Optimal. Leaf size=76 \[ \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.03, 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 = {382, 79, 53, 65,
214} \begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 214
Rule 382
Rubi steps
\begin {align*} \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=-\text {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 ) \text {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) \text {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) \text {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 [A]
time = 0.14, size = 70, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} x (3 b c-2 a d+a c x)}{a^2 (b+a x)}+\frac {(-3 b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs.
\(2(66)=132\).
time = 0.06, size = 387, normalized size = 5.09
method | result | size |
risch | \(\frac {c \left (a x +b \right )}{a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (\frac {\ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) d}{a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) b c}{2 a^{\frac {5}{2}}}-\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}\, d}{a^{2} \left (x +\frac {b}{a}\right )}+\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}\, b c}{a^{3} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(187\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-4 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} d \,x^{2}+6 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b c \,x^{2}+2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b d \,x^{2}-3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,x^{2}+4 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} d -4 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} b c -8 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b d x +12 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2} c x +4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} d x -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c x -4 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2} d +6 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{3} c +2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} d -3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} c \right )}{2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{2}}\) | \(387\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (66) = 132\).
time = 0.49, size = 144, normalized size = 1.89 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.16, size = 210, normalized size = 2.76 \begin {gather*} \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 ] \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. 224 vs.
\(2 (65) = 130\).
time = 20.95, size = 224, normalized size = 2.95 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (66) = 132\).
time = 1.60, size = 148, normalized size = 1.95 \begin {gather*} -\frac {{\left (3 \, b c \log \left ({\left | b \right |}\right ) - 2 \, a d \log \left ({\left | b \right |}\right ) + 4 \, b c - 4 \, a d\right )} \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} + \frac {\sqrt {a x^{2} + b x} c}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {{\left (3 \, b c - 2 \, a d\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (b^{2} c - a b d\right )}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a + \sqrt {a} b\right )} a^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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