Optimal. Leaf size=84 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{3/2}}-\frac {b \sqrt {a x^2+b x^3}}{4 a x^2}-\frac {\sqrt {a x^2+b x^3}}{2 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2008, 206} \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{3/2}}-\frac {b \sqrt {a x^2+b x^3}}{4 a x^2}-\frac {\sqrt {a x^2+b x^3}}{2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2008
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3}}{x^4} \, dx &=-\frac {\sqrt {a x^2+b x^3}}{2 x^3}+\frac {1}{4} b \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3}}{4 a x^2}-\frac {b^2 \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{8 a}\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3}}{4 a x^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{4 a}\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3}}{4 a x^2}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 42, normalized size = 0.50 \[ -\frac {2 b^2 \left (x^2 (a+b x)\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x}{a}+1\right )}{3 a^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 149, normalized size = 1.77 \[ \left [\frac {\sqrt {a} b^{2} x^{3} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, \sqrt {b x^{3} + a x^{2}} {\left (a b x + 2 \, a^{2}\right )}}{8 \, a^{2} x^{3}}, -\frac {\sqrt {-a} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (a b x + 2 \, a^{2}\right )}}{4 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 72, normalized size = 0.86 \[ -\frac {\frac {b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a} a} + \frac {{\left (b x + a\right )}^{\frac {3}{2}} b^{3} \mathrm {sgn}\relax (x) + \sqrt {b x + a} a b^{3} \mathrm {sgn}\relax (x)}{a b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 73, normalized size = 0.87 \[ -\frac {\sqrt {b \,x^{3}+a \,x^{2}}\, \left (-a \,b^{2} x^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\sqrt {b x +a}\, a^{\frac {5}{2}}+\left (b x +a \right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{4 \sqrt {b x +a}\, a^{\frac {5}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{3} + a x^{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {b\,x^3+a\,x^2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} \left (a + b x\right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________