Optimal. Leaf size=66 \[ -3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 a \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {49, 52, 65, 223,
209} \begin {gather*} -3 a \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 b \sqrt {x} \sqrt {a-b x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx &=-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 b) \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx\\ &=-3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=-3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 a b) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-(3 a b) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=-3 b \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 a \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 61, normalized size = 0.92 \begin {gather*} \frac {(-2 a-b x) \sqrt {a-b x}}{\sqrt {x}}-3 a \sqrt {-b} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 74, normalized size = 1.12
method | result | size |
risch | \(-\frac {\sqrt {-b x +a}\, \left (b x +2 a \right )}{\sqrt {x}}-\frac {3 a \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{2 \sqrt {x}\, \sqrt {-b x +a}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 68, normalized size = 1.03 \begin {gather*} 3 \, a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + a} a}{\sqrt {x}} - \frac {\sqrt {-b x + a} a b}{{\left (b - \frac {b x - a}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.63, size = 109, normalized size = 1.65 \begin {gather*} \left [\frac {3 \, a \sqrt {-b} x \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (b x + 2 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{2 \, x}, \frac {3 \, a \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (b x + 2 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.67, size = 197, normalized size = 2.98 \begin {gather*} \begin {cases} \frac {2 i a^{\frac {3}{2}}}{\sqrt {x} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} b \sqrt {x}}{\sqrt {-1 + \frac {b x}{a}}} + 3 i a \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {i b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 a^{\frac {3}{2}}}{\sqrt {x} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} b \sqrt {x}}{\sqrt {1 - \frac {b x}{a}}} - 3 a \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {b^{2} x^{\frac {3}{2}}}{\sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a-b\,x\right )}^{3/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________