Optimal. Leaf size=274 \[ \frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^5}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {\sqrt {a+c x^2}}{4 d x^4} \]
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Rubi [A] time = 0.30, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {961, 266, 47, 51, 63, 208, 264, 277, 217, 206, 50, 735, 844, 725} \[ \frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}+\frac {e^3 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^5}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {\sqrt {a+c x^2}}{4 d x^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 51
Rule 63
Rule 206
Rule 208
Rule 217
Rule 264
Rule 266
Rule 277
Rule 725
Rule 735
Rule 844
Rule 961
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx &=\int \left (\frac {\sqrt {a+c x^2}}{d x^5}-\frac {e \sqrt {a+c x^2}}{d^2 x^4}+\frac {e^2 \sqrt {a+c x^2}}{d^3 x^3}-\frac {e^3 \sqrt {a+c x^2}}{d^4 x^2}+\frac {e^4 \sqrt {a+c x^2}}{d^5 x}-\frac {e^5 \sqrt {a+c x^2}}{d^5 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {a+c x^2}}{x^5} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x^4} \, dx}{d^2}+\frac {e^2 \int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{d^3}-\frac {e^3 \int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d^4}+\frac {e^4 \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^5}-\frac {e^5 \int \frac {\sqrt {a+c x^2}}{d+e x} \, dx}{d^5}\\ &=-\frac {e^4 \sqrt {a+c x^2}}{d^5}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^3} \, dx,x,x^2\right )}{2 d}+\frac {e^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (c e^3\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^4}+\frac {e^4 \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^5}-\frac {e^4 \int \frac {a e-c d x}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^5}\\ &=-\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {c \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{8 d}+\frac {\left (c e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 d^3}+\frac {\left (c e^3\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^4}-\frac {\left (c e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^4}+\frac {\left (a e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^5}-\frac {\left (e^3 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^5}\\ &=-\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac {\sqrt {c} e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d^4}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{16 a d}+\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 d^3}+\frac {\left (c e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^4}+\frac {\left (a e^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^5}+\frac {\left (e^3 \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d^5}\\ &=-\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^5}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}-\frac {c \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{8 a d}\\ &=-\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^5}+\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 344, normalized size = 1.26 \[ \frac {-\frac {2 c^2 d^4 \left (a+c x^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {c x^2}{a}+1\right )}{a^3}+\frac {2 d^3 e \left (a+c x^2\right )^{3/2}}{a x^3}-\frac {3 d^2 e^2 \left (c x^2 \sqrt {\frac {c x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )+a+c x^2\right )}{x^2 \sqrt {a+c x^2}}+6 e^3 \left (\sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )+\sqrt {c} d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )\right )+\frac {6 d e^3 \left (-\sqrt {a} \sqrt {c} x \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )+a+c x^2\right )}{x \sqrt {a+c x^2}}-6 e^4 \sqrt {a+c x^2}+6 e^4 \left (\sqrt {a+c x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\right )}{6 d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 1007, normalized size = 3.68 \[ \left [\frac {24 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {a} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{2} d^{5} x^{4}}, \frac {48 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {a} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{2} d^{5} x^{4}}, \frac {12 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{2} d^{5} x^{4}}, \frac {24 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{2} d^{5} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 596, normalized size = 2.18 \[ -\frac {2 \, {\left (c d^{2} e^{3} + a e^{5}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} d^{5}} - \frac {{\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a d^{5}} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} c^{2} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a c^{\frac {3}{2}} d^{2} e + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a c d e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} d^{2} e + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{2} c d e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{2} \sqrt {c} e^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{3} c d e^{2} + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{3} \sqrt {c} e^{3} + 8 \, a^{4} c^{\frac {3}{2}} d^{2} e + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{4} c d e^{2} - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{4} \sqrt {c} e^{3} + 24 \, a^{5} \sqrt {c} e^{3}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4} a d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 703, normalized size = 2.57 \[ \frac {a \,e^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{5}}+\frac {c \,e^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{3}}-\frac {\sqrt {a}\, e^{4} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{d^{5}}-\frac {c \,e^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}\, d^{3}}+\frac {c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}} d}-\frac {\sqrt {c}\, e^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{d^{4}}+\frac {\sqrt {c}\, e^{3} \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{d^{4}}-\frac {\sqrt {c \,x^{2}+a}\, c \,e^{3} x}{a \,d^{4}}+\frac {\sqrt {c \,x^{2}+a}\, c \,e^{2}}{2 a \,d^{3}}-\frac {\sqrt {c \,x^{2}+a}\, c^{2}}{8 a^{2} d}+\frac {\sqrt {c \,x^{2}+a}\, e^{4}}{d^{5}}-\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{4}}{d^{5}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e^{3}}{a \,d^{4} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e^{2}}{2 a \,d^{3} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} c}{8 a^{2} d \,x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} e}{3 a \,d^{2} x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 a d \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c\,x^2+a}}{x^5\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {a + c x^{2}}}{x^{5} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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