Optimal. Leaf size=93 \[ -\frac {15}{4} a^2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x} \]
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Rubi [A] time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {15}{4} a^2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x} \]
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)^{5/2}}{x^{3/2}} \, dx &=-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-(5 b) \int \frac {(a-b x)^{3/2}}{\sqrt {x}} \, dx\\ &=-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {1}{4} (15 a b) \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx\\ &=-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {1}{8} \left (15 a^2 b\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {1}{4} \left (15 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {1}{4} \left (15 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=-\frac {15}{4} a b \sqrt {x} \sqrt {a-b x}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {15}{4} a^2 \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 = 49, normalized size = 0.53 \[ -\frac {2 a^2 \sqrt {a-b x} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {b x}{a}\right )}{\sqrt {x} \sqrt {1-\frac {b x}{a}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 137, normalized size = 1.47 \[ \left [\frac {15 \, a^{2} \sqrt {-b} x \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x^{2} - 9 \, a b x - 8 \, a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, x}, \frac {15 \, a^{2} \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (2 \, b^{2} x^{2} - 9 \, a b x - 8 \, a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, x}\right ] \]
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 [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b x +a \right )^{\frac {5}{2}}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 112, normalized size = 1.20 \[ \frac {15}{4} \, a^{2} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + a} a^{2}}{\sqrt {x}} - \frac {\frac {7 \, \sqrt {-b x + a} a^{2} b^{2}}{\sqrt {x}} + \frac {9 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2} b}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x - a\right )} b}{x} + \frac {{\left (b x - a\right )}^{2}}{x^{2}}\right )}} \]
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 \[ \int \frac {{\left (a-b\,x\right )}^{5/2}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.22, size = 267, normalized size = 2.87 \[ \begin {cases} \frac {2 i a^{\frac {5}{2}}}{\sqrt {x} \sqrt {-1 + \frac {b x}{a}}} + \frac {i a^{\frac {3}{2}} b \sqrt {x}}{4 \sqrt {-1 + \frac {b x}{a}}} - \frac {11 i \sqrt {a} b^{2} x^{\frac {3}{2}}}{4 \sqrt {-1 + \frac {b x}{a}}} + \frac {15 i a^{2} \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4} + \frac {i b^{3} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1 - \frac {b x}{a}}} - \frac {a^{\frac {3}{2}} b \sqrt {x}}{4 \sqrt {1 - \frac {b x}{a}}} + \frac {11 \sqrt {a} b^{2} x^{\frac {3}{2}}}{4 \sqrt {1 - \frac {b x}{a}}} - \frac {15 a^{2} \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4} - \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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