Optimal. Leaf size=95 \[ \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {5 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 95, 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 {5 a \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}+\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}} \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)^{5/2}} \, dx &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx}{3 b}\\ &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{b^2}\\ &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b^3}\\ &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b^3}\\ &=\frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a-b x}}-\frac {5 \sqrt {x} \sqrt {a-b x}}{b^3}+\frac {5 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 78, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x} \left (15 a^2-20 a b x+3 b^2 x^2\right )}{3 b^3 (a-b x)^{3/2}}+\frac {5 a \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs.
\(2(71)=142\).
time = 0.12, size = 160, normalized size = 1.68
method | result | size |
risch | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b^{3}}+\frac {\left (\frac {5 a \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {7}{2}}}+\frac {2 a^{2} \sqrt {-\left (-\frac {a}{b}+x \right )^{2} b -a \left (-\frac {a}{b}+x \right )}}{3 b^{5} \left (-\frac {a}{b}+x \right )^{2}}+\frac {14 a \sqrt {-\left (-\frac {a}{b}+x \right )^{2} b -a \left (-\frac {a}{b}+x \right )}}{3 b^{4} \left (-\frac {a}{b}+x \right )}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 94, normalized size = 0.99 \begin {gather*} \frac {2 \, a b^{2} + \frac {10 \, {\left (b x - a\right )} a b}{x} - \frac {15 \, {\left (b x - a\right )}^{2} a}{x^{2}}}{3 \, {\left (\frac {{\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 {5 \, a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.70, size = 215, normalized size = 2.26 \begin {gather*} \left [-\frac {15 \, {\left (a b^{2} x^{2} - 2 \, 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 (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.33, size = 971, normalized size = 10.22 \begin {gather*} \begin {cases} - \frac {30 i a^{\frac {81}{2}} b^{22} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {15 \pi a^{\frac {81}{2}} b^{22} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {30 i a^{\frac {79}{2}} b^{23} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {15 \pi a^{\frac {79}{2}} b^{23} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {30 i a^{40} b^{\frac {45}{2}} x^{26}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} - \frac {40 i a^{39} b^{\frac {47}{2}} x^{27}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} + \frac {6 i a^{38} b^{\frac {49}{2}} x^{28}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {-1 + \frac {b x}{a}} - 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {15 a^{\frac {81}{2}} b^{22} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {15 a^{\frac {79}{2}} b^{23} x^{\frac {53}{2}} \sqrt {1 - \frac {b x}{a}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {15 a^{40} b^{\frac {45}{2}} x^{26}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 - \frac {b x}{a}}} + \frac {20 a^{39} b^{\frac {47}{2}} x^{27}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 - \frac {b x}{a}}} - \frac {3 a^{38} b^{\frac {49}{2}} x^{28}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 - \frac {b x}{a}} - 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \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. 221 vs.
\(2 (71) = 142\).
time = 20.98, size = 221, normalized size = 2.33 \begin {gather*} \frac {{\left (\frac {15 \, a \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} - \frac {6 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b^{3}} - \frac {8 \, {\left (9 \, a^{2} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} - 12 \, a^{3} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} b + 7 \, a^{4} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} \sqrt {-b} b}\right )} {\left | b \right |}}{6 \, b^{2}} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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