Optimal. Leaf size=352 \[ \frac {\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}-\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]
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Rubi [A] time = 0.33, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {851, 832, 779, 612, 621, 206} \[ -\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac {\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rule 851
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 (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^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (7 c d^2-3 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}{5 c d e}\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c^2 d^2 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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}{256 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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 )}{128 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\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}}{128 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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 )}{256 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 2.77, size = 497, normalized size = 1.41 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {\frac {5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (8 c^3 d^3 e^3 \sqrt {c d} \sqrt {c d^2-a e^2} (a e+c d x)^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-c d \left (c d^2-a e^2\right ) \left (-3 c^{5/2} d^{5/2} \sqrt {e} \left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )-2 e^2 (c d)^{5/2} \sqrt {c d^2-a e^2} (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+3 e (c d)^{5/2} \left (c d^2-a e^2\right )^{3/2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}\right )\right )}{\sqrt {c d} \sqrt {c d^2-a e^2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-48 c^4 d^4 e^3 (d+e x) \left (5 a e^2+7 c d^2\right ) (a e+c d x)^3}{384 c^5 d^5 e^4 (a e+c d x)}+x (d+e x) (a e+c d x)^2\right )}{5 c d e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 846, normalized size = 2.40 \[ \left [-\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, c^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, 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 = 0.02, size = 1560, normalized size = 4.43 \[ \text {result too large to display} \]
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 \[ \int \frac {x^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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