Optimal. Leaf size=66 \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {(a+b x)^{5/2}}{x}+\frac {5}{3} b (a+b x)^{3/2}+5 a b \sqrt {a+b x} \]
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {(a+b x)^{5/2}}{x}+\frac {5}{3} b (a+b x)^{3/2}+5 a b \sqrt {a+b x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^2} \, dx &=-\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} (5 b) \int \frac {(a+b x)^{3/2}}{x} \, dx\\ &=\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} (5 a b) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\frac {1}{2} \left (5 a^2 b\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}+\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=5 a b \sqrt {a+b x}+\frac {5}{3} b (a+b x)^{3/2}-\frac {(a+b x)^{5/2}}{x}-5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.50 \[ \frac {2 b (a+b x)^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b x}{a}+1\right )}{7 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 126, normalized size = 1.91 \[ \left [\frac {15 \, a^{\frac {3}{2}} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x + a}}{6 \, x}, \frac {15 \, \sqrt {-a} a b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x + a}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 74, normalized size = 1.12 \[ \frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} + 12 \, \sqrt {b x + a} a b^{2} - \frac {3 \, \sqrt {b x + a} a^{2} b}{x}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.92 \[ 2 \left (\left (-\frac {5 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {b x +a}}{2 b x}\right ) a^{2}+2 \sqrt {b x +a}\, a +\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}\right ) b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 71, normalized size = 1.08 \[ \frac {5}{2} \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} b + 4 \, \sqrt {b x + a} a b - \frac {\sqrt {b x + a} a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 58, normalized size = 0.88 \[ \frac {2\,b\,{\left (a+b\,x\right )}^{3/2}}{3}-\frac {a^2\,\sqrt {a+b\,x}}{x}+4\,a\,b\,\sqrt {a+b\,x}+a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.75, size = 99, normalized size = 1.50 \[ - \frac {a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{x} + \frac {14 a^{\frac {3}{2}} b \sqrt {1 + \frac {b x}{a}}}{3} + \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b x}{a} \right )}}{2} - 5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )} + \frac {2 \sqrt {a} b^{2} x \sqrt {1 + \frac {b x}{a}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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