Optimal. Leaf size=100 \[ \frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {15 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, 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 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{7/2}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}+\frac {2 x^{5/2}}{b \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 {x^{5/2}}{(a-b x)^{3/2}} \, dx &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}-\frac {5 \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{b}\\ &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {(15 a) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{4 b^2}\\ &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {\left (15 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{8 b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {\left (15 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {\left (15 a^2\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^3}\\ &=\frac {2 x^{5/2}}{b \sqrt {a-b x}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}-\frac {15 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 81, normalized size = 0.81 \begin {gather*} \frac {1}{4} \left (\frac {\sqrt {x} \left (15 a^2-5 a b x-2 b^2 x^2\right )}{b^3 \sqrt {a-b x}}-\frac {15 a^2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{7/2}}\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 = 7.52, size = 201, normalized size = 2.01 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (15 a^{\frac {7}{2}} b^6 \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}-15 a^2 b^{\frac {13}{2}} \sqrt {x} \left (-a+b x\right )+5 a b^{\frac {15}{2}} x^{\frac {3}{2}} \left (-a+b x\right )+2 b^{\frac {17}{2}} x^{\frac {5}{2}} \left (-a+b x\right )\right )}{4 a^{\frac {3}{2}} b^{\frac {19}{2}} \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {-15 a^2 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{4 b^{\frac {7}{2}}}+\frac {15 a^{\frac {3}{2}} \sqrt {x}}{4 b^3 \sqrt {1-\frac {b x}{a}}}-\frac {5 \sqrt {a} x^{\frac {3}{2}}}{4 b^2 \sqrt {1-\frac {b x}{a}}}-\frac {x^{\frac {5}{2}}}{2 \sqrt {a} b \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 127, normalized size = 1.27
method | result | size |
risch | \(\frac {\left (2 b x +7 a \right ) \sqrt {x}\, \sqrt {-b x +a}}{4 b^{3}}+\frac {\left (-\frac {15 a^{2} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{8 b^{\frac {7}{2}}}-\frac {2 a^{2} \sqrt {-\left (-\frac {a}{b}+x \right )^{2} b -a \left (-\frac {a}{b}+x \right )}}{b^{4} \left (-\frac {a}{b}+x \right )}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 118, normalized size = 1.18 \begin {gather*} \frac {8 \, a^{2} b^{2} - \frac {25 \, {\left (b x - a\right )} a^{2} b}{x} + \frac {15 \, {\left (b x - a\right )}^{2} a^{2}}{x^{2}}}{4 \, {\left (\frac {\sqrt {-b x + a} b^{5}}{\sqrt {x}} + \frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + a\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} + \frac {15 \, a^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 181, normalized size = 1.81 \begin {gather*} \left [-\frac {15 \, {\left (a^{2} b x - a^{3}\right )} \sqrt {-b} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (2 \, b^{3} x^{2} + 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, {\left (b^{5} x - a b^{4}\right )}}, \frac {15 \, {\left (a^{2} b x - a^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (2 \, b^{3} x^{2} + 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, {\left (b^{5} x - a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.73, size = 224, normalized size = 2.24 \begin {gather*} \begin {cases} - \frac {15 i a^{\frac {3}{2}} \sqrt {x}}{4 b^{3} \sqrt {-1 + \frac {b x}{a}}} + \frac {5 i \sqrt {a} x^{\frac {3}{2}}}{4 b^{2} \sqrt {-1 + \frac {b x}{a}}} + \frac {15 i a^{2} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {7}{2}}} + \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} b \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {15 a^{\frac {3}{2}} \sqrt {x}}{4 b^{3} \sqrt {1 - \frac {b x}{a}}} - \frac {5 \sqrt {a} x^{\frac {3}{2}}}{4 b^{2} \sqrt {1 - \frac {b x}{a}}} - \frac {15 a^{2} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {7}{2}}} - \frac {x^{\frac {5}{2}}}{2 \sqrt {a} b \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 = 0.01, size = 136, normalized size = 1.36 \begin {gather*} 2 \left (\frac {2 \left (\left (-\frac {\frac {1}{16}\cdot 2 b^{4} \sqrt {x} \sqrt {x}}{b^{5}}-\frac {\frac {1}{16}\cdot 5 b^{3} a}{b^{5}}\right ) \sqrt {x} \sqrt {x}+\frac {\frac {1}{16}\cdot 15 b^{2} a^{2}}{b^{5}}\right ) \sqrt {x} \sqrt {a-b x}}{a-b x}+\frac {30 a^{2} \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{16 b^{3} \sqrt {-b}}\right ) \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 {x^{5/2}}{{\left (a-b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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