Optimal. Leaf size=664 \[ \frac {2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}+\frac {5 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}+\frac {\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}-\frac {\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 1.17, antiderivative size = 664, 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 = {851, 822, 834, 806, 724, 206} \begin {gather*} -\frac {\left (33 a^2 c d^2 e^4-21 a^3 e^6-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}-\frac {\left (-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}+\frac {\left (-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}+\frac {2 \left (c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4-9 a^3 e^6+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {5 \left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \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 )}{16 a^{9/2} d^{11/2} e^{9/2}}-\frac {2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 834
Rule 851
Rubi steps
\begin {align*} \int \frac {1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac {a e+c d x}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} a e \left (c d^2-3 a e^2\right ) \left (c d^2-a e^2\right )+5 a c d e^2 \left (c d^2-a e^2\right ) x}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {3}{4} a e \left (c d^2-a e^2\right ) \left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right )+\frac {3}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}-\frac {4 \int \frac {\frac {3}{8} a e \left (c d^2-a e^2\right ) \left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right )+\frac {3}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{9 a^3 d^3 e^3 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac {\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}+\frac {2 \int \frac {\frac {3}{16} a e \left (c d^2-a e^2\right ) \left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right )+\frac {3}{8} a c d e^2 \left (c d^2-a e^2\right ) \left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{9 a^4 d^4 e^4 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac {\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac {\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}-\frac {\left (5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\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}{16 a^4 d^5 e^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac {\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac {\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}+\frac {\left (5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\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 )}{8 a^4 d^5 e^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+13 a^2 c d^2 e^4-9 a^3 e^6+c d e \left (3 c^2 d^4+14 a c d^2 e^2-9 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (7 c^3 d^6-3 a c^2 d^4 e^2+33 a^2 c d^2 e^4-21 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x^3}+\frac {\left (35 c^4 d^8-16 a c^3 d^6 e^2-18 a^2 c^2 d^4 e^4+168 a^3 c d^2 e^6-105 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 d^4 e^3 \left (c d^2-a e^2\right )^3 x^2}-\frac {\left (105 c^5 d^{10}-55 a c^4 d^8 e^2-54 a^2 c^3 d^6 e^4-78 a^3 c^2 d^4 e^6+525 a^4 c d^2 e^8-315 a^5 e^{10}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d^5 e^4 \left (c d^2-a e^2\right )^3 x}+\frac {5 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\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 )}{16 a^{9/2} d^{11/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.39, size = 593, normalized size = 0.89 \begin {gather*} \frac {(a e+c d x) \left (24 a^{7/2} d^{9/2} e^{7/2} \left (a e^2-c d^2\right )^3+x \left (6 a^{5/2} d^{7/2} e^{5/2} \left (c d^2-a e^2\right )^3 \left (9 a e^2+7 c d^2\right )+3 a^{3/2} d^{5/2} e^{3/2} x \left (a e^2-c d^2\right )^3 \left (63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right )+x^2 \left (9 \sqrt {a} c d^{7/2} \sqrt {e} \left (c d^2-a e^2\right )^2 \left (21 a^3 e^6+3 a^2 c d^2 e^4-5 a c^2 d^4 e^2-35 c^3 d^6\right )+3 \sqrt {a} d^{3/2} \sqrt {e} \left (a e^2-c d^2\right ) \left (105 a^4 e^9-84 a^3 c d^2 e^7-42 a^2 c^2 d^4 e^5-20 a c^3 d^6 e^3+105 c^4 d^8 e\right ) (a e+c d x)+(d+e x) \sqrt {a e+c d x} \left (45 \sqrt {d+e x} \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+3 \sqrt {a} \sqrt {d} \sqrt {e} \left (315 a^5 e^{11}-525 a^4 c d^2 e^9+78 a^3 c^2 d^4 e^7+54 a^2 c^3 d^6 e^5+55 a c^4 d^8 e^3-105 c^5 d^{10} e\right ) \sqrt {a e+c d x}\right )\right )\right )\right )}{72 a^{9/2} d^{11/2} e^{9/2} x^3 \left (c d^2-a e^2\right )^3 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.13, size = 711, normalized size = 1.07 \begin {gather*} \frac {\sqrt {a d e+a e^2 x+c d^2 x+c d e x^2} \left (-8 a^6 d^4 e^9+18 a^6 d^3 e^{10} x-63 a^6 d^2 e^{11} x^2-420 a^6 d e^{12} x^3-315 a^6 e^{13} x^4+24 a^5 c d^6 e^7-40 a^5 c d^5 e^8 x+135 a^5 c d^4 e^9 x^2+651 a^5 c d^3 e^{10} x^3+105 a^5 c d^2 e^{11} x^4-315 a^5 c d e^{12} x^5-24 a^4 c^2 d^8 e^5+12 a^4 c^2 d^7 e^6 x-62 a^4 c^2 d^6 e^7 x^2-3 a^4 c^2 d^5 e^8 x^3+636 a^4 c^2 d^4 e^9 x^4+525 a^4 c^2 d^3 e^{10} x^5+8 a^3 c^3 d^{10} e^3+24 a^3 c^3 d^9 e^4 x+6 a^3 c^3 d^8 e^5 x^2-106 a^3 c^3 d^7 e^6 x^3-174 a^3 c^3 d^6 e^7 x^4-78 a^3 c^3 d^5 e^8 x^5-14 a^2 c^4 d^{11} e^2 x-51 a^2 c^4 d^{10} e^3 x^2-114 a^2 c^4 d^9 e^4 x^3-131 a^2 c^4 d^8 e^5 x^4-54 a^2 c^4 d^7 e^6 x^5+35 a c^5 d^{12} e x^2+15 a c^5 d^{11} e^2 x^3-75 a c^5 d^{10} e^3 x^4-55 a c^5 d^9 e^4 x^5+105 c^6 d^{13} x^3+210 c^6 d^{12} e x^4+105 c^6 d^{11} e^2 x^5\right )}{24 a^4 d^5 e^4 x^3 (d+e x)^2 \left (a e^2-c d^2\right )^3 (a e+c d x)}-\frac {5 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac {x \sqrt {c d e}-\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {a} \sqrt {d} \sqrt {e}}\right )}{8 a^{9/2} d^{11/2} e^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 93.16, size = 2526, normalized size = 3.80
result too large to display
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1705, normalized size = 2.57
result 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 \begin {gather*} \int \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{4}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^4\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{4} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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