Optimal. Leaf size=174 \[ -\frac {b d^2 (4 a c+3 b) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{8 c^{5/2} (a c+b)^{3/2}}+\frac {d (4 a c+5 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^2 x^4} \]
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Rubi [A] time = 0.51, antiderivative size = 218, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6722, 1975, 446, 96, 94, 93, 208} \[ -\frac {b d^2 (4 a c+3 b) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a c+b} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a \left (c+d x^2\right )+b}}\right )}{8 c^{5/2} (a c+b)^{3/2} \sqrt {a \left (c+d x^2\right )+b}}+\frac {d (4 a c+3 b) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 x^2 (a c+b)}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{4 c x^4 (a c+b)} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rule 446
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^5} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a \left (c+d x^2\right )}}{x^5 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{x^5 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{x^3 \sqrt {c+d x}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac {\left ((3 b+4 a c) d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right )}{8 c (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac {\left (b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{16 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}+\frac {\left (b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{8 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {(3 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^2 (b+a c) x^2}-\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{4 c (b+a c) x^4}-\frac {b (3 b+4 a c) d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{8 c^{5/2} (b+a c)^{3/2} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 278, normalized size = 1.60 \[ -\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (2 \sqrt {c (a c+b)} \left (c+d x^2\right ) \left (2 a^2 c \left (c^2-d^2 x^4\right )+a b \left (4 c^2-3 c d x^2-3 d^2 x^4\right )+b^2 \left (2 c-3 d x^2\right )\right )-2 b d^2 x^4 \log (x) (4 a c+3 b) \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}+b d^2 x^4 (4 a c+3 b) \sqrt {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )} \log \left (2 \sqrt {c (a c+b)} \sqrt {\left (c+d x^2\right ) \left (a c+a d x^2+b\right )}+2 a c \left (c+d x^2\right )+b \left (2 c+d x^2\right )\right )\right )}{16 c x^4 (c (a c+b))^{3/2} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 577, normalized size = 3.32 \[ \left [\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}, \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 5 \, a b c^{2} + 3 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 713, normalized size = 4.10 \[ \frac {1}{8} \, {\left (\frac {{\left (4 \, a b c d^{2} + 3 \, b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}}{\sqrt {-a c^{2} - b c}}\right )}{{\left (a c^{3} + b c^{2}\right )} \sqrt {-a c^{2} - b c}} + \frac {8 \, a^{\frac {7}{2}} c^{5} d {\left | d \right |} + 16 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{4} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c^{3} d {\left | d \right |} + 24 \, a^{\frac {5}{2}} b c^{4} d {\left | d \right |} + 36 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c^{3} d^{2} + 8 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b c^{2} d {\left | d \right |} + 24 \, a^{\frac {3}{2}} b^{2} c^{3} d {\left | d \right |} - 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a b c d^{2} + 25 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} c^{2} d^{2} + 8 \, \sqrt {a} b^{3} c^{2} d {\left | d \right |} - 3 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} b^{2} d^{2} + 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} b^{3} c d^{2}}{{\left (a c^{3} + b c^{2}\right )} {\left (a c^{2} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} + b c\right )}^{2}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 923, normalized size = 5.30 \[ \frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-4 a^{3} b \,c^{5} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-11 a^{2} b^{2} c^{4} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-10 a \,b^{3} c^{3} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-3 b^{4} c^{2} d^{2} x^{4} \ln \left (\frac {2 a c d \,x^{2}+b d \,x^{2}+2 a \,c^{2}+2 b c +2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}{x^{2}}\right )-12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a^{2} c \,d^{3} x^{6}-10 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a b \,d^{3} x^{6}-20 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a^{2} c^{2} d^{2} x^{4}-28 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a b c \,d^{2} x^{4}-10 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b^{2} d^{2} x^{4}+12 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a c d \,x^{2}+10 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b d \,x^{2}-4 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} a \,c^{2}-4 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} b c \right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \left (a c +b \right )^{2} \left (a \,c^{2}+b c \right )^{\frac {3}{2}} c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 322, normalized size = 1.85 \[ \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, {\left (a c^{3} + b c^{2}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {{\left (4 \, a b c^{2} + 5 \, b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 7 \, a b^{2} c + 3 \, b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{3} c^{5} + 3 \, a^{2} b c^{4} + 3 \, a b^{2} c^{3} + b^{3} c^{2} + \frac {{\left (a c^{5} + b c^{4}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \]
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 \[ \int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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