Optimal. Leaf size=500 \[ \frac {\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6} \]
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Rubi [A] time = 0.64, antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 834, 806, 720, 724, 206} \[ \frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}+\frac {\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rule 849
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^8 (d+e x)} \, dx &=\int \frac {(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}}{x^8} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-9 a e^2\right )+2 a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7} \, dx}{7 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\int \frac {\left (-\frac {1}{4} a e \left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right )-\frac {1}{2} a c d e^2 \left (5 c d^2-9 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}}{x^6} \, dx}{42 a^2 d^2 e^2}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{48 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{256 a^3 d^4 e^3}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 a^4 d^5 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^4 d^5 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \left (2 a d e+\left (c d^2+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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 408, normalized size = 0.82 \[ \frac {((d+e x) (a e+c d x))^{3/2} \left (\frac {7 \left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (5 x \left (c d^2-a e^2\right ) \left (\frac {x \left (c d^2-a e^2\right ) \left (\frac {x \left (a e^2-c d^2\right ) \left (3 x^2 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {d+e x} \sqrt {a e+c d x} \left (a e (2 d+5 e x)-3 c d^2 x\right )\right )}{a^{5/2} \sqrt {d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt {a e+c d x}\right )}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right )-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right )}{15360 a d^3 e x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac {(d+e x) \left (9 a e^2+7 c d^2\right ) (a e+c d x)^2}{12 a d e x^6}-\frac {(d+e x) (a e+c d x)^2}{x^7}\right )}{7 a d e} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 5353, normalized size = 10.71 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^8\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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