Optimal. Leaf size=271 \[ \frac {3 \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}}-\frac {3 \left (a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^3}+\frac {(d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e^3}-\frac {2 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.34, antiderivative size = 298, normalized size of antiderivative = 1.10, 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 = {849, 818, 779, 621, 206} \[ \frac {3 \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )-2 c d e x \left (5 c d^2-a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 818
Rule 849
Rubi steps
\begin {align*} \int \frac {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 {x^3 (a e+c d 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 x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \int \frac {x \left (2 a c d^2 e \left (c d^2-a e^2\right )+\frac {1}{2} c d \left (c d^2-a e^2\right ) \left (5 c d^2-a e^2\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {\left (3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^2 d^2 e^3}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {\left (3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\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 )}{4 c^2 d^2 e^3}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )-2 c d e \left (5 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e^3 \left (c d^2-a e^2\right )}+\frac {3 \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\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 )}{8 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 331, normalized size = 1.22 \[ \frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (-a^3 e^6-a^2 c d^2 e^4-3 a c^2 d^4 e^2+5 c^3 d^6\right ) \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )+c^{3/2} d^{3/2} \sqrt {e} \left (3 a^3 e^5 (d+e x)+a^2 c d e^3 \left (4 d^2+5 d e x+e^2 x^2\right )-a c^2 d^2 e \left (15 d^3+d^2 e x-4 d e^2 x^2+2 e^3 x^3\right )+c^3 d^4 x \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{4 c^{7/2} d^{7/2} e^{7/2} \left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.29, size = 758, normalized size = 2.80 \[ \left [\frac {3 \, {\left (5 \, c^{3} d^{7} - 3 \, a c^{2} d^{5} e^{2} - a^{2} c d^{3} e^{4} - a^{3} d e^{6} + {\left (5 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} - a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (15 \, c^{3} d^{6} e - 4 \, a c^{2} d^{4} e^{3} - 3 \, a^{2} c d^{2} e^{5} - 2 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (5 \, c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} - 3 \, a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (c^{4} d^{6} e^{4} - a c^{3} d^{4} e^{6} + {\left (c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (5 \, c^{3} d^{7} - 3 \, a c^{2} d^{5} e^{2} - a^{2} c d^{3} e^{4} - a^{3} d e^{6} + {\left (5 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} - a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (15 \, c^{3} d^{6} e - 4 \, a c^{2} d^{4} e^{3} - 3 \, a^{2} c d^{2} e^{5} - 2 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + {\left (5 \, c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} - 3 \, a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (c^{4} d^{6} e^{4} - a c^{3} d^{4} e^{6} + {\left (c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x\right )}}\right ] \]
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 = 0.02, size = 391, normalized size = 1.44 \[ \frac {3 a^{2} e \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{8 \sqrt {c d e}\, c^{2} d^{2}}+\frac {3 a \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{4 \sqrt {c d e}\, c e}+\frac {15 d^{2} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{8 \sqrt {c d e}\, e^{3}}+\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, d^{3}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) e^{4}}+\frac {\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, x}{2 c d \,e^{2}}-\frac {3 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a}{4 c^{2} d^{2} e}-\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{4 c \,e^{3}} \]
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 = -0.00 \[ \int \frac {x^3}{\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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