Optimal. Leaf size=83 \[ \frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}-\frac {2 b^3 \sqrt {x}}{a^4}+\frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b^3 \sqrt {x}}{a^4}+\frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{a+\frac {b}{x}} \, dx &=\int \frac {x^{7/2}}{b+a x} \, dx\\ &=\frac {2 x^{7/2}}{7 a}-\frac {b \int \frac {x^{5/2}}{b+a x} \, dx}{a}\\ &=-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a}+\frac {b^2 \int \frac {x^{3/2}}{b+a x} \, dx}{a^2}\\ &=\frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a}-\frac {b^3 \int \frac {\sqrt {x}}{b+a x} \, dx}{a^3}\\ &=-\frac {2 b^3 \sqrt {x}}{a^4}+\frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a}+\frac {b^4 \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{a^4}\\ &=-\frac {2 b^3 \sqrt {x}}{a^4}+\frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a}+\frac {\left (2 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=-\frac {2 b^3 \sqrt {x}}{a^4}+\frac {2 b^2 x^{3/2}}{3 a^3}-\frac {2 b x^{5/2}}{5 a^2}+\frac {2 x^{7/2}}{7 a}+\frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.87 \[ \frac {2 b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}+\frac {2 \sqrt {x} \left (15 a^3 x^3-21 a^2 b x^2+35 a b^2 x-105 b^3\right )}{105 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 153, normalized size = 1.84 \[ \left [\frac {105 \, b^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt {x}}{105 \, a^{4}}, \frac {2 \, {\left (105 \, b^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt {x}\right )}}{105 \, a^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 0.84 \[ \frac {2 \, b^{4} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {2 \, {\left (15 \, a^{6} x^{\frac {7}{2}} - 21 \, a^{5} b x^{\frac {5}{2}} + 35 \, a^{4} b^{2} x^{\frac {3}{2}} - 105 \, a^{3} b^{3} \sqrt {x}\right )}}{105 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.78 \[ \frac {2 x^{\frac {7}{2}}}{7 a}-\frac {2 b \,x^{\frac {5}{2}}}{5 a^{2}}+\frac {2 b^{4} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}+\frac {2 b^{2} x^{\frac {3}{2}}}{3 a^{3}}-\frac {2 b^{3} \sqrt {x}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.29, size = 65, normalized size = 0.78 \[ \frac {2 \, {\left (15 \, a^{3} - \frac {21 \, a^{2} b}{x} + \frac {35 \, a b^{2}}{x^{2}} - \frac {105 \, b^{3}}{x^{3}}\right )} x^{\frac {7}{2}}}{105 \, a^{4}} - \frac {2 \, b^{4} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 59, normalized size = 0.71 \[ \frac {2\,x^{7/2}}{7\,a}-\frac {2\,b\,x^{5/2}}{5\,a^2}+\frac {2\,b^2\,x^{3/2}}{3\,a^3}-\frac {2\,b^3\,\sqrt {x}}{a^4}+\frac {2\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.58, size = 136, normalized size = 1.64 \[ \begin {cases} \frac {2 x^{\frac {7}{2}}}{7 a} - \frac {2 b x^{\frac {5}{2}}}{5 a^{2}} + \frac {2 b^{2} x^{\frac {3}{2}}}{3 a^{3}} - \frac {2 b^{3} \sqrt {x}}{a^{4}} - \frac {i b^{\frac {7}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{5} \sqrt {\frac {1}{a}}} + \frac {i b^{\frac {7}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{5} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\\frac {2 x^{\frac {9}{2}}}{9 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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