Optimal. Leaf size=103 \[ -\frac {(5 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(5 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}} \]
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 )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {\left (-\frac {5 b c}{2}+a d\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a^2}\\ &=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3}\\ &=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 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^3 b}\\ &=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(5 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 60, normalized size = 0.58 \[ \frac {x \left ((5 b c-2 a d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b}{a x}+1\right )+3 a c x\right )}{3 a^2 \sqrt {a+\frac {b}{x}} (a x+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 331, normalized size = 3.21 \[ \left [-\frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}, \frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 145, normalized size = 1.41 \[ -\frac {\frac {3 \, b^{2} c \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a^{3}} - \frac {3 \, {\left (5 \, b^{2} c - 2 \, a b d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} - \frac {2 \, {\left (a b^{2} c - a^{2} b d + \frac {6 \, {\left (a x + b\right )} b^{2} c}{x} - \frac {3 \, {\left (a x + b\right )} a b d}{x}\right )} x}{{\left (a x + b\right )} a^{3} \sqrt {\frac {a x + b}{x}}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 541, normalized size = 5.25 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (6 a^{4} b d \,x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-15 a^{3} b^{2} c \,x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+18 a^{3} b^{2} d \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-45 a^{2} b^{3} c \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {9}{2}} d \,x^{3}+30 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b c \,x^{3}+18 a^{2} b^{3} d x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-45 a \,b^{4} c x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-36 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b d \,x^{2}+90 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{2} c \,x^{2}+6 a \,b^{4} d \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-15 b^{5} c \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-36 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{2} d x +90 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{3} c x +12 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} d x -24 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b c x -12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{3} d +30 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{4} c +8 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b d -20 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} c \right ) x}{6 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{3} a^{\frac {7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 170, normalized size = 1.65 \[ \frac {1}{6} \, c {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {1}{3} \, d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.91, size = 87, normalized size = 0.84 \[ \frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,d}{3\,a}+\frac {2\,d\,\left (a+\frac {b}{x}\right )}{a^2}}{{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {7}{2};\ \frac {9}{2};\ -\frac {a\,x}{b}\right )}{7\,{\left (a+\frac {b}{x}\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 155.82, size = 1479, normalized size = 14.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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