Optimal. Leaf size=628 \[ -\frac{3 \left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 a^5 d^6 e^5 x^2}+\frac{\left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (15 a^2 c d^2 e^4-231 a^3 e^6+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{\left (-33 a^2 e^4+10 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 a^2 d^3 e^2 x^6}+\frac{3 \left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 a^{11/2} d^{13/2} e^{11/2}}-\frac{\left (\frac{5 c}{a e}-\frac{11 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 x^7}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8} \]
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Rubi [A] time = 0.885273, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {849, 834, 806, 720, 724, 206} \[ -\frac{3 \left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 a^5 d^6 e^5 x^2}+\frac{\left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (15 a^2 c d^2 e^4-231 a^3 e^6+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{\left (-33 a^2 e^4+10 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 a^2 d^3 e^2 x^6}+\frac{3 \left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 a^{11/2} d^{13/2} e^{11/2}}-\frac{\left (\frac{5 c}{a e}-\frac{11 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 x^7}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8} \]
Antiderivative was successfully verified.
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Rule 849
Rule 834
Rule 806
Rule 720
Rule 724
Rule 206
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^9 (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^9} \, dx\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\int \frac{\left (-\frac{1}{2} a e \left (5 c d^2-11 a e^2\right )+3 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^8} \, dx}{8 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}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\int \frac{\left (-\frac{3}{4} a e \left (15 c^2 d^4+10 a c d^2 e^2-33 a^2 e^4\right )-a c d e^2 \left (5 c d^2-11 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^7} \, dx}{56 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}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\int \frac{\left (-\frac{3}{8} a e \left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right )-\frac{3}{4} a c d e^2 \left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{x^6} \, dx}{336 a^3 d^3 e^3}\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}-\frac{\left (\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}{256 a^3 d^4 e^3}\\ &=\frac{\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{\left (3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}{4096 a^4 d^5 e^4}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac{\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}-\frac{\left (3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 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}{32768 a^5 d^6 e^5}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac{\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{\left (3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 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 )}{16384 a^5 d^6 e^5}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac{\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac{\left (\frac{5 c}{a e}-\frac{11 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}}{112 x^7}+\frac{\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac{\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac{3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 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 )}{32768 a^{11/2} d^{13/2} e^{11/2}}\\ \end{align*}
Mathematica [A] time = 1.50025, size = 512, normalized size = 0.82 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (-\frac{(d+e x) \left (33 a^2 e^4+34 a c d^2 e^2+21 c^2 d^4\right ) (a e+c d x)^2}{56 a^2 d^2 e^2 x^6}+\frac{\left (45 a^2 c d^2 e^4+33 a^3 e^6+35 a c^2 d^4 e^2+15 c^3 d^6\right ) \left (5 x \left (c d^2-a e^2\right ) \left (x \left (c d^2-a e^2\right ) \left (8 a^{5/2} \sqrt{d} e^{5/2} (d+e x)^{5/2} \sqrt{a e+c d x}+x \left (c d^2-a e^2\right ) \left (3 x^2 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a e (2 d+5 e x)-3 c d^2 x\right )\right )\right )+16 a^{5/2} d^{3/2} e^{5/2} (d+e x)^{5/2} (a e+c d x)^{3/2}\right )+128 a^{5/2} d^{5/2} e^{5/2} (d+e x)^{5/2} (a e+c d x)^{5/2}\right )}{10240 a^{9/2} d^{11/2} e^{9/2} x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac{(d+e x) \left (11 a e^2+9 c d^2\right ) (a e+c d x)^2}{14 a d e x^7}-\frac{(d+e x) (a e+c d x)^2}{x^8}\right )}{8 a d e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.201, size = 6030, normalized size = 9.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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