Optimal. Leaf size=91 \[ 2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {662, 620, 206} \[ -\frac {2 b^2 \sqrt {a x+b x^2}}{x}+2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^6} \, dx &=-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^2 \int \frac {\sqrt {a x+b x^2}}{x^2} \, dx\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^3 \int \frac {1}{\sqrt {a x+b x^2}} \, dx\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.55 \[ -\frac {2 a^2 \sqrt {x (a+b x)} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {b x}{a}\right )}{5 x^3 \sqrt {\frac {b x}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 144, normalized size = 1.58 \[ \left [\frac {15 \, b^{\frac {5}{2}} x^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{2} + a x}}{15 \, x^{3}}, -\frac {2 \, {\left (15 \, \sqrt {-b} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{2} + a x}\right )}}{15 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 175, normalized size = 1.92 \[ -b^{\frac {5}{2}} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right ) + \frac {2 \, {\left (45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b^{2} + 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} b^{\frac {3}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} b + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{4} \sqrt {b} + 3 \, a^{5}\right )}}{15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 232, normalized size = 2.55 \[ b^{\frac {5}{2}} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )-\frac {4 \sqrt {b \,x^{2}+a x}\, b^{4} x}{a^{2}}-\frac {2 \sqrt {b \,x^{2}+a x}\, b^{3}}{a}+\frac {32 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} b^{5} x}{3 a^{4}}+\frac {16 \left (b \,x^{2}+a x \right )^{\frac {3}{2}} b^{4}}{3 a^{3}}+\frac {256 \left (b \,x^{2}+a x \right )^{\frac {5}{2}} b^{5}}{15 a^{5}}-\frac {256 \left (b \,x^{2}+a x \right )^{\frac {7}{2}} b^{4}}{15 a^{5} x^{2}}+\frac {32 \left (b \,x^{2}+a x \right )^{\frac {7}{2}} b^{3}}{5 a^{4} x^{3}}-\frac {16 \left (b \,x^{2}+a x \right )^{\frac {7}{2}} b^{2}}{15 a^{3} x^{4}}-\frac {4 \left (b \,x^{2}+a x \right )^{\frac {7}{2}} b}{15 a^{2} x^{5}}-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{5 a \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 134, normalized size = 1.47 \[ b^{\frac {5}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {38 \, \sqrt {b x^{2} + a x} b^{2}}{15 \, x} - \frac {7 \, \sqrt {b x^{2} + a x} a b}{30 \, x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{10 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b}{3 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{2 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{5 \, x^{5}} \]
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 \[ \int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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