Integrand size = 17, antiderivative size = 71 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\frac {5}{4} a \sqrt {a x+b x^2}+\frac {1}{2} b x \sqrt {a x+b x^2}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 \sqrt {b}} \] Output:
5/4*a*(b*x^2+a*x)^(1/2)+1/2*b*x*(b*x^2+a*x)^(1/2)+3/4*a^2*arctanh(b^(1/2)* x/(b*x^2+a*x)^(1/2))/b^(1/2)
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{4} \sqrt {x (a+b x)} \left (5 a+2 b x-\frac {3 a^2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b} \sqrt {x} \sqrt {a+b x}}\right ) \] Input:
Integrate[(a*x + b*x^2)^(3/2)/x^2,x]
Output:
(Sqrt[x*(a + b*x)]*(5*a + 2*b*x - (3*a^2*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x])))/4
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1131, 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.
\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 1131 |
\(\displaystyle \frac {3}{4} a \int \frac {\sqrt {b x^2+a x}}{x}dx+\frac {\left (a x+b x^2\right )^{3/2}}{2 x}\) |
\(\Big \downarrow \) 1131 |
\(\displaystyle \frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a x}}dx+\sqrt {a x+b x^2}\right )+\frac {\left (a x+b x^2\right )^{3/2}}{2 x}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {3}{4} a \left (a \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}+\sqrt {a x+b x^2}\right )+\frac {\left (a x+b x^2\right )^{3/2}}{2 x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}}+\sqrt {a x+b x^2}\right )+\frac {\left (a x+b x^2\right )^{3/2}}{2 x}\) |
Input:
Int[(a*x + b*x^2)^(3/2)/x^2,x]
Output:
(a*x + b*x^2)^(3/2)/(2*x) + (3*a*(Sqrt[a*x + b*x^2] + (a*ArcTanh[(Sqrt[b]* x)/Sqrt[a*x + b*x^2]])/Sqrt[b]))/4
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]
Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\left (2 b x +5 a \right ) x \left (b x +a \right )}{4 \sqrt {x \left (b x +a \right )}}+\frac {3 a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 \sqrt {b}}\) | \(59\) |
pseudoelliptic | \(\frac {2 b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, x +3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) a^{2}+5 a \sqrt {b}\, \sqrt {x \left (b x +a \right )}}{4 \sqrt {b}}\) | \(59\) |
default | \(\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\) | \(99\) |
Input:
int((b*x^2+a*x)^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
1/4*(2*b*x+5*a)*x*(b*x+a)/(x*(b*x+a))^(1/2)+3/8*a^2*ln((1/2*a+b*x)/b^(1/2) +(b*x^2+a*x)^(1/2))/b^(1/2)
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (2 \, b^{2} x + 5 \, a b\right )} \sqrt {b x^{2} + a x}}{8 \, b}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (2 \, b^{2} x + 5 \, a b\right )} \sqrt {b x^{2} + a x}}{4 \, b}\right ] \] Input:
integrate((b*x^2+a*x)^(3/2)/x^2,x, algorithm="fricas")
Output:
[1/8*(3*a^2*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(2*b^ 2*x + 5*a*b)*sqrt(b*x^2 + a*x))/b, -1/4*(3*a^2*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (2*b^2*x + 5*a*b)*sqrt(b*x^2 + a*x))/b]
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate((b*x**2+a*x)**(3/2)/x**2,x)
Output:
Integral((x*(a + b*x))**(3/2)/x**2, x)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\frac {3 \, a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, \sqrt {b}} + \frac {3}{4} \, \sqrt {b x^{2} + a x} a + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{2 \, x} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^2,x, algorithm="maxima")
Output:
3/8*a^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/sqrt(b) + 3/4*sqrt(b* x^2 + a*x)*a + 1/2*(b*x^2 + a*x)^(3/2)/x
Time = 0.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=-\frac {3 \, a^{2} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{8 \, \sqrt {b}} + \frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (2 \, b x + 5 \, a\right )} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^2,x, algorithm="giac")
Output:
-3/8*a^2*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/sqrt(b) + 1/4*sqrt(b*x^2 + a*x)*(2*b*x + 5*a)
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}}{x^2} \,d x \] Input:
int((a*x + b*x^2)^(3/2)/x^2,x)
Output:
int((a*x + b*x^2)^(3/2)/x^2, x)
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx=\frac {5 \sqrt {x}\, \sqrt {b x +a}\, a b +2 \sqrt {x}\, \sqrt {b x +a}\, b^{2} x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2}}{4 b} \] Input:
int((b*x^2+a*x)^(3/2)/x^2,x)
Output:
(5*sqrt(x)*sqrt(a + b*x)*a*b + 2*sqrt(x)*sqrt(a + b*x)*b**2*x + 3*sqrt(b)* log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2)/(4*b)