Optimal. Leaf size=140 \[ -\frac{c^2 \sqrt{a+c x^2} (5 A+16 B x)}{16 x^2}-\frac{\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac{c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac{5 A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}+B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
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Rubi [A] time = 0.355361, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{c^2 \sqrt{a+c x^2} (5 A+16 B x)}{16 x^2}-\frac{\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac{c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac{5 A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}+B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(5/2))/x^7,x]
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Rubi in Sympy [A] time = 48.4669, size = 131, normalized size = 0.94 \[ - \frac{5 A c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} + B c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{c^{2} \left (60 A + 192 B x\right ) \sqrt{a + c x^{2}}}{192 x^{2}} - \frac{c \left (30 A + 48 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{144 x^{4}} - \frac{\left (5 A + 6 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{30 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.524852, size = 140, normalized size = 1. \[ -\frac{\sqrt{a+c x^2} \left (8 a^2 (5 A+6 B x)+2 a c x^2 (65 A+88 B x)+c^2 x^4 (165 A+368 B x)\right )}{240 x^6}-\frac{5 A c^3 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{16 \sqrt{a}}+\frac{5 A c^3 \log (x)}{16 \sqrt{a}}+B c^{5/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^7,x]
[Out]
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Maple [B] time = 0.02, size = 281, normalized size = 2. \[ -{\frac{A}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Ac}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{c}^{2}}{16\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{3}}{16\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{c}^{3}}{48\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,A{c}^{3}}{16\,a}\sqrt{c{x}^{2}+a}}-{\frac{B}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Bc}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,B{c}^{2}}{15\,{a}^{3}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,B{c}^{3}x}{15\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,B{c}^{3}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{3}x}{a}\sqrt{c{x}^{2}+a}}+B{c}^{{\frac{5}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(5/2)/x^7,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.352274, size = 1, normalized size = 0.01 \[ \left [\frac{240 \, B \sqrt{a} c^{\frac{5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 75 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, \sqrt{a} x^{6}}, \frac{480 \, B \sqrt{a} \sqrt{-c} c^{2} x^{6} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 75 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, \sqrt{a} x^{6}}, -\frac{75 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - 120 \, B \sqrt{-a} c^{\frac{5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) +{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} x^{6}}, \frac{240 \, B \sqrt{-a} \sqrt{-c} c^{2} x^{6} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 75 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^7,x, algorithm="fricas")
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Sympy [A] time = 47.1005, size = 299, normalized size = 2.14 \[ - \frac{A a^{3}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 A a^{2} \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{35 A a c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{3 A c^{\frac{5}{2}}}{16 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 \sqrt{a}} - \frac{B \sqrt{a} c^{2}}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{11 B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{8 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15} + B c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{B c^{3} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(5/2)/x**7,x)
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GIAC/XCAS [A] time = 0.295582, size = 536, normalized size = 3.83 \[ \frac{5 \, A c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a}} - B c^{\frac{5}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A c^{3} + 720 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a c^{\frac{5}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a c^{3} - 2160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{2} c^{\frac{5}{2}} + 450 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 3680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{3} c^{\frac{5}{2}} + 450 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 3360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{4} c^{\frac{5}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 1488 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{5} c^{\frac{5}{2}} + 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{5} c^{3} - 368 \, B a^{6} c^{\frac{5}{2}}}{120 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^7,x, algorithm="giac")
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