Optimal. Leaf size=90 \[ 5 a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )+5 b^2 \sqrt {x} \sqrt {a-b x}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 90, 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 = {47, 50, 63, 217, 203} \begin {gather*} 5 b^2 \sqrt {x} \sqrt {a-b x}+5 a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {(a-b x)^{5/2}}{x^{5/2}} \, dx &=-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}-\frac {1}{3} (5 b) \int \frac {(a-b x)^{3/2}}{x^{3/2}} \, dx\\ &=\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx\\ &=5 b^2 \sqrt {x} \sqrt {a-b x}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac {1}{2} \left (5 a b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=5 b^2 \sqrt {x} \sqrt {a-b x}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=5 b^2 \sqrt {x} \sqrt {a-b x}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=5 b^2 \sqrt {x} \sqrt {a-b x}+\frac {10 b (a-b x)^{3/2}}{3 \sqrt {x}}-\frac {2 (a-b x)^{5/2}}{3 x^{3/2}}+5 a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.57 \begin {gather*} -\frac {2 a^2 \sqrt {a-b x} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {b x}{a}\right )}{3 x^{3/2} \sqrt {1-\frac {b x}{a}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 76, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a-b x} \left (-2 a^2+14 a b x+3 b^2 x^2\right )}{3 x^{3/2}}+5 a \sqrt {-b} b \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 139, normalized size = 1.54 \begin {gather*} \left [\frac {15 \, a \sqrt {-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{6 \, x^{2}}, -\frac {15 \, a b^{\frac {3}{2}} x^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b x +a \right )^{\frac {5}{2}}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 84, normalized size = 0.93 \begin {gather*} -5 \, a b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \frac {4 \, \sqrt {-b x + a} a b}{\sqrt {x}} + \frac {\sqrt {-b x + a} a b^{2}}{{\left (b - \frac {b x - a}{x}\right )} \sqrt {x}} - \frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}} a}{3 \, x^{\frac {3}{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 {{\left (a-b\,x\right )}^{5/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.85, size = 245, normalized size = 2.72 \begin {gather*} \begin {cases} - \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3} - 5 i a b^{\frac {3}{2}} \log {\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {5 i a b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )}}{2} + 5 a b^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + b^{\frac {5}{2}} x \sqrt {\frac {a}{b x} - 1} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i a^{2} \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {14 i a b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3} + \frac {5 i a b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )}}{2} - 5 i a b^{\frac {3}{2}} \log {\left (\sqrt {- \frac {a}{b x} + 1} + 1 \right )} + i b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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