Optimal. Leaf size=81 \[ -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {5 b^2 \sqrt {a+b x}}{8 x}-\frac {(a+b x)^{5/2}}{3 x^3}-\frac {5 b (a+b x)^{3/2}}{12 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {47, 63, 208} \begin {gather*} -\frac {5 b^2 \sqrt {a+b x}}{8 x}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {5 b (a+b x)^{3/2}}{12 x^2}-\frac {(a+b x)^{5/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^4} \, dx &=-\frac {(a+b x)^{5/2}}{3 x^3}+\frac {1}{6} (5 b) \int \frac {(a+b x)^{3/2}}{x^3} \, dx\\ &=-\frac {5 b (a+b x)^{3/2}}{12 x^2}-\frac {(a+b x)^{5/2}}{3 x^3}+\frac {1}{8} \left (5 b^2\right ) \int \frac {\sqrt {a+b x}}{x^2} \, dx\\ &=-\frac {5 b^2 \sqrt {a+b x}}{8 x}-\frac {5 b (a+b x)^{3/2}}{12 x^2}-\frac {(a+b x)^{5/2}}{3 x^3}+\frac {1}{16} \left (5 b^3\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=-\frac {5 b^2 \sqrt {a+b x}}{8 x}-\frac {5 b (a+b x)^{3/2}}{12 x^2}-\frac {(a+b x)^{5/2}}{3 x^3}+\frac {1}{8} \left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {5 b^2 \sqrt {a+b x}}{8 x}-\frac {5 b (a+b x)^{3/2}}{12 x^2}-\frac {(a+b x)^{5/2}}{3 x^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 79, normalized size = 0.98 \begin {gather*} -\frac {8 a^3+34 a^2 b x+15 b^3 x^3 \sqrt {\frac {b x}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x}{a}+1}\right )+59 a b^2 x^2+33 b^3 x^3}{24 x^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 68, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {a+b x} \left (15 a^2-40 a (a+b x)+33 (a+b x)^2\right )}{24 x^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 146, normalized size = 1.80 \begin {gather*} \left [\frac {15 \, \sqrt {a} b^{3} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (33 \, a b^{2} x^{2} + 26 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{48 \, a x^{3}}, \frac {15 \, \sqrt {-a} b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (33 \, a b^{2} x^{2} + 26 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{24 \, a x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 79, normalized size = 0.98 \begin {gather*} \frac {\frac {15 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {33 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} + 15 \, \sqrt {b x + a} a^{2} b^{4}}{b^{3} x^{3}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.78 \begin {gather*} 2 \left (-\frac {5 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {-\frac {5 \sqrt {b x +a}\, a^{2}}{16}+\frac {5 \left (b x +a \right )^{\frac {3}{2}} a}{6}-\frac {11 \left (b x +a \right )^{\frac {5}{2}}}{16}}{b^{3} x^{3}}\right ) b^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 115, normalized size = 1.42 \begin {gather*} \frac {5 \, b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, \sqrt {a}} - \frac {33 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 15 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2} - a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 64, normalized size = 0.79 \begin {gather*} \frac {5\,a\,{\left (a+b\,x\right )}^{3/2}}{3\,x^3}-\frac {5\,a^2\,\sqrt {a+b\,x}}{8\,x^3}-\frac {11\,{\left (a+b\,x\right )}^{5/2}}{8\,x^3}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.16, size = 104, normalized size = 1.28 \begin {gather*} - \frac {a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x^{\frac {5}{2}}} - \frac {13 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{12 x^{\frac {3}{2}}} - \frac {11 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{8 \sqrt {x}} - \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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