Optimal. Leaf size=121 \[ -\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {5 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223,
209} \begin {gather*} \frac {5 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{3/2}}-\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \sqrt {x} (a-b x)^{5/2} \, dx &=\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {1}{8} (5 a) \int \sqrt {x} (a-b x)^{3/2} \, dx\\ &=\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {1}{16} \left (5 a^2\right ) \int \sqrt {x} \sqrt {a-b x} \, dx\\ &=\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {1}{64} \left (5 a^3\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx\\ &=-\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{128 b}\\ &=-\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{64 b}\\ &=-\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{64 b}\\ &=-\frac {5 a^3 \sqrt {x} \sqrt {a-b x}}{64 b}+\frac {5}{32} a^2 x^{3/2} \sqrt {a-b x}+\frac {5}{24} a x^{3/2} (a-b x)^{3/2}+\frac {1}{4} x^{3/2} (a-b x)^{5/2}+\frac {5 a^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{64 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 93, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x} \sqrt {a-b x} \left (-15 a^3+118 a^2 b x-136 a b^2 x^2+48 b^3 x^3\right )}{192 b}+\frac {5 a^4 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{64 (-b)^{3/2}} \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 = 10.81, size = 256, normalized size = 2.12 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-15 a^{\frac {11}{2}} b \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}+15 a^4 b^{\frac {3}{2}} \sqrt {x} \left (-a+b x\right )+a b^{\frac {5}{2}} x^{\frac {3}{2}} \left (-a+b x\right ) \left (-133 a^2+254 a b x-184 b^2 x^2\right )+48 b^{\frac {11}{2}} x^{\frac {9}{2}} \left (-a+b x\right )\right )}{192 a^{\frac {3}{2}} b^{\frac {5}{2}} \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {5 a^4 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{64 b^{\frac {3}{2}}}-\frac {5 a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {1-\frac {b x}{a}}}+\frac {133 a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {1-\frac {b x}{a}}}-\frac {127 a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {1-\frac {b x}{a}}}+\frac {23 \sqrt {a} b^2 x^{\frac {7}{2}}}{24 \sqrt {1-\frac {b x}{a}}}-\frac {b^3 x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 121, normalized size = 1.00
method | result | size |
risch | \(-\frac {\left (-48 b^{3} x^{3}+136 a \,b^{2} x^{2}-118 a^{2} b x +15 a^{3}\right ) \sqrt {x}\, \sqrt {-b x +a}}{192 b}+\frac {5 a^{4} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{128 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(102\) |
default | \(\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {5}{2}}}{4}+\frac {5 a \left (\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2}+\frac {a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4}\right )}{2}\right )}{8}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 168, normalized size = 1.39 \begin {gather*} -\frac {5 \, a^{4} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{64 \, b^{\frac {3}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{4} b^{3}}{\sqrt {x}} + \frac {55 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {3}{2}}} + \frac {73 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {5}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {7}{2}}}}{192 \, {\left (b^{5} - \frac {4 \, {\left (b x - a\right )} b^{4}}{x} + \frac {6 \, {\left (b x - a\right )}^{2} b^{3}}{x^{2}} - \frac {4 \, {\left (b x - a\right )}^{3} b^{2}}{x^{3}} + \frac {{\left (b x - a\right )}^{4} b}{x^{4}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 164, normalized size = 1.36 \begin {gather*} \left [-\frac {15 \, a^{4} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, b^{4} x^{3} - 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x - 15 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{384 \, b^{2}}, -\frac {15 \, a^{4} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (48 \, b^{4} x^{3} - 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x - 15 \, a^{3} b\right )} \sqrt {-b x + a} \sqrt {x}}{192 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.94, size = 326, normalized size = 2.69 \begin {gather*} \begin {cases} \frac {5 i a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {-1 + \frac {b x}{a}}} - \frac {133 i a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {-1 + \frac {b x}{a}}} + \frac {127 i a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {-1 + \frac {b x}{a}}} - \frac {23 i \sqrt {a} b^{2} x^{\frac {7}{2}}}{24 \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{4} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {i b^{3} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {1 - \frac {b x}{a}}} + \frac {133 a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {1 - \frac {b x}{a}}} - \frac {127 a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {1 - \frac {b x}{a}}} + \frac {23 \sqrt {a} b^{2} x^{\frac {7}{2}}}{24 \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{4} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} - \frac {b^{3} x^{\frac {9}{2}}}{4 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 417 vs.
\(2 (87) = 174\).
time = 41.91, size = 629, normalized size = 5.20 \begin {gather*} \frac {\frac {2 b^{3} \left |b\right | \left (2 \left (\left (\left (-\frac {\frac {1}{23040}\cdot 1440 b^{11} \sqrt {a-b x} \sqrt {a-b x}}{b^{14}}+\frac {\frac {1}{23040}\cdot 6000 b^{11} a}{b^{14}}\right ) \sqrt {a-b x} \sqrt {a-b x}-\frac {\frac {1}{23040}\cdot 9780 b^{11} a^{2}}{b^{14}}\right ) \sqrt {a-b x} \sqrt {a-b x}+\frac {\frac {1}{23040}\cdot 8370 b^{11} a^{3}}{b^{14}}\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}-\frac {70 a^{4} \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{256 b^{2} \sqrt {-b}}\right )}{b^{2}}-\frac {6 a b^{2} \left |b\right | \left (2 \left (\left (\frac {\frac {1}{2304}\cdot 192 b^{5} \sqrt {a-b x} \sqrt {a-b x}}{b^{7}}-\frac {\frac {1}{2304}\cdot 624 b^{5} a}{b^{7}}\right ) \sqrt {a-b x} \sqrt {a-b x}+\frac {\frac {1}{2304}\cdot 792 b^{5} a^{2}}{b^{7}}\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}-\frac {10 a^{3} \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{32 b \sqrt {-b}}\right )}{b^{2}}-\frac {6 a^{2} b \left |b\right | \left (2 \left (\frac {1}{8} \sqrt {a-b x} \sqrt {a-b x}-\frac {10}{32} a\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}+\frac {6 a^{2} b \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{16 \sqrt {-b}}\right )}{b^{2} b}-\frac {2 a^{3} \left |b\right | \left (\frac {1}{2} \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}-\frac {2 a b \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{4 \sqrt {-b}}\right )}{b^{2}}}{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 \sqrt {x}\,{\left (a-b\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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