Optimal. Leaf size=171 \[ \frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2} \]
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Rubi [A] time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, 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^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}+\frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int x^{5/2} (a-b x)^{5/2} \, dx &=\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{12} (5 a) \int x^{5/2} (a-b x)^{3/2} \, dx\\ &=\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{8} a^2 \int x^{5/2} \sqrt {a-b x} \, dx\\ &=\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {1}{64} a^3 \int \frac {x^{5/2}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^4\right ) \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx}{384 b}\\ &=-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^5\right ) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{512 b^2}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{1024 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{512 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {\left (5 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^3}\\ &=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {1}{32} a^2 x^{7/2} \sqrt {a-b x}+\frac {1}{12} a x^{7/2} (a-b x)^{3/2}+\frac {1}{6} x^{7/2} (a-b x)^{5/2}+\frac {5 a^6 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 120, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a-b x} \left (\frac {15 a^{11/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {1-\frac {b x}{a}}}+\sqrt {b} \sqrt {x} \left (-15 a^5-10 a^4 b x-8 a^3 b^2 x^2+432 a^2 b^3 x^3-640 a b^4 x^4+256 b^5 x^5\right )\right )}{1536 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 130, normalized size = 0.76 \begin {gather*} \frac {5 a^6 \sqrt {-b} \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )}{512 b^4}+\frac {\sqrt {a-b x} \left (-15 a^5 \sqrt {x}-10 a^4 b x^{3/2}-8 a^3 b^2 x^{5/2}+432 a^2 b^3 x^{7/2}-640 a b^4 x^{9/2}+256 b^5 x^{11/2}\right )}{1536 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 208, normalized size = 1.22 \begin {gather*} \left [-\frac {15 \, a^{6} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{3072 \, b^{4}}, -\frac {15 \, a^{6} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{1536 \, 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 = 165, normalized size = 0.96 \begin {gather*} \frac {5 \sqrt {\left (-b x +a \right ) x}\, a^{6} \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{1024 \sqrt {-b x +a}\, b^{\frac {7}{2}} \sqrt {x}}+\frac {5 \sqrt {-b x +a}\, a^{5} \sqrt {x}}{512 b^{3}}+\frac {5 \left (-b x +a \right )^{\frac {3}{2}} a^{4} \sqrt {x}}{768 b^{3}}-\frac {\left (-b x +a \right )^{\frac {7}{2}} x^{\frac {5}{2}}}{6 b}+\frac {\left (-b x +a \right )^{\frac {5}{2}} a^{3} \sqrt {x}}{192 b^{3}}-\frac {\left (-b x +a \right )^{\frac {7}{2}} a \,x^{\frac {3}{2}}}{12 b^{2}}-\frac {\left (-b x +a \right )^{\frac {7}{2}} a^{2} \sqrt {x}}{32 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 242, normalized size = 1.42 \begin {gather*} -\frac {5 \, a^{6} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{512 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{6} b^{5}}{\sqrt {x}} + \frac {85 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} - \frac {198 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (-b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x - a\right )} b^{8}}{x} + \frac {15 \, {\left (b x - a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x - a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x - a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x - a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x - a\right )}^{6} b^{3}}{x^{6}}\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 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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sympy [A] time = 25.96, size = 435, normalized size = 2.54 \begin {gather*} \begin {cases} \frac {5 i a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {-1 + \frac {b x}{a}}} - \frac {55 i a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {-1 + \frac {b x}{a}}} + \frac {67 i a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {-1 + \frac {b x}{a}}} - \frac {7 i \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{6} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} + \frac {i b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {1 - \frac {b x}{a}}} + \frac {55 a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {1 - \frac {b x}{a}}} - \frac {67 a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {1 - \frac {b x}{a}}} + \frac {7 \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{6} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} - \frac {b^{3} x^{\frac {13}{2}}}{6 \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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