Optimal. Leaf size=72 \[ -\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {2 x^3}{b \sqrt {a x^2+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 1588} \[ \frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x}-\frac {2 x^3}{b \sqrt {a x^2+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1588
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {4 \int \frac {x^2}{\sqrt {a x^2+b x^3}} \, dx}{b}\\ &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {(8 a) \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx}{3 b^2}\\ &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 39, normalized size = 0.54 \[ \frac {2 x \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 49, normalized size = 0.68 \[ \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{3 \, {\left (b^{4} x^{2} + a b^{3} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 46, normalized size = 0.64 \[ -\frac {2 \left (b x +a \right ) \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right ) x^{3}}{3 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.44, size = 30, normalized size = 0.42 \[ \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )}}{3 \, \sqrt {b x + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.22, size = 47, normalized size = 0.65 \[ -\frac {2\,\sqrt {b\,x^3+a\,x^2}\,\left (8\,a^2+4\,a\,b\,x-b^2\,x^2\right )}{3\,b^3\,x\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________