Optimal. Leaf size=76 \[ -\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {375, 78, 51, 63, 208} \[ \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 375
Rubi steps
\begin {align*} \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {\left (-\frac {3 b c}{2}+a d\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^2}\\ &=\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^2 b}\\ &=\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.63 \[ \frac {(3 b c-2 a d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b}{a x}+1\right )+a c x}{a^2 \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 210, normalized size = 2.76 \[ \left [-\frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 127, normalized size = 1.67 \[ \frac {\frac {{\left (3 \, b^{2} c - 2 \, a b d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, a b^{2} c - 2 \, a^{2} b d - \frac {3 \, {\left (a x + b\right )} b^{2} c}{x} + \frac {2 \, {\left (a x + b\right )} a b d}{x}}{{\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )} a^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 387, normalized size = 5.09 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (2 a^{3} b d \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 a^{2} b^{2} c \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+4 a^{2} b^{2} d x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a \,b^{3} c x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-4 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} d \,x^{2}+6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b c \,x^{2}+2 a \,b^{3} d \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 b^{4} c \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-8 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b d x +12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{2} c x -4 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{2} d +6 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{3} c +4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d -4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{2} a^{\frac {5}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 144, normalized size = 1.89 \[ \frac {1}{2} \, c {\left (\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} - d {\left (\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.44, size = 71, normalized size = 0.93 \[ \frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2\,d}{a\,\sqrt {a+\frac {b}{x}}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 81.34, size = 224, normalized size = 2.95 \[ c \left (\frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}}\right ) + d \left (- \frac {2 a^{3} x \sqrt {1 + \frac {b}{a x}}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{3} x \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{3} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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