Optimal. Leaf size=100 \[ -\frac {15 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{7/2}}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}+\frac {2 x^{5/2}}{b \sqrt {a-b x}} \]
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Rubi [A] time = 0.03, 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 = {47, 50, 63, 217, 203} \begin {gather*} -\frac {15 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{7/2}}+\frac {5 x^{3/2} \sqrt {a-b x}}{2 b^2}+\frac {15 a \sqrt {x} \sqrt {a-b x}}{4 b^3}+\frac {2 x^{5/2}}{b \sqrt {a-b 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 {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 ) \operatorname {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 ) \operatorname {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 [C] time = 0.01, size = 51, normalized size = 0.51 \begin {gather*} \frac {2 x^{7/2} \sqrt {1-\frac {b x}{a}} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {b x}{a}\right )}{7 a \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 100, normalized size = 1.00 \begin {gather*} -\frac {15 a^2 \sqrt {-b} \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )}{4 b^4}-\frac {\sqrt {a-b x} \left (15 a^2 \sqrt {x}-5 a b x^{3/2}-2 b^2 x^{5/2}\right )}{4 b^3 (b x-a)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.37, 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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giac [B] time = 98.07, size = 154, normalized size = 1.54 \begin {gather*} \frac {{\left (2 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a} {\left (\frac {2 \, {\left (b x - a\right )}}{b^{3}} + \frac {9 \, a}{b^{3}}\right )} + \frac {32 \, a^{3}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} \sqrt {-b} b} - \frac {15 \, a^{2} \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}}\right )} {\left | b \right |}}{8 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 127, normalized size = 1.27 \begin {gather*} \frac {\left (-\frac {15 a^{2} \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{8 b^{\frac {7}{2}}}-\frac {2 \sqrt {-\left (x -\frac {a}{b}\right ) a -\left (x -\frac {a}{b}\right )^{2} b}\, a^{2}}{\left (x -\frac {a}{b}\right ) b^{4}}\right ) \sqrt {\left (-b x +a \right ) x}}{\sqrt {-b x +a}\, \sqrt {x}}+\frac {\left (2 b x +7 a \right ) \sqrt {-b x +a}\, \sqrt {x}}{4 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, 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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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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sympy [A] time = 8.03, 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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