Optimal. Leaf size=195 \[ -\frac {\left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} e^{5/2}}+\frac {2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x) \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d e^2} \]
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Rubi [A] time = 0.35, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1638, 792, 621, 206} \[ -\frac {\left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} e^{5/2}}+\frac {2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x) \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d e^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 792
Rule 1638
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d e^2}+\frac {\int \frac {-\frac {1}{2} d e \left (c d^2+a e^2\right )-\frac {1}{2} e^2 \left (3 c d^2+a e^2\right ) x}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e^3}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d e^2}+\frac {2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)}-\frac {1}{2} \left (\frac {a}{c d}+\frac {3 d}{e^2}\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d e^2}+\frac {2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)}-\left (\frac {a}{c d}+\frac {3 d}{e^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d e^2}+\frac {2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 \left (c d^2-a e^2\right ) (d+e x)}-\frac {\left (3 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 255, normalized size = 1.31 \[ \frac {c^{3/2} d^{3/2} \sqrt {e} \left (-a^2 e^3 (d+e x)+a c d e \left (3 d^2-e^2 x^2\right )+c^2 d^3 x (3 d+e x)\right )-\sqrt {c d} \sqrt {c d^2-a e^2} \left (-a^2 e^4-2 a c d^2 e^2+3 c^2 d^4\right ) \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{c^{5/2} d^{5/2} e^{5/2} \left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 586, normalized size = 3.01 \[ \left [\frac {{\left (3 \, c^{2} d^{5} - 2 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left (c^{3} d^{5} e^{3} - a c^{2} d^{3} e^{5} + {\left (c^{3} d^{4} e^{4} - a c^{2} d^{2} e^{6}\right )} x\right )}}, \frac {{\left (3 \, c^{2} d^{5} - 2 \, a c d^{3} e^{2} - a^{2} d e^{4} + {\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (3 \, c^{2} d^{4} e - a c d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (c^{3} d^{5} e^{3} - a c^{2} d^{3} e^{5} + {\left (c^{3} d^{4} e^{4} - a c^{2} d^{2} e^{6}\right )} x\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 241, normalized size = 1.24 \[ -\frac {a \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{2 \sqrt {c d e}\, c d}-\frac {3 d \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{2 \sqrt {c d e}\, e^{2}}-\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, d^{2}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) e^{3}}+\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{c d \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^2}{\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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