Optimal. Leaf size=32 \[ -\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^{5/2}} \, dx &=\int \left (\frac {a^2}{x^{5/2}}+\frac {2 a b}{x^{3/2}}+\frac {b^2}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 26, normalized size = 0.81 \begin {gather*} -\frac {2 \left (a^2+6 a b x-3 b^2 x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 25, normalized size = 0.78
method | result | size |
gosper | \(-\frac {2 \left (-3 x^{2} b^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(23\) |
trager | \(-\frac {2 \left (-3 x^{2} b^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(23\) |
risch | \(-\frac {2 \left (-3 x^{2} b^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\) | \(23\) |
derivativedivides | \(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 b^{2} \sqrt {x}\) | \(25\) |
default | \(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 b^{2} \sqrt {x}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 23, normalized size = 0.72 \begin {gather*} 2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.44, size = 24, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.22, size = 31, normalized size = 0.97 \begin {gather*} - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 a b}{\sqrt {x}} + 2 b^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.55, size = 23, normalized size = 0.72 \begin {gather*} 2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 24, normalized size = 0.75 \begin {gather*} -\frac {2\,a^2+12\,a\,b\,x-6\,b^2\,x^2}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________