Optimal. Leaf size=105 \[ \frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}}-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {50, 63, 217, 203} \begin {gather*} -\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx &=-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {(5 a) \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{6 b}\\ &=-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^2\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{8 b^2}\\ &=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{16 b^3}\\ &=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3}\\ &=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {\left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^3}\\ &=-\frac {5 a^2 \sqrt {x} \sqrt {a-b x}}{8 b^3}-\frac {5 a x^{3/2} \sqrt {a-b x}}{12 b^2}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 88, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a-b x} \left (\frac {15 a^{5/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {1-\frac {b x}{a}}}-\sqrt {b} \sqrt {x} \left (15 a^2+10 a b x+8 b^2 x^2\right )\right )}{24 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 91, normalized size = 0.87 \begin {gather*} \frac {5 a^3 \sqrt {-b} \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )}{8 b^4}+\frac {\sqrt {a-b x} \left (-15 a^2 \sqrt {x}-10 a b x^{3/2}-8 b^2 x^{5/2}\right )}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 141, normalized size = 1.34 \begin {gather*} \left [-\frac {15 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{4}}, -\frac {15 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{4}}\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 [A] time = 0.01, size = 108, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {-b x +a}\, x^{\frac {5}{2}}}{3 b}-\frac {5 \sqrt {-b x +a}\, a \,x^{\frac {3}{2}}}{12 b^{2}}+\frac {5 \sqrt {\left (-b x +a \right ) x}\, a^{3} \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{16 \sqrt {-b x +a}\, b^{\frac {7}{2}} \sqrt {x}}-\frac {5 \sqrt {-b x +a}\, a^{2} \sqrt {x}}{8 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 135, normalized size = 1.29 \begin {gather*} -\frac {5 \, a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {\frac {33 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} + \frac {40 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{6} - \frac {3 \, {\left (b x - a\right )} b^{5}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{4}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b^{3}}{x^{3}}\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}}{\sqrt {a-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.43, size = 270, normalized size = 2.57 \begin {gather*} \begin {cases} \frac {5 i a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \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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