Optimal. Leaf size=48 \[ -\frac {4 b \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{35 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{7 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270}
\begin {gather*} \frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{7 a}-\frac {4 b x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{35 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 277
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{3/2} x^{5/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{7 a}-\frac {(2 b) \int \left (a+\frac {b}{x}\right )^{3/2} x^{3/2} \, dx}{7 a}\\ &=-\frac {4 b \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{35 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{7 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 5.12, size = 38, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (b+a x)^2 (-2 b+5 a x)}{35 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 35, normalized size = 0.73
method | result | size |
gosper | \(\frac {2 \left (a x +b \right ) \left (5 a x -2 b \right ) x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{35 a^{2}}\) | \(33\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +b \right )^{2} \left (5 a x -2 b \right )}{35 a^{2}}\) | \(35\) |
risch | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (5 a^{3} x^{3}+8 a^{2} b \,x^{2}+a \,b^{2} x -2 b^{3}\right )}{35 a^{2}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 35, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 7 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}}\right )}}{35 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (5 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} + a b^{2} x - 2 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{35 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (39) = 78\).
time = 12.23, size = 88, normalized size = 1.83 \begin {gather*} \frac {2 a \sqrt {b} x^{3} \sqrt {\frac {a x}{b} + 1}}{7} + \frac {16 b^{\frac {3}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{35} + \frac {2 b^{\frac {5}{2}} x \sqrt {\frac {a x}{b} + 1}}{35 a} - \frac {4 b^{\frac {7}{2}} \sqrt {\frac {a x}{b} + 1}}{35 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (36) = 72\).
time = 1.07, size = 90, normalized size = 1.88 \begin {gather*} \frac {2}{15} \, b {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\left (x\right ) - \frac {2}{105} \, a {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.43, size = 45, normalized size = 0.94 \begin {gather*} \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{7/2}}{7}+\frac {16\,b\,x^{5/2}}{35}+\frac {2\,b^2\,x^{3/2}}{35\,a}-\frac {4\,b^3\,\sqrt {x}}{35\,a^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________