Optimal. Leaf size=96 \[ \frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 \sqrt {b}}+\frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {50, 63, 217, 203} \[ \frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 \sqrt {b}}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {(a-b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {1}{6} (5 a) \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a-b x}+\frac {5}{12} a \sqrt {x} (a-b x)^{3/2}+\frac {1}{3} \sqrt {x} (a-b x)^{5/2}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.85 \[ \frac {1}{24} \sqrt {a-b x} \left (\frac {15 a^{5/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {1-\frac {b x}{a}}}+\sqrt {x} \left (33 a^2-26 a b x+8 b^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 142, normalized size = 1.48 \[ \left [-\frac {15 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, b^{3} x^{2} - 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b}, -\frac {15 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (8 \, b^{3} x^{2} - 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 99, normalized size = 1.03 \[ \frac {5 \sqrt {\left (-b x +a \right ) x}\, a^{3} \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{16 \sqrt {-b x +a}\, \sqrt {b}\, \sqrt {x}}+\frac {5 \sqrt {-b x +a}\, a^{2} \sqrt {x}}{8}+\frac {5 \left (-b x +a \right )^{\frac {3}{2}} a \sqrt {x}}{12}+\frac {\left (-b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 130, normalized size = 1.35 \[ -\frac {5 \, a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, \sqrt {b}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{3} - \frac {3 \, {\left (b x - a\right )} b^{2}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b}{x^{2}} - \frac {{\left (b x - a\right )}^{3}}{x^{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.23, size = 246, normalized size = 2.56 \[ \begin {cases} - \frac {11 i a^{\frac {5}{2}} \sqrt {x}}{8 \sqrt {-1 + \frac {b x}{a}}} + \frac {59 i a^{\frac {3}{2}} b x^{\frac {3}{2}}}{24 \sqrt {-1 + \frac {b x}{a}}} - \frac {17 i \sqrt {a} b^{2} x^{\frac {5}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {i b^{3} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {11 a^{\frac {5}{2}} \sqrt {x} \sqrt {1 - \frac {b x}{a}}}{8} - \frac {13 a^{\frac {3}{2}} b x^{\frac {3}{2}} \sqrt {1 - \frac {b x}{a}}}{12} + \frac {\sqrt {a} b^{2} x^{\frac {5}{2}} \sqrt {1 - \frac {b x}{a}}}{3} + \frac {5 a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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