Optimal. Leaf size=449 \[ -\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}}+\frac {\left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} x^2 \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e} \]
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Rubi [A] time = 0.57, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 832, 779, 612, 621, 206} \[ -\frac {\left (-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-35 a^3 e^6-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}+\frac {\left (-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8+21 c^4 d^8\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^5}-\frac {\left (15 a^2 c d^2 e^4+7 a^3 e^6+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e}+\frac {1}{20} x^2 \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rule 849
Rubi steps
\begin {align*} \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx &=\int 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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}+\frac {\int x^2 \left (-3 a c d^2 e-\frac {3}{2} c d \left (3 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}{6 c d e}\\ &=\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}+\frac {\int x \left (3 a c d^2 e \left (3 c d^2-a e^2\right )+\frac {3}{4} c d \left (21 c^2 d^4-6 a c d^2 e^2-7 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}{30 c^2 d^2 e^2}\\ &=\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}+\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \int \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 {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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}{1024 c^4 d^4 e^5}\\ &=\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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 )}{512 c^4 d^4 e^5}\\ &=\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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 )}{1024 c^{9/2} d^{9/2} e^{11/2}}\\ \end {align*}
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Mathematica [A] time = 2.29, size = 425, normalized size = 0.95 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+5 a^4 c d e^8 (11 d+14 e x)+2 a^3 c^2 d^2 e^6 \left (27 d^2-16 d e x-28 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (13 d^3-6 d^2 e x+4 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (-525 d^4+336 d^3 e x-264 d^2 e^2 x^2+224 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (315 d^5-210 d^4 e x+168 d^3 e^2 x^2-144 d^2 e^3 x^3+128 d e^4 x^4+1280 e^5 x^5\right )\right )-\frac {15 \sqrt {c d} \left (c d^2-a e^2\right )^{5/2} \left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{7680 c^{9/2} d^{9/2} e^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 1044, normalized size = 2.33 \[ \left [-\frac {15 \, {\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\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 (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e - 525 \, a c^{5} d^{9} e^{3} + 78 \, a^{2} c^{4} d^{7} e^{5} + 54 \, a^{3} c^{3} d^{5} e^{7} + 55 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} - 16 \, {\left (9 \, c^{6} d^{8} e^{4} - 14 \, a c^{5} d^{6} e^{6} - 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (21 \, c^{6} d^{9} e^{3} - 33 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (105 \, c^{6} d^{10} e^{2} - 168 \, a c^{5} d^{8} e^{4} + 18 \, a^{2} c^{4} d^{6} e^{6} + 16 \, a^{3} c^{3} d^{4} e^{8} - 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{5} d^{5} e^{6}}, \frac {15 \, {\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\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 (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e - 525 \, a c^{5} d^{9} e^{3} + 78 \, a^{2} c^{4} d^{7} e^{5} + 54 \, a^{3} c^{3} d^{5} e^{7} + 55 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} - 16 \, {\left (9 \, c^{6} d^{8} e^{4} - 14 \, a c^{5} d^{6} e^{6} - 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (21 \, c^{6} d^{9} e^{3} - 33 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (105 \, c^{6} d^{10} e^{2} - 168 \, a c^{5} d^{8} e^{4} + 18 \, a^{2} c^{4} d^{6} e^{6} + 16 \, a^{3} c^{3} d^{4} e^{8} - 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{5} d^{5} e^{6}}\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 = 1883, normalized size = 4.19 \[ \text {result too large to display} \]
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 \[ \int \frac {x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{3} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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