Optimal. Leaf size=138 \[ \frac {2 \left (3 a e^2+c d^2\right ) \left (a e^2+c d^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 d}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {792, 613} \[ \frac {2 \left (3 a e^2+c d^2\right ) \left (a e^2+c d^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 d}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 792
Rubi steps
\begin {align*} \int \frac {x}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 d}{3 e \left (c d^2-a e^2\right ) (d+e 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 ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 e \left (c d^2-a e^2\right )}\\ &=-\frac {2 d}{3 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \left (c d^2+3 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 100, normalized size = 0.72 \[ \frac {2 \left (a^2 e^3 (2 d+3 e x)+2 a c d e \left (3 d^2+5 d e x+3 e^2 x^2\right )+c^2 d^3 x (3 d+2 e x)\right )}{3 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 4.13, size = 314, normalized size = 2.28 \[ \frac {2 \, {\left (6 \, a c d^{3} e + 2 \, a^{2} d e^{3} + 2 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x^{2} + {\left (3 \, c^{2} d^{4} + 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3 \, {\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} + {\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} + {\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} + {\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \]
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 = 149, normalized size = 1.08 \[ -\frac {2 \left (c d x +a e \right ) \left (6 a c d \,e^{3} x^{2}+2 c^{2} d^{3} e \,x^{2}+3 a^{2} e^{4} x +10 a c \,d^{2} e^{2} x +3 c^{2} d^{4} x +2 a^{2} d \,e^{3}+6 a c \,d^{3} e \right )}{3 \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \]
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 [B] time = 3.32, size = 499, normalized size = 3.62 \[ \frac {4\,a^2\,d\,e^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+6\,a^2\,e^4\,x\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+6\,c^2\,d^4\,x\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+4\,c^2\,d^3\,e\,x^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+12\,a\,c\,d^3\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+20\,a\,c\,d^2\,e^2\,x\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}+12\,a\,c\,d\,e^3\,x^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{-3\,a^4\,d^2\,e^7-6\,a^4\,d\,e^8\,x-3\,a^4\,e^9\,x^2+9\,a^3\,c\,d^4\,e^5+15\,a^3\,c\,d^3\,e^6\,x+3\,a^3\,c\,d^2\,e^7\,x^2-3\,a^3\,c\,d\,e^8\,x^3-9\,a^2\,c^2\,d^6\,e^3-9\,a^2\,c^2\,d^5\,e^4\,x+9\,a^2\,c^2\,d^4\,e^5\,x^2+9\,a^2\,c^2\,d^3\,e^6\,x^3+3\,a\,c^3\,d^8\,e-3\,a\,c^3\,d^7\,e^2\,x-15\,a\,c^3\,d^6\,e^3\,x^2-9\,a\,c^3\,d^5\,e^4\,x^3+3\,c^4\,d^9\,x+6\,c^4\,d^8\,e\,x^2+3\,c^4\,d^7\,e^2\,x^3} \]
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 \[ \int \frac {x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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