Optimal. Leaf size=329 \[ \frac {\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 d^3 e^2 x \left (c d^2-a e^2\right )}-\frac {3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{5/2} d^{7/2} e^{5/2}}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2 \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{d x^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.51, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {851, 822, 834, 806, 724, 206} \[ -\frac {3 \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{5/2} d^{7/2} e^{5/2}}+\frac {\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a^2 d^3 e^2 x \left (c d^2-a e^2\right )}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a d^2 e x^2 \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{d x^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 834
Rule 851
Rubi steps
\begin {align*} \int \frac {1}{x^3 (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^3 \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^2 \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^2-5 a e^2\right ) \left (c d^2-a e^2\right )+2 a c d e^2 \left (c d^2-a e^2\right ) x}{x^3 \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^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac {\int \frac {-\frac {1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )-\frac {1}{2} a c d e^2 \left (c d^2-5 a e^2\right ) \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^2 d^2 e^2 \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^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}+\frac {\left (3 \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}-\frac {\left (3 \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\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 )}{4 a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a d^2 e \left (c d^2-a e^2\right ) x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{4 a^2 d^3 e^2 \left (c d^2-a e^2\right ) x}-\frac {3 \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\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 )}{8 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 283, normalized size = 0.86 \[ \frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (a^3 e^4 \left (2 d^2-5 d e x-15 e^2 x^2\right )-a^2 c d e^2 \left (2 d^3-4 d^2 e x+d e^2 x^2+15 e^3 x^3\right )+a c^2 d^3 e x \left (d^2+5 d e x+4 e^2 x^2\right )+3 c^3 d^5 x^2 (d+e x)\right )-3 x^2 \sqrt {d+e x} \left (-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right ) \sqrt {a e+c d x} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{4 a^{5/2} d^{7/2} e^{5/2} x^2 \left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 7.60, size = 792, normalized size = 2.41 \[ \left [\frac {3 \, {\left ({\left (c^{3} d^{6} e + a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - 5 \, a^{3} e^{7}\right )} x^{3} + {\left (c^{3} d^{7} + a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} - 5 \, a^{3} d e^{6}\right )} x^{2}\right )} \sqrt {a d e} \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 (2 \, a^{2} c d^{5} e^{2} - 2 \, a^{3} d^{3} e^{4} - {\left (3 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{3} e^{4} - 15 \, a^{3} d e^{6}\right )} x^{2} - {\left (3 \, a c^{2} d^{6} e + 2 \, a^{2} c d^{4} e^{3} - 5 \, a^{3} d^{2} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left ({\left (a^{3} c d^{6} e^{4} - a^{4} d^{4} e^{6}\right )} x^{3} + {\left (a^{3} c d^{7} e^{3} - a^{4} d^{5} e^{5}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (c^{3} d^{6} e + a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - 5 \, a^{3} e^{7}\right )} x^{3} + {\left (c^{3} d^{7} + a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} - 5 \, a^{3} d e^{6}\right )} x^{2}\right )} \sqrt {-a d e} \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 (2 \, a^{2} c d^{5} e^{2} - 2 \, a^{3} d^{3} e^{4} - {\left (3 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{3} e^{4} - 15 \, a^{3} d e^{6}\right )} x^{2} - {\left (3 \, a c^{2} d^{6} e + 2 \, a^{2} c d^{4} e^{3} - 5 \, a^{3} d^{2} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left ({\left (a^{3} c d^{6} e^{4} - a^{4} d^{4} e^{6}\right )} x^{3} + {\left (a^{3} c d^{7} e^{3} - a^{4} d^{5} e^{5}\right )} x^{2}\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 = 414, normalized size = 1.26 \[ -\frac {3 c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{4 \sqrt {a d e}\, a d}-\frac {3 c^{2} d \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{8 \sqrt {a d e}\, a^{2} e^{2}}-\frac {15 e^{2} \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{8 \sqrt {a d e}\, d^{3}}+\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{2}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) d^{3}}+\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{4 a \,d^{3} x}+\frac {3 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c}{4 a^{2} d \,e^{2} x}-\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{2 a \,d^{2} e \,x^{2}} \]
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 \[ \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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 \[ \int \frac {1}{x^{3} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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