Integrand size = 17, antiderivative size = 64 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=b \sqrt {a x+b x^2}-\frac {2 a \sqrt {a x+b x^2}}{x}+3 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \] Output:
b*(b*x^2+a*x)^(1/2)-2*a*(b*x^2+a*x)^(1/2)/x+3*a*b^(1/2)*arctanh(b^(1/2)*x/ (b*x^2+a*x)^(1/2))
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {a+b x} \left ((-2 a+b x) \sqrt {a+b x}+6 a \sqrt {b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{\sqrt {x (a+b x)}} \] Input:
Integrate[(a*x + b*x^2)^(3/2)/x^3,x]
Output:
(Sqrt[a + b*x]*((-2*a + b*x)*Sqrt[a + b*x] + 6*a*Sqrt[b]*Sqrt[x]*ArcTanh[( Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/Sqrt[x*(a + b*x)]
Time = 0.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1125, 25, 27, 1160, 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.
\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 1125 |
\(\displaystyle -\int -\frac {b (2 a+b x)}{\sqrt {b x^2+a x}}dx-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {b (2 a+b x)}{\sqrt {b x^2+a x}}dx-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {2 a+b x}{\sqrt {b x^2+a x}}dx-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle b \left (\frac {3}{2} a \int \frac {1}{\sqrt {b x^2+a x}}dx+\sqrt {a x+b x^2}\right )-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle b \left (3 a \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}+\sqrt {a x+b x^2}\right )-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle b \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}}+\sqrt {a x+b x^2}\right )-\frac {2 a \sqrt {a x+b x^2}}{x}\) |
Input:
Int[(a*x + b*x^2)^(3/2)/x^3,x]
Output:
(-2*a*Sqrt[a*x + b*x^2])/x + b*(Sqrt[a*x + b*x^2] + (3*a*ArcTanh[(Sqrt[b]* x)/Sqrt[a*x + b*x^2]])/Sqrt[b])
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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {\left (b x +a \right ) \left (-b x +2 a \right )}{\sqrt {x \left (b x +a \right )}}+\frac {3 a \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2}\) | \(56\) |
pseudoelliptic | \(\frac {b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, x +3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) a b x -2 a \sqrt {b}\, \sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\) | \(60\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{3}}+\frac {4 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\right )}{a}\) | \(125\) |
Input:
int((b*x^2+a*x)^(3/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-(b*x+a)*(-b*x+2*a)/(x*(b*x+a))^(1/2)+3/2*a*b^(1/2)*ln((1/2*a+b*x)/b^(1/2) +(b*x^2+a*x)^(1/2))
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\left [\frac {3 \, a \sqrt {b} x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, \sqrt {b x^{2} + a x} {\left (b x - 2 \, a\right )}}{2 \, x}, -\frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - \sqrt {b x^{2} + a x} {\left (b x - 2 \, a\right )}}{x}\right ] \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3,x, algorithm="fricas")
Output:
[1/2*(3*a*sqrt(b)*x*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*sqrt( b*x^2 + a*x)*(b*x - 2*a))/x, -(3*a*sqrt(-b)*x*arctan(sqrt(b*x^2 + a*x)*sqr t(-b)/(b*x + a)) - sqrt(b*x^2 + a*x)*(b*x - 2*a))/x]
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \] Input:
integrate((b*x**2+a*x)**(3/2)/x**3,x)
Output:
Integral((x*(a + b*x))**(3/2)/x**3, x)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\frac {3}{2} \, a \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {3 \, \sqrt {b x^{2} + a x} a}{x} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{x^{2}} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3,x, algorithm="maxima")
Output:
3/2*a*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 3*sqrt(b*x^2 + a*x)*a/x + (b*x^2 + a*x)^(3/2)/x^2
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=-\frac {3}{2} \, a \sqrt {b} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right ) + \sqrt {b x^{2} + a x} b + \frac {2 \, a^{2}}{\sqrt {b} x - \sqrt {b x^{2} + a x}} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^3,x, algorithm="giac")
Output:
-3/2*a*sqrt(b)*log(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a)) + sqrt(b*x^2 + a*x)*b + 2*a^2/(sqrt(b)*x - sqrt(b*x^2 + a*x))
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}}{x^3} \,d x \] Input:
int((a*x + b*x^2)^(3/2)/x^3,x)
Output:
int((a*x + b*x^2)^(3/2)/x^3, x)
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^3} \, dx=\frac {-8 \sqrt {x}\, \sqrt {b x +a}\, a +4 \sqrt {x}\, \sqrt {b x +a}\, b x +12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a x -9 \sqrt {b}\, a x}{4 x} \] Input:
int((b*x^2+a*x)^(3/2)/x^3,x)
Output:
( - 8*sqrt(x)*sqrt(a + b*x)*a + 4*sqrt(x)*sqrt(a + b*x)*b*x + 12*sqrt(b)*l og((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*x - 9*sqrt(b)*a*x)/(4*x)