Optimal. Leaf size=69 \[ -\frac {2 x^{3/2}}{3 b (a+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, 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 = {49, 65, 223,
212} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 212
Rule 223
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 \text {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 \text {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 = 60, normalized size = 0.87 \begin {gather*} -\frac {2 \sqrt {x} (3 a+4 b x)}{3 b^2 (a+b x)^{3/2}}-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 5.87, size = 102, normalized size = 1.48 \begin {gather*} \frac {2 \left (3 a b^{\frac {9}{2}} \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \sqrt {\frac {a+b x}{a}}-3 \sqrt {a} b^5 \sqrt {x}+3 b^{\frac {11}{2}} x \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \sqrt {\frac {a+b x}{a}}-\frac {4 b^6 x^{\frac {3}{2}}}{\sqrt {a}}\right )}{3 b^7 \sqrt {\frac {a+b x}{a}} \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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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 = 0.35, size = 69, normalized size = 1.00 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 186, normalized size = 2.70 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs.
\(2 (63) = 126\).
time = 2.15, size = 328, normalized size = 4.75 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 106, normalized size = 1.54 \begin {gather*} 2 \left (\frac {2 \left (-\frac {\frac {1}{18}\cdot 12 b^{2} a \sqrt {x} \sqrt {x}}{b^{3} a}-\frac {\frac {1}{18}\cdot 9 b a^{2}}{b^{3} a}\right ) \sqrt {x} \sqrt {a+b x}}{\left (a+b x\right )^{2}}-\frac {\ln \left |\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right |}{b^{2} \sqrt {b}}\right ) \end {gather*}
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 \begin {gather*} \int \frac {x^{3/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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