Optimal. Leaf size=93 \[ -\frac {15}{4} a b \sqrt {x} \sqrt {a-b x}-\frac {5}{2} b \sqrt {x} (a-b x)^{3/2}-\frac {2 (a-b x)^{5/2}}{\sqrt {x}}-\frac {15}{4} a^2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]
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Rubi [A]
time = 0.02, 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 = {49, 52, 65, 223,
209} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 209
Rule 223
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 ) \text {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 ) \text {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 [A]
time = 0.12, size = 79, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a-b x} \left (-8 a^2-9 a b x+2 b^2 x^2\right )}{4 \sqrt {x}}-\frac {15}{4} a^2 \sqrt {-b} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 5.86, size = 216, normalized size = 2.32 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (8 a^3 \left (-a+b x\right )+15 a^{\frac {7}{2}} \sqrt {b} \sqrt {x} \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}+a b x \left (a-11 b x\right ) \left (-a+b x\right )+2 b^3 x^3 \left (-a+b x\right )\right )}{4 a^{\frac {3}{2}} \sqrt {x} \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {-2 a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1-\frac {b x}{a}}}-\frac {15 a^2 \sqrt {b} \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{4}-\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 {b^3 x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 88, normalized size = 0.95
method | result | size |
risch | \(-\frac {\sqrt {-b x +a}\, \left (-2 x^{2} b^{2}+9 a b x +8 a^{2}\right )}{4 \sqrt {x}}-\frac {15 a^{2} \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{8 \sqrt {x}\, \sqrt {-b x +a}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 112, normalized size = 1.20 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 137, normalized size = 1.47 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.98, size = 267, normalized size = 2.87 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 10.45, size = 167, normalized size = 1.80 \begin {gather*} -\frac {b b^{2} \left (\frac {2 \left (\left (-\frac {1}{4} \sqrt {a-b x} \sqrt {a-b x}-\frac {5}{8} a\right ) \sqrt {a-b x} \sqrt {a-b x}+\frac {15}{8} a^{2}\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}}{a b-b \left (a-b x\right )}+\frac {30 a^{2} \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{8 \sqrt {-b}}\right )}{\left |b\right | b} \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 = -0.01 \begin {gather*} \int \frac {{\left (a-b\,x\right )}^{5/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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