Optimal. Leaf size=91 \[ -\frac {5 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {47, 50, 63, 217, 206} \begin {gather*} -\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {5 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}+\frac {5 \int \frac {x^{3/2}}{(a+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {(5 a) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b^3}\\ &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {5 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.55 \begin {gather*} \frac {2 x^{7/2} \sqrt {\frac {b x}{a}+1} \, _2F_1\left (\frac {5}{2},\frac {7}{2};\frac {9}{2};-\frac {b x}{a}\right )}{7 a^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 78, normalized size = 0.86 \begin {gather*} \frac {15 a^2 \sqrt {x}+20 a b x^{3/2}+3 b^2 x^{5/2}}{3 b^3 (a+b x)^{3/2}}+\frac {5 a \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 214, normalized size = 2.35 \begin {gather*} \left [\frac {15 \, {\left (a b^{2} x^{2} + 2 \, 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 (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {15 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} + 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 92.46, size = 197, normalized size = 2.16 \begin {gather*} \frac {{\left (\frac {15 \, a \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac {5}{2}}} + \frac {6 \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}}{b^{3}} + \frac {8 \, {\left (9 \, a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} + 12 \, a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} + 7 \, a^{4} b^{\frac {5}{2}}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{2}}\right )} {\left | b \right |}}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 147, normalized size = 1.62 \begin {gather*} \frac {\left (-\frac {5 a \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {7}{2}}}-\frac {2 \sqrt {-\left (x +\frac {a}{b}\right ) a +\left (x +\frac {a}{b}\right )^{2} b}\, a^{2}}{3 \left (x +\frac {a}{b}\right )^{2} b^{5}}+\frac {14 \sqrt {-\left (x +\frac {a}{b}\right ) a +\left (x +\frac {a}{b}\right )^{2} b}\, a}{3 \left (x +\frac {a}{b}\right ) b^{4}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}+\frac {\sqrt {b x +a}\, \sqrt {x}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 109, normalized size = 1.20 \begin {gather*} \frac {2 \, a b^{2} + \frac {10 \, {\left (b x + a\right )} a b}{x} - \frac {15 \, {\left (b x + a\right )}^{2} a}{x^{2}}}{3 \, {\left (\frac {{\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 {5 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{2 \, 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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.61, size = 396, normalized size = 4.35 \begin {gather*} - \frac {15 a^{\frac {81}{2}} b^{22} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {15 a^{\frac {79}{2}} b^{23} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{40} b^{\frac {45}{2}} x^{26}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {20 a^{39} b^{\frac {47}{2}} x^{27}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{38} b^{\frac {49}{2}} x^{28}}{3 a^{\frac {79}{2}} b^{\frac {51}{2}} x^{\frac {51}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{\frac {53}{2}} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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