Optimal. Leaf size=345 \[ -\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac {1}{24} x^2 \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e} \]
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Rubi [A] time = 0.51, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {849, 832, 779, 621, 206} \[ -\frac {\left (-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-15 a^3 e^6-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (9 a^2 c d^2 e^4+5 a^3 e^6+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}+\frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac {1}{24} x^2 \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rule 849
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx &=\int \frac {x^3 (a e+c d x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x^2 \left (-3 a c d^2 e-\frac {1}{2} c d \left (7 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}{4 c d e}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x \left (a c d^2 e \left (7 c d^2-a e^2\right )+\frac {1}{4} c d \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 c^2 d^2 e^2}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c^3 d^3 e^4}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{64 c^3 d^3 e^4}\\ &=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{128 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 304, normalized size = 0.88 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\sqrt {c} \sqrt {d} \sqrt {e} \left (-15 a^3 e^6+a^2 c d e^4 (10 e x-17 d)+a c^2 d^2 e^2 \left (-25 d^2+12 d e x-8 e^2 x^2\right )+c^3 d^3 \left (105 d^3-70 d^2 e x+56 d e^2 x^2-48 e^3 x^3\right )\right )\right )}{192 c^{7/2} d^{7/2} e^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 678, normalized size = 1.97 \[ \left [-\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{5}}, -\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{5}}\right ] \]
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 [B] time = 0.03, size = 946, normalized size = 2.74 \[ -\frac {5 a^{4} e^{4} \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 )}{128 \sqrt {c d e}\, c^{3} d^{3}}-\frac {a^{3} e^{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 )}{32 \sqrt {c d e}\, c^{2} d}-\frac {3 a^{2} 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 )}{64 \sqrt {c d e}\, c}-\frac {a \,d^{3} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\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 )}\right )}{2 \sqrt {c d e}\, e^{2}}+\frac {11 a \,d^{3} \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 )}{32 \sqrt {c d e}\, e^{2}}+\frac {c \,d^{5} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\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 )}\right )}{2 \sqrt {c d e}\, e^{4}}-\frac {29 c \,d^{5} \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 )}{128 \sqrt {c d e}\, e^{4}}+\frac {5 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} e x}{32 c^{2} d^{2}}+\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a x}{16 c e}+\frac {29 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{2} x}{32 e^{3}}+\frac {5 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{3} e^{2}}{64 c^{3} d^{3}}+\frac {19 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2}}{64 c^{2} d}+\frac {43 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a d}{64 c \,e^{2}}-\frac {\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}}{e^{4}}+\frac {29 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{3}}{64 e^{4}}+\frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} x}{4 c d \,e^{2}}-\frac {5 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} a}{24 c^{2} d^{2} e}-\frac {13 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}}}{24 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 \[ \int \frac {x^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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