Optimal. Leaf size=92 \[ \frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223,
212} \begin {gather*} \frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}}+\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{6} (5 a) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{16} \left (5 a^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {1}{8} \left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {5}{8} a^2 \sqrt {x} \sqrt {a+b x}+\frac {5}{12} a \sqrt {x} (a+b x)^{3/2}+\frac {1}{3} \sqrt {x} (a+b x)^{5/2}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 73, normalized size = 0.79 \begin {gather*} \frac {1}{24} \sqrt {x} \sqrt {a+b x} \left (33 a^2+26 a b x+8 b^2 x^2\right )-\frac {5 a^3 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{8 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 5.63, size = 62, normalized size = 0.67 \begin {gather*} \frac {15 a^3 \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]+\sqrt {a} \sqrt {b} \sqrt {x} \left (33 a^2+26 a b x+8 b^2 x^2\right ) \sqrt {\frac {a+b x}{a}}}{24 \sqrt {b}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 94, normalized size = 1.02
method | result | size |
risch | \(\frac {\left (8 x^{2} b^{2}+26 a b x +33 a^{2}\right ) \sqrt {x}\, \sqrt {b x +a}}{24}+\frac {5 a^{3} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{16 \sqrt {b}\, \sqrt {x}\, \sqrt {b x +a}}\) | \(84\) |
default | \(\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 a \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (64) = 128\).
time = 0.34, size = 141, normalized size = 1.53 \begin {gather*} -\frac {5 \, a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{16 \, \sqrt {b}} - \frac {\frac {15 \, \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 {33 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{3} - \frac {3 \, {\left (b x + a\right )} b^{2}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b}{x^{2}} - \frac {{\left (b x + a\right )}^{3}}{x^{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 141, normalized size = 1.53 \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} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b}, -\frac {15 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.04, size = 102, normalized size = 1.11 \begin {gather*} \frac {11 a^{\frac {5}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{8} + \frac {13 a^{\frac {3}{2}} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{12} + \frac {\sqrt {a} b^{2} x^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{3} + \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 10.43, size = 150, normalized size = 1.63 \begin {gather*} \frac {b^{2} \left (2 \left (\left (\frac {\frac {1}{144}\cdot 24 \sqrt {a+b x} \sqrt {a+b x}}{b}+\frac {\frac {1}{144}\cdot 30 a}{b}\right ) \sqrt {a+b x} \sqrt {a+b x}+\frac {\frac {1}{144}\cdot 45 a^{2}}{b}\right ) \sqrt {a+b x} \sqrt {-a b+b \left (a+b x\right )}-\frac {10 a^{3} \ln \left |\sqrt {-a b+b \left (a+b x\right )}-\sqrt {b} \sqrt {a+b x}\right |}{16 \sqrt {b}}\right )}{\left |b\right | b} \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 {{\left (a+b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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