Optimal. Leaf size=125 \[ a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )-\frac {\left (a+\frac {b}{x}\right )^{5/2} (2 a d+5 b c)}{5 a}-\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} (2 a d+5 b c)-a \sqrt {a+\frac {b}{x}} (2 a d+5 b c)+\frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {375, 78, 50, 63, 208} \[ a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )-\frac {\left (a+\frac {b}{x}\right )^{5/2} (2 a d+5 b c)}{5 a}-\frac {1}{3} \left (a+\frac {b}{x}\right )^{3/2} (2 a d+5 b c)-a \sqrt {a+\frac {b}{x}} (2 a d+5 b c)+\frac {c x \left (a+\frac {b}{x}\right )^{7/2}}{a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rule 375
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right ) \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {\left (\frac {5 b c}{2}+a d\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {(5 b c+2 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}+\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} (5 b c+2 a d) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{3} (5 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {(5 b c+2 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}+\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} (a (5 b c+2 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-a (5 b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} (5 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {(5 b c+2 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}+\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} \left (a^2 (5 b c+2 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-a (5 b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} (5 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {(5 b c+2 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}+\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {\left (a^2 (5 b c+2 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b}\\ &=-a (5 b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} (5 b c+2 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {(5 b c+2 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}+\frac {c \left (a+\frac {b}{x}\right )^{7/2} x}{a}+a^{3/2} (5 b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 94, normalized size = 0.75 \[ a^{3/2} (2 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {\sqrt {a+\frac {b}{x}} \left (a^2 x^2 (15 c x-46 d)-2 a b x (35 c x+11 d)-2 b^2 (5 c x+3 d)\right )}{15 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 222, normalized size = 1.78 \[ \left [\frac {15 \, {\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt {a} x^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \, {\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \, {\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{30 \, x^{2}}, -\frac {15 \, {\left (5 \, a b c + 2 \, a^{2} d\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (15 \, a^{2} c x^{3} - 6 \, b^{2} d - 2 \, {\left (35 \, a b c + 23 \, a^{2} d\right )} x^{2} - 2 \, {\left (5 \, b^{2} c + 11 \, a b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{15 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 253, normalized size = 2.02 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-30 a^{3} b d \,x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-75 a^{2} b^{2} c \,x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-60 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} d \,x^{4}-150 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b c \,x^{4}+60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d \,x^{2}+120 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \,x^{2}+32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b d x +20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c x +12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d \right )}{30 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 161, normalized size = 1.29 \[ \frac {1}{6} \, {\left (6 \, \sqrt {a + \frac {b}{x}} a^{2} x - 15 \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b - 24 \, \sqrt {a + \frac {b}{x}} a b\right )} c - \frac {1}{15} \, {\left (15 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} + 10 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a + 30 \, \sqrt {a + \frac {b}{x}} a^{2}\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 99, normalized size = 0.79 \[ -\frac {2\,d\,{\left (a+\frac {b}{x}\right )}^{5/2}}{5}-2\,a^2\,d\,\sqrt {a+\frac {b}{x}}-\frac {2\,a\,d\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}-\frac {2\,c\,x\,{\left (a+\frac {b}{x}\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {3}{2};\ -\frac {1}{2};\ -\frac {a\,x}{b}\right )}{3\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}}-a^{5/2}\,d\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 82.77, size = 520, normalized size = 4.16 \[ \frac {4 a^{\frac {11}{2}} b^{\frac {7}{2}} d x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {9}{2}} d x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {11}{2}} d x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {13}{2}} d \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} + a^{\frac {3}{2}} b c \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )} - \frac {4 a^{6} b^{3} d x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {4 a^{5} b^{4} d x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} b^{3} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{4} x^{\frac {5}{2}}} - \frac {2 a^{3} d \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + a^{2} \sqrt {b} c \sqrt {x} \sqrt {\frac {a x}{b} + 1} - \frac {4 a^{2} b c \operatorname {atan}{\left (\frac {\sqrt {a + \frac {b}{x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} - 2 a^{2} d \sqrt {a + \frac {b}{x}} - 4 a b c \sqrt {a + \frac {b}{x}} + 2 a b d \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + b^{2} c \left (\begin {cases} - \frac {\sqrt {a}}{x} & \text {for}\: b = 0 \\- \frac {2 \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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