Optimal. Leaf size=56 \[ \frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}}-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}}-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}}-\frac {(2 b) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}}+\frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 31, normalized size = 0.55 \[ -\frac {2 (a-2 b x) \sqrt {x^2 (a+b x)}}{3 a^2 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 29, normalized size = 0.52 \[ \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{3 \, a^{2} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 55, normalized size = 0.98 \[ \frac {8 \, {\left (3 \, {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 33, normalized size = 0.59 \[ -\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {b \,x^{3}+a \,x^{2}}\, a^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{3} + a x^{2}} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^{3/2}\,\sqrt {b\,x^3+a\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {3}{2}} \sqrt {x^{2} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________