Optimal. Leaf size=91 \[ -\frac {2 x^{5/2}}{3 b (a+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {a+b x}}+\frac {5 \sqrt {x} \sqrt {a+b x}}{b^3}-\frac {5 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.02, 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 = {49, 52, 65, 223,
212} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
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) \text {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) \text {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 [A]
time = 0.13, size = 72, normalized size = 0.79 \begin {gather*} \frac {\sqrt {x} \left (15 a^2+20 a b x+3 b^2 x^2\right )}{3 b^3 (a+b x)^{3/2}}+\frac {5 a \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 8.51, size = 131, normalized size = 1.44 \begin {gather*} \frac {-15 a^{\frac {7}{2}} b^{\frac {17}{2}} \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}+15 a^2 b^9 \sqrt {x} \left (a+b x\right )-15 a^{\frac {5}{2}} b^{\frac {19}{2}} x \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}+20 a b^{10} x^{\frac {3}{2}} \left (a+b x\right )+3 b^{11} x^{\frac {5}{2}} \left (a+b x\right )}{3 a^{\frac {3}{2}} b^{12} \left (\frac {a+b x}{a}\right )^{\frac {3}{2}} \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs.
\(2(67)=134\).
time = 0.14, size = 147, normalized size = 1.62
method | result | size |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +a}}{b^{3}}+\frac {\left (-\frac {5 a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 b^{\frac {7}{2}}}-\frac {2 a^{2} \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{3 b^{5} \left (x +\frac {a}{b}\right )^{2}}+\frac {14 a \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{3 b^{4} \left (x +\frac {a}{b}\right )}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {x}\, \sqrt {b x +a}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, 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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Fricas [A]
time = 0.32, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs.
\(2 (85) = 170\).
time = 5.02, 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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Giac [A]
time = 0.01, size = 138, normalized size = 1.52 \begin {gather*} 2 \left (\frac {2 \left (\left (\frac {\frac {1}{36}\cdot 9 b^{4} a \sqrt {x} \sqrt {x}}{b^{5} a}+\frac {\frac {1}{36}\cdot 60 b^{3} a^{2}}{b^{5} a}\right ) \sqrt {x} \sqrt {x}+\frac {\frac {1}{36}\cdot 45 b^{2} a^{3}}{b^{5} a}\right ) \sqrt {x} \sqrt {a+b x}}{\left (a+b x\right )^{2}}+\frac {10 a \ln \left |\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right |}{4 b^{3} \sqrt {b}}\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}}{{\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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