Optimal. Leaf size=69 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {47, 63, 217, 206} \[ -\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(a+b x)^{5/2}} \, dx &=-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}+\frac {\int \frac {\sqrt {x}}{(a+b x)^{3/2}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 80, normalized size = 1.16 \[ \frac {6 \sqrt {a} (a+b x) \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )-2 \sqrt {b} \sqrt {x} (3 a+4 b x)}{3 b^{5/2} (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 186, normalized size = 2.70 \[ \left [\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (4 \, b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 105.59, size = 165, normalized size = 2.39 \[ -\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt {b}} + \frac {8 \, {\left (3 \, a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} + 3 \, a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} + 2 \, a^{3} b^{\frac {5}{2}}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3}}\right )} {\left | b \right |}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {3}{2}}}{\left (b x +a \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.83, size = 69, normalized size = 1.00 \[ -\frac {2 \, {\left (b + \frac {3 \, {\left (b x + a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} - \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{b^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{3/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.03, size = 328, normalized size = 4.75 \[ \frac {6 a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {6 a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {6 a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {8 a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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