Optimal. Leaf size=66 \[ -\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 b \sqrt {x} \sqrt {a-b x}-3 a \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
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Rubi [A] time = 0.02, 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 = {47, 50, 63, 217, 203} \begin {gather*} -\frac {2 (a-b x)^{3/2}}{\sqrt {x}}-3 b \sqrt {x} \sqrt {a-b x}-3 a \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 203
Rule 217
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) \operatorname {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) \operatorname {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*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.71 \begin {gather*} -\frac {2 a \sqrt {a-b x} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {b x}{a}\right )}{\sqrt {x} \sqrt {1-\frac {b x}{a}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, 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 {a-b x}-\sqrt {-b} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, 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.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b x +a \right )^{\frac {3}{2}}}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.85, 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.
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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.
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sympy [A] time = 2.88, 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.
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