/x+1/4*a*(-a *d+4*b*c)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(3/2)
Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\frac {\sqrt {x (a+b x)} \left (\sqrt {b} (4 b c+a d+2 b d x)+\frac {a (-4 b c+a d) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {x} \sqrt {a+b x}}\right )}{4 b^{3/2}} \] Input:
Integrate[((c + d*x)*Sqrt[a*x + b*x^2])/x,x]
Output:
(Sqrt[x*(a + b*x)]*(Sqrt[b]*(4*b*c + a*d + 2*b*d*x) + (a*(-4*b*c + a*d)*Lo g[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(Sqrt[x]*Sqrt[a + b*x])))/(4*b^(3/2 ))
Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1221, 1131, 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.
\(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)}{x} \, dx\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle \frac {(4 b c-a d) \int \frac {\sqrt {b x^2+a x}}{x}dx}{4 b}+\frac {d \left (a x+b x^2\right )^{3/2}}{2 b x}\) |
\(\Big \downarrow \) 1131 |
\(\displaystyle \frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a x}}dx+\sqrt {a x+b x^2}\right )}{4 b}+\frac {d \left (a x+b x^2\right )^{3/2}}{2 b x}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {(4 b c-a d) \left (a \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}+\sqrt {a x+b x^2}\right )}{4 b}+\frac {d \left (a x+b x^2\right )^{3/2}}{2 b x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}}+\sqrt {a x+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d \left (a x+b x^2\right )^{3/2}}{2 b x}\) |
Input:
Int[((c + d*x)*Sqrt[a*x + b*x^2])/x,x]
Output:
(d*(a*x + b*x^2)^(3/2))/(2*b*x) + ((4*b*c - a*d)*(Sqrt[a*x + b*x^2] + (a*A rcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/Sqrt[b]))/(4*b)
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_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b *d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne Q[m + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) 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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(-\frac {\left (a^{2} d -4 a b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\left (2 d x +4 c \right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) \sqrt {x \left (b x +a \right )}}{4 b^{\frac {3}{2}}}\) | \(66\) |
risch | \(\frac {\left (2 b d x +a d +4 b c \right ) x \left (b x +a \right )}{4 b \sqrt {x \left (b x +a \right )}}-\frac {a \left (a d -4 b c \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\) | \(73\) |
default | \(d \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )+c \left (\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}\right )\) | \(103\) |
Input:
int((d*x+c)*(b*x^2+a*x)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/4/b^(3/2)*((a^2*d-4*a*b*c)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-((2*d*x +4*c)*b^(3/2)+b^(1/2)*a*d)*(x*(b*x+a))^(1/2))
Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\left [-\frac {{\left (4 \, a b c - a^{2} d\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (2 \, b^{2} d x + 4 \, b^{2} c + a b d\right )} \sqrt {b x^{2} + a x}}{8 \, b^{2}}, -\frac {{\left (4 \, a b c - a^{2} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (2 \, b^{2} d x + 4 \, b^{2} c + a b d\right )} \sqrt {b x^{2} + a x}}{4 \, b^{2}}\right ] \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x,x, algorithm="fricas")
Output:
[-1/8*((4*a*b*c - a^2*d)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt( b)) - 2*(2*b^2*d*x + 4*b^2*c + a*b*d)*sqrt(b*x^2 + a*x))/b^2, -1/4*((4*a*b *c - a^2*d)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (2*b^2 *d*x + 4*b^2*c + a*b*d)*sqrt(b*x^2 + a*x))/b^2]
\[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (c + d x\right )}{x}\, dx \] Input:
integrate((d*x+c)*(b*x**2+a*x)**(1/2)/x,x)
Output:
Integral(sqrt(x*(a + b*x))*(c + d*x)/x, x)
Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a x} d x + \frac {a c \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, \sqrt {b}} - \frac {a^{2} d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} + \sqrt {b x^{2} + a x} c + \frac {\sqrt {b x^{2} + a x} a d}{4 \, b} \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x,x, algorithm="maxima")
Output:
1/2*sqrt(b*x^2 + a*x)*d*x + 1/2*a*c*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sq rt(b))/sqrt(b) - 1/8*a^2*d*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^ (3/2) + sqrt(b*x^2 + a*x)*c + 1/4*sqrt(b*x^2 + a*x)*a*d/b
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (2 \, d x + \frac {4 \, b c + a d}{b}\right )} - \frac {{\left (4 \, a b c - a^{2} d\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{8 \, b^{\frac {3}{2}}} \] Input:
integrate((d*x+c)*(b*x^2+a*x)^(1/2)/x,x, algorithm="giac")
Output:
1/4*sqrt(b*x^2 + a*x)*(2*d*x + (4*b*c + a*d)/b) - 1/8*(4*a*b*c - a^2*d)*lo g(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(3/2)
Time = 8.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=c\,\sqrt {b\,x^2+a\,x}+d\,\sqrt {b\,x^2+a\,x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )-\frac {a^2\,d\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{8\,b^{3/2}}+\frac {a\,c\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{2\,\sqrt {b}} \] Input:
int(((a*x + b*x^2)^(1/2)*(c + d*x))/x,x)
Output:
c*(a*x + b*x^2)^(1/2) + d*(a*x + b*x^2)^(1/2)*(x/2 + a/(4*b)) - (a^2*d*log ((a/2 + b*x)/b^(1/2) + (a*x + b*x^2)^(1/2)))/(8*b^(3/2)) + (a*c*log((a/2 + b*x)/b^(1/2) + (a*x + b*x^2)^(1/2)))/(2*b^(1/2))
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x) \sqrt {a x+b x^2}}{x} \, dx=\frac {\sqrt {x}\, \sqrt {b x +a}\, a b d +4 \sqrt {x}\, \sqrt {b x +a}\, b^{2} c +2 \sqrt {x}\, \sqrt {b x +a}\, b^{2} d x -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} d +4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a b c}{4 b^{2}} \] Input:
int((d*x+c)*(b*x^2+a*x)^(1/2)/x,x)
Output:
(sqrt(x)*sqrt(a + b*x)*a*b*d + 4*sqrt(x)*sqrt(a + b*x)*b**2*c + 2*sqrt(x)* sqrt(a + b*x)*b**2*d*x - sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqr t(a))*a**2*d + 4*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a* b*c)/(4*b**2)