Optimal. Leaf size=121 \[ \frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 613} \[ \frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{(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}{3 \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 {(4 c d) \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 \left (c d^2-a e^2\right )}\\ &=\frac {2}{3 \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 {8 c d \left (c d^2+a e^2+2 c d e x\right )}{3 \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 = 0.03, size = 95, normalized size = 0.79 \[ \frac {2 a^2 e^4-4 a c d e^2 (3 d+2 e x)-2 c^2 d^2 \left (3 d^2+12 d e x+8 e^2 x^2\right )}{3 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.73, size = 306, normalized size = 2.53 \[ -\frac {2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 4 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3 \, {\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} + {\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} + {\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} + {\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \]
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 = 138, normalized size = 1.14 \[ -\frac {2 \left (c d x +a e \right ) \left (-8 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x -12 c^{2} d^{3} e x +a^{2} e^{4}-6 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right )}{3 \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \]
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 = 2.88, size = 120, normalized size = 0.99 \[ \frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (-a^2\,e^4+6\,a\,c\,d^2\,e^2+4\,a\,c\,d\,e^3\,x+3\,c^2\,d^4+12\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{3\,\left (a\,e+c\,d\,x\right )\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^2} \]
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 \[ \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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