Optimal. Leaf size=187 \[ -\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}+\frac{55 e^9 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.260319, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1807, 807, 266, 47, 63, 208} \[ -\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}+\frac{55 e^9 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1807
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-27 d^4 e-29 d^3 e^2 x-9 d^2 e^3 x^2\right )}{x^9} \, dx}{9 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+\frac{\int \frac{\left (232 d^5 e^2+99 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{72 d^4}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{1}{8} \left (11 e^3\right ) \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{1}{16} \left (11 e^3\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac{1}{96} \left (55 e^5\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{1}{128} \left (55 e^7\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac{1}{256} \left (55 e^9\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{1}{128} \left (55 e^7\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )\\ &=-\frac{55 e^7 \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac{55 e^9 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d}\\ \end{align*}
Mathematica [C] time = 0.183328, size = 218, normalized size = 1.17 \[ \frac{-16 d^8 e^2 x^2-168 d^7 e^3 x^3+1184 d^6 e^4 x^4+714 d^5 e^5 x^5-2336 d^4 e^6 x^6-1239 d^3 e^7 x^7+1744 d^2 e^8 x^8+315 d e^9 x^9 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )-112 d^{10}+693 d e^9 x^9-464 e^{10} x^{10}}{1008 d x^9 \sqrt{d^2-e^2 x^2}}-\frac{3 e^9 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )}{7 d^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.253, size = 250, normalized size = 1.3 \begin{align*} -{\frac{3\,e}{8\,{x}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{3}}{48\,{d}^{2}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{e}^{5}}{192\,{d}^{4}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{7}}{128\,{d}^{6}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{9}}{128\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{55\,{e}^{9}}{384\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{55\,{e}^{9}}{128\,{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{55\,{e}^{9}}{128}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{29\,{e}^{2}}{63\,d{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{d}{9\,{x}^{9}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.41347, size = 328, normalized size = 1.75 \begin{align*} -\frac{3465 \, e^{9} x^{9} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (3712 \, e^{8} x^{8} - 4599 \, d e^{7} x^{7} - 10240 \, d^{2} e^{6} x^{6} - 3066 \, d^{3} e^{5} x^{5} + 8448 \, d^{4} e^{4} x^{4} + 7224 \, d^{5} e^{3} x^{3} - 1024 \, d^{6} e^{2} x^{2} - 3024 \, d^{7} e x - 896 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8064 \, d x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 34.8065, size = 1914, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26285, size = 837, normalized size = 4.48 \begin{align*} \frac{x^{9}{\left (\frac{189 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{18}}{x} + \frac{324 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{16}}{x^{2}} - \frac{672 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{14}}{x^{3}} - \frac{3024 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{12}}{x^{4}} - \frac{1512 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{10}}{x^{5}} + \frac{9744 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{8}}{x^{6}} + \frac{18144 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{6}}{x^{7}} - \frac{16632 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{4}}{x^{8}} + 28 \, e^{20}\right )} e^{7}}{129024 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{9} d} + \frac{55 \, e^{9} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{128 \, d} + \frac{{\left (\frac{16632 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{8} e^{106}}{x} - \frac{18144 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8} e^{104}}{x^{2}} - \frac{9744 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8} e^{102}}{x^{3}} + \frac{1512 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{8} e^{100}}{x^{4}} + \frac{3024 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{8} e^{98}}{x^{5}} + \frac{672 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{8} e^{96}}{x^{6}} - \frac{324 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{8} e^{94}}{x^{7}} - \frac{189 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{8} e^{92}}{x^{8}} - \frac{28 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{8} e^{90}}{x^{9}}\right )} e^{\left (-99\right )}}{129024 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]