Optimal. Leaf size=100 \[ \frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 43, 65, 211}
\begin {gather*} \frac {(3 a B+A b) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}-\frac {\sqrt {x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac {x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 65
Rule 79
Rule 211
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{(a+b x)^3} \, dx &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}+\frac {(A b+3 a B) \int \frac {\sqrt {x}}{(a+b x)^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a b^2}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a b^2}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 86, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {x} \left (a A b+3 a^2 B-A b^2 x+5 a b B x\right )}{4 a b^2 (a+b x)^2}+\frac {(A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 79, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\left (A b -5 B a \right ) x^{\frac {3}{2}}}{4 a b}-\frac {\left (A b +3 B a \right ) \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}}+\frac {\left (A b +3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} a \sqrt {a b}}\) | \(79\) |
default | \(\frac {\frac {\left (A b -5 B a \right ) x^{\frac {3}{2}}}{4 a b}-\frac {\left (A b +3 B a \right ) \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}}+\frac {\left (A b +3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} a \sqrt {a b}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 94, normalized size = 0.94 \begin {gather*} -\frac {{\left (5 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} + {\left (3 \, B a^{2} + A a b\right )} \sqrt {x}}{4 \, {\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} + \frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.23, size = 291, normalized size = 2.91 \begin {gather*} \left [-\frac {{\left (3 \, B a^{3} + A a^{2} b + {\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (3 \, B a^{3} b + A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, -\frac {{\left (3 \, B a^{3} + A a^{2} b + {\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (3 \, B a^{3} b + A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \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. 1316 vs.
\(2 (90) = 180\).
time = 8.90, size = 1316, normalized size = 13.16 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{b^{3}} & \text {for}\: a = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{a^{3}} & \text {for}\: b = 0 \\\frac {A a^{2} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {A a^{2} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 A a b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 A a b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 A a b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 A b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {A b^{3} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {A b^{3} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 B a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 B a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 B a^{2} b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 B a^{2} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 B a^{2} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {10 B a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 B a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 B a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{3} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{4} x \sqrt {- \frac {a}{b}} + 8 a b^{5} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.47, size = 82, normalized size = 0.82 \begin {gather*} \frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b^{2}} - \frac {5 \, B a b x^{\frac {3}{2}} - A b^{2} x^{\frac {3}{2}} + 3 \, B a^{2} \sqrt {x} + A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.45, size = 84, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,a^{3/2}\,b^{5/2}}-\frac {\frac {\sqrt {x}\,\left (A\,b+3\,B\,a\right )}{4\,b^2}-\frac {x^{3/2}\,\left (A\,b-5\,B\,a\right )}{4\,a\,b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________