Optimal. Leaf size=85 \[ -\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}+\frac {7 b^2 \sqrt {x}}{a^4}-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (a x+b)} \]
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Rubi [A] time = 0.03, antiderivative size = 85, 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 = {263, 47, 50, 63, 205} \[ \frac {7 b^2 \sqrt {x}}{a^4}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (a x+b)} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^{7/2}}{(b+a x)^2} \, dx\\ &=-\frac {x^{7/2}}{a (b+a x)}+\frac {7 \int \frac {x^{5/2}}{b+a x} \, dx}{2 a}\\ &=\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (b+a x)}-\frac {(7 b) \int \frac {x^{3/2}}{b+a x} \, dx}{2 a^2}\\ &=-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (b+a x)}+\frac {\left (7 b^2\right ) \int \frac {\sqrt {x}}{b+a x} \, dx}{2 a^3}\\ &=\frac {7 b^2 \sqrt {x}}{a^4}-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (b+a x)}-\frac {\left (7 b^3\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 a^4}\\ &=\frac {7 b^2 \sqrt {x}}{a^4}-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (b+a x)}-\frac {\left (7 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=\frac {7 b^2 \sqrt {x}}{a^4}-\frac {7 b x^{3/2}}{3 a^3}+\frac {7 x^{5/2}}{5 a^2}-\frac {x^{7/2}}{a (b+a x)}-\frac {7 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.32 \[ \frac {2 x^{9/2} \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {a x}{b}\right )}{9 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 188, normalized size = 2.21 \[ \left [\frac {105 \, {\left (a b^{2} x + b^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt {x}}{30 \, {\left (a^{5} x + a^{4} b\right )}}, -\frac {105 \, {\left (a b^{2} x + b^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - {\left (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt {x}}{15 \, {\left (a^{5} x + a^{4} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 76, normalized size = 0.89 \[ -\frac {7 \, b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {b^{3} \sqrt {x}}{{\left (a x + b\right )} a^{4}} + \frac {2 \, {\left (3 \, a^{8} x^{\frac {5}{2}} - 10 \, a^{7} b x^{\frac {3}{2}} + 45 \, a^{6} b^{2} \sqrt {x}\right )}}{15 \, a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.84 \[ \frac {2 x^{\frac {5}{2}}}{5 a^{2}}-\frac {7 b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {4 b \,x^{\frac {3}{2}}}{3 a^{3}}+\frac {b^{3} \sqrt {x}}{\left (a x +b \right ) a^{4}}+\frac {6 b^{2} \sqrt {x}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.56, size = 77, normalized size = 0.91 \[ \frac {6 \, a^{3} - \frac {14 \, a^{2} b}{x} + \frac {70 \, a b^{2}}{x^{2}} + \frac {105 \, b^{3}}{x^{3}}}{15 \, {\left (\frac {a^{5}}{x^{\frac {5}{2}}} + \frac {a^{4} b}{x^{\frac {7}{2}}}\right )}} + \frac {7 \, b^{3} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 68, normalized size = 0.80 \[ \frac {2\,x^{5/2}}{5\,a^2}-\frac {4\,b\,x^{3/2}}{3\,a^3}+\frac {6\,b^2\,\sqrt {x}}{a^4}+\frac {b^3\,\sqrt {x}}{x\,a^5+b\,a^4}-\frac {7\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 57.53, size = 542, normalized size = 6.38 \[ \begin {cases} \tilde {\infty } x^{\frac {9}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: b = 0 \\\frac {12 i a^{4} \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {1}{a}}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {28 i a^{3} b^{\frac {3}{2}} x^{\frac {5}{2}} \sqrt {\frac {1}{a}}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {140 i a^{2} b^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{a}}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {210 i a b^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{a}}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {105 a b^{3} x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {105 a b^{3} x \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {105 b^{4} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {105 b^{4} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{30 i a^{6} \sqrt {b} x \sqrt {\frac {1}{a}} + 30 i a^{5} b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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