Optimal. Leaf size=229 \[ \frac {\left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d^2 e x \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{d x \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.29, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {851, 822, 806, 724, 206} \[ \frac {\left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d^2 e x \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{d x \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 851
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\int \frac {a e+c d x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \int \frac {-\frac {1}{2} a e \left (c d^2-3 a e^2\right ) \left (c d^2-a e^2\right )+a c d e^2 \left (c d^2-a e^2\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a d^2 e \left (c d^2-a e^2\right ) x}-\frac {1}{2} \left (\frac {c}{a e}+\frac {3 e}{d^2}\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a d^2 e \left (c d^2-a e^2\right ) x}-\left (-\frac {c}{a e}-\frac {3 e}{d^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a d^2 e \left (c d^2-a e^2\right ) x}+\frac {\left (c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 201, normalized size = 0.88 \[ \frac {x \sqrt {d+e x} \left (-3 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \sqrt {a e+c d x} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+\sqrt {a} \sqrt {d} \sqrt {e} \left (a^2 e^3 (d+3 e x)-a c d e \left (d^2-3 e^2 x^2\right )-c^2 d^3 x (d+e x)\right )}{a^{3/2} d^{5/2} e^{3/2} x \left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.30, size = 610, normalized size = 2.66 \[ \left [\frac {\sqrt {a d e} {\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3} + {\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} + {\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}, -\frac {\sqrt {-a d e} {\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3} + {\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} + {\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}\right ] \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 253, normalized size = 1.10 \[ \frac {c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, a e}+\frac {3 e \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, d^{2}}-\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) d^{2}}-\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{a \,d^{2} e x} \]
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 \[ \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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