Optimal. Leaf size=111 \[ -\frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{64 a^{3/2}}+\frac {5 b^3 x \sqrt {a+\frac {b}{x}}}{64 a}+\frac {5}{32} b^2 x^2 \sqrt {a+\frac {b}{x}}+\frac {1}{4} x^4 \left (a+\frac {b}{x}\right )^{5/2}+\frac {5}{24} b x^3 \left (a+\frac {b}{x}\right )^{3/2} \]
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Rubi [A] time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ -\frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{64 a^{3/2}}+\frac {5}{32} b^2 x^2 \sqrt {a+\frac {b}{x}}+\frac {5 b^3 x \sqrt {a+\frac {b}{x}}}{64 a}+\frac {5}{24} b x^3 \left (a+\frac {b}{x}\right )^{3/2}+\frac {1}{4} x^4 \left (a+\frac {b}{x}\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} x^3 \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4-\frac {1}{8} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{24} b \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4-\frac {1}{16} \left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{32} b^2 \sqrt {a+\frac {b}{x}} x^2+\frac {5}{24} b \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4-\frac {1}{64} \left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5 b^3 \sqrt {a+\frac {b}{x}} x}{64 a}+\frac {5}{32} b^2 \sqrt {a+\frac {b}{x}} x^2+\frac {5}{24} b \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4+\frac {\left (5 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{128 a}\\ &=\frac {5 b^3 \sqrt {a+\frac {b}{x}} x}{64 a}+\frac {5}{32} b^2 \sqrt {a+\frac {b}{x}} x^2+\frac {5}{24} b \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4+\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{64 a}\\ &=\frac {5 b^3 \sqrt {a+\frac {b}{x}} x}{64 a}+\frac {5}{32} b^2 \sqrt {a+\frac {b}{x}} x^2+\frac {5}{24} b \left (a+\frac {b}{x}\right )^{3/2} x^3+\frac {1}{4} \left (a+\frac {b}{x}\right )^{5/2} x^4-\frac {5 b^4 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{64 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 39, normalized size = 0.35 \[ \frac {2 b^4 \left (a+\frac {b}{x}\right )^{7/2} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {b}{a x}+1\right )}{7 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 173, normalized size = 1.56 \[ \left [\frac {15 \, \sqrt {a} b^{4} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (48 \, a^{4} x^{4} + 136 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 15 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{384 \, a^{2}}, \frac {15 \, \sqrt {-a} b^{4} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (48 \, a^{4} x^{4} + 136 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 15 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{192 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 107, normalized size = 0.96 \[ \frac {5 \, b^{4} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right ) \mathrm {sgn}\relax (x)}{128 \, a^{\frac {3}{2}}} - \frac {5 \, b^{4} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{128 \, a^{\frac {3}{2}}} + \frac {1}{192} \, \sqrt {a x^{2} + b x} {\left (\frac {15 \, b^{3} \mathrm {sgn}\relax (x)}{a} + 2 \, {\left (59 \, b^{2} \mathrm {sgn}\relax (x) + 4 \, {\left (6 \, a^{2} x \mathrm {sgn}\relax (x) + 17 \, a b \mathrm {sgn}\relax (x)\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 135, normalized size = 1.22 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-15 a \,b^{4} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+60 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{2} x +96 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x +30 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{3}+176 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b \right ) x}{384 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.29, size = 164, normalized size = 1.48 \[ \frac {5 \, b^{4} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{128 \, a^{\frac {3}{2}}} + \frac {15 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{4} + 73 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a b^{4} - 55 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} b^{4} + 15 \, \sqrt {a + \frac {b}{x}} a^{3} b^{4}}{192 \, {\left ({\left (a + \frac {b}{x}\right )}^{4} a - 4 \, {\left (a + \frac {b}{x}\right )}^{3} a^{2} + 6 \, {\left (a + \frac {b}{x}\right )}^{2} a^{3} - 4 \, {\left (a + \frac {b}{x}\right )} a^{4} + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 89, normalized size = 0.80 \[ \frac {73\,x^4\,{\left (a+\frac {b}{x}\right )}^{5/2}}{192}-\frac {55\,a\,x^4\,{\left (a+\frac {b}{x}\right )}^{3/2}}{192}+\frac {5\,a^2\,x^4\,\sqrt {a+\frac {b}{x}}}{64}+\frac {5\,x^4\,{\left (a+\frac {b}{x}\right )}^{7/2}}{64\,a}+\frac {b^4\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{64\,a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.59, size = 155, normalized size = 1.40 \[ \frac {a^{3} x^{\frac {9}{2}}}{4 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {23 a^{2} \sqrt {b} x^{\frac {7}{2}}}{24 \sqrt {\frac {a x}{b} + 1}} + \frac {127 a b^{\frac {3}{2}} x^{\frac {5}{2}}}{96 \sqrt {\frac {a x}{b} + 1}} + \frac {133 b^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {\frac {a x}{b} + 1}} + \frac {5 b^{\frac {7}{2}} \sqrt {x}}{64 a \sqrt {\frac {a x}{b} + 1}} - \frac {5 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{64 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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