Optimal. Leaf size=88 \[ \frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 x^{3/2}}-\frac {32 b \sqrt {a-b x}}{3 a^4 \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37}
\begin {gather*} -\frac {32 b \sqrt {a-b x}}{3 a^4 \sqrt {x}}-\frac {16 \sqrt {a-b x}}{3 a^3 x^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}+\frac {2}{3 a x^{3/2} (a-b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx &=\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {2 \int \frac {1}{x^{5/2} (a-b x)^{3/2}} \, dx}{a}\\ &=\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}+\frac {8 \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx}{a^2}\\ &=\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 x^{3/2}}+\frac {(16 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a^3}\\ &=\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 x^{3/2}}-\frac {32 b \sqrt {a-b x}}{3 a^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 50, normalized size = 0.57 \begin {gather*} -\frac {2 \left (a^3+6 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )}{3 a^4 x^{3/2} (a-b x)^{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 2 in
optimal.
time = 59.41, size = 447, normalized size = 5.08 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a^4-5 a^3 b x+30 a^2 b^2 x^2-40 a b^3 x^3+16 b^4 x^4\right ) \sqrt {\frac {a-b x}{b x}}}{3 a^4 x \left (a^3-3 a^2 b x+3 a b^2 x^2-b^3 x^3\right )},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I a^4 b^{\frac {19}{2}} \sqrt {1-\frac {a}{b x}}}{3 a^7 b^9 x-9 a^6 b^{10} x^2+9 a^5 b^{11} x^3-3 a^4 b^{12} x^4}-\frac {10 I a^3 b^{\frac {21}{2}} x \sqrt {1-\frac {a}{b x}}}{3 a^7 b^9 x-9 a^6 b^{10} x^2+9 a^5 b^{11} x^3-3 a^4 b^{12} x^4}+\frac {I 60 a^2 b^{\frac {23}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{3 a^7 b^9 x-9 a^6 b^{10} x^2+9 a^5 b^{11} x^3-3 a^4 b^{12} x^4}-\frac {80 I a b^{\frac {25}{2}} x^3 \sqrt {1-\frac {a}{b x}}}{3 a^7 b^9 x-9 a^6 b^{10} x^2+9 a^5 b^{11} x^3-3 a^4 b^{12} x^4}+\frac {I 32 b^{\frac {27}{2}} x^4 \sqrt {1-\frac {a}{b x}}}{3 a^7 b^9 x-9 a^6 b^{10} x^2+9 a^5 b^{11} x^3-3 a^4 b^{12} x^4}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 80, normalized size = 0.91
method | result | size |
gosper | \(-\frac {2 \left (16 b^{3} x^{3}-24 a \,b^{2} x^{2}+6 a^{2} b x +a^{3}\right )}{3 x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(45\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (8 b x +a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {2 b^{2} \left (-8 b x +9 a \right ) \sqrt {x}}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(51\) |
default | \(-\frac {2}{3 a \,x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}+\frac {2 b \left (-\frac {2}{a \left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}+\frac {4 b \left (\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\right )}{a}\right )}{a}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 68, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (\frac {9 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{4}} + \frac {2 \, {\left (b^{3} - \frac {9 \, {\left (b x - a\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 70, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\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 = 5.98, size = 688, normalized size = 7.82 \begin {gather*} \begin {cases} \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a^{4} b^{\frac {19}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 i a^{3} b^{\frac {21}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {60 i a^{2} b^{\frac {23}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 i a b^{\frac {25}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {32 i b^{\frac {27}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \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. 147 vs.
\(2 (66) = 132\).
time = 0.01, size = 200, normalized size = 2.27 \begin {gather*} 2 \left (\frac {2 \left (-\frac {\frac {1}{18}\cdot 24 b^{4} a^{3} \sqrt {x} \sqrt {x}}{b a^{7}}+\frac {\frac {1}{18}\cdot 27 b^{3} a^{4}}{b a^{7}}\right ) \sqrt {x} \sqrt {a-b x}}{\left (a-b x\right )^{2}}+\frac {2 \left (6 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{4}-18 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2} a+8 b \sqrt {-b} a^{2}\right )}{3 a^{3} \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 92, normalized size = 1.05 \begin {gather*} \frac {2\,a^3\,\sqrt {a-b\,x}+32\,b^3\,x^3\,\sqrt {a-b\,x}+12\,a^2\,b\,x\,\sqrt {a-b\,x}-48\,a\,b^2\,x^2\,\sqrt {a-b\,x}}{x^{3/2}\,\left (x\,\left (6\,a^5\,b-3\,a^4\,b^2\,x\right )-3\,a^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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