Optimal. Leaf size=119 \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.298727, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]
[Out]
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Rubi in Sympy [A] time = 17.8514, size = 112, normalized size = 0.94 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{7 b x^{7}} + 2 B c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 B c^{2} \sqrt{b x + c x^{2}}}{x} - \frac{2 B c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{2 B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.348266, size = 101, normalized size = 0.85 \[ \frac{2 \sqrt{x (b+c x)} \left (-\frac{15 A (b+c x)^3}{b}-7 B x \left (3 b^2+11 b c x+23 c^2 x^2\right )+\frac{105 B c^{5/2} x^{7/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{b+c x}}\right )}{105 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^7,x]
[Out]
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Maple [B] time = 0.015, size = 263, normalized size = 2.2 \[ -{\frac{2\,A}{7\,b{x}^{7}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,B}{5\,b{x}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Bc}{15\,{b}^{2}{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{16\,B{c}^{2}}{15\,{b}^{3}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{32\,B{c}^{3}}{5\,{b}^{4}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{256\,B{c}^{4}}{15\,{b}^{5}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{256\,B{c}^{5}}{15\,{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{32\,B{c}^{5}x}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{16\,B{c}^{4}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{B{c}^{4}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{B{c}^{3}\sqrt{c{x}^{2}+bx}}{b}}+B{c}^{{\frac{5}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(5/2)/x^7,x)
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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 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.293969, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B b c^{\frac{5}{2}} x^{4} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, b x^{4}}, \frac{2 \,{\left (105 \, B b \sqrt{-c} c^{2} x^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) -{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}\right )}}{105 \, b x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.296678, size = 527, normalized size = 4.43 \[ -B c^{\frac{5}{2}}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A c^{\frac{7}{2}} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{2} c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b c^{3} + 245 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{3} c^{\frac{3}{2}} + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{4} c + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{3} c^{2} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{5} \sqrt{c} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{4} c^{\frac{3}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{5} c + 15 \, A b^{6} \sqrt{c}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^7,x, algorithm="giac")
[Out]