Optimal. Leaf size=218 \[ -\frac{7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac{7 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{11/2}}-\frac{x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac{7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c} \]
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Rubi [A] time = 0.381084, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2034, 794, 670, 640, 612, 620, 206} \[ -\frac{7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac{7 b^3 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{11/2}}-\frac{x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac{7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 794
Rule 670
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x^7 \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (A+B x) \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac{\left (3 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int x^3 \sqrt{b x+c x^2} \, dx,x,x^2\right )}{12 c}\\ &=-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac{(7 b (3 b B-4 A c)) \operatorname{Subst}\left (\int x^2 \sqrt{b x+c x^2} \, dx,x,x^2\right )}{80 c^2}\\ &=\frac{7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac{\left (7 b^2 (3 b B-4 A c)\right ) \operatorname{Subst}\left (\int x \sqrt{b x+c x^2} \, dx,x,x^2\right )}{128 c^3}\\ &=-\frac{7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac{7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac{\left (7 b^3 (3 b B-4 A c)\right ) \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )}{256 c^4}\\ &=\frac{7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{1024 c^5}-\frac{7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac{7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac{\left (7 b^5 (3 b B-4 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{2048 c^5}\\ &=\frac{7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{1024 c^5}-\frac{7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac{7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac{\left (7 b^5 (3 b B-4 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^5}\\ &=\frac{7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{1024 c^5}-\frac{7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac{7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac{(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac{B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{1024 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.293925, size = 193, normalized size = 0.89 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left (-16 b^2 c^3 x^4 \left (14 A+9 B x^2\right )+56 b^3 c^2 x^2 \left (5 A+3 B x^2\right )-210 b^4 c \left (2 A+B x^2\right )+64 b c^4 x^6 \left (3 A+2 B x^2\right )+256 c^5 x^8 \left (6 A+5 B x^2\right )+315 b^5 B\right )-105 b^{9/2} (3 b B-4 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{15360 c^{11/2} x \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 290, normalized size = 1.3 \begin{align*}{\frac{1}{15360\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 1280\,B \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{9/2}{x}^{9}+1536\,A \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{9/2}{x}^{7}-1152\,B \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{7/2}{x}^{7}b-1344\,A \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{7/2}{x}^{5}b+1008\,B \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{5/2}{x}^{5}{b}^{2}+1120\,A \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{5/2}{x}^{3}{b}^{2}-840\,B \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{3/2}{x}^{3}{b}^{3}-840\,A \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{3/2}x{b}^{3}+630\,B \left ( c{x}^{2}+b \right ) ^{3/2}\sqrt{c}x{b}^{4}+420\,A\sqrt{c{x}^{2}+b}{c}^{3/2}x{b}^{4}-315\,B\sqrt{c{x}^{2}+b}\sqrt{c}x{b}^{5}+420\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{5}c-315\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{6} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.69133, size = 844, normalized size = 3.87 \begin{align*} \left [-\frac{105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \,{\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \,{\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \,{\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{30720 \, c^{6}}, \frac{105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (1280 \, B c^{6} x^{10} + 128 \,{\left (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \,{\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \,{\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \,{\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15360 \, c^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{7} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17702, size = 331, normalized size = 1.52 \begin{align*} \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B x^{2} \mathrm{sgn}\left (x\right ) + \frac{B b c^{9} \mathrm{sgn}\left (x\right ) + 12 \, A c^{10} \mathrm{sgn}\left (x\right )}{c^{10}}\right )} x^{2} - \frac{3 \,{\left (3 \, B b^{2} c^{8} \mathrm{sgn}\left (x\right ) - 4 \, A b c^{9} \mathrm{sgn}\left (x\right )\right )}}{c^{10}}\right )} x^{2} + \frac{7 \,{\left (3 \, B b^{3} c^{7} \mathrm{sgn}\left (x\right ) - 4 \, A b^{2} c^{8} \mathrm{sgn}\left (x\right )\right )}}{c^{10}}\right )} x^{2} - \frac{35 \,{\left (3 \, B b^{4} c^{6} \mathrm{sgn}\left (x\right ) - 4 \, A b^{3} c^{7} \mathrm{sgn}\left (x\right )\right )}}{c^{10}}\right )} x^{2} + \frac{105 \,{\left (3 \, B b^{5} c^{5} \mathrm{sgn}\left (x\right ) - 4 \, A b^{4} c^{6} \mathrm{sgn}\left (x\right )\right )}}{c^{10}}\right )} \sqrt{c x^{2} + b} x + \frac{7 \,{\left (3 \, B b^{6} \mathrm{sgn}\left (x\right ) - 4 \, A b^{5} c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{1024 \, c^{\frac{11}{2}}} - \frac{7 \,{\left (3 \, B b^{6} \log \left ({\left | b \right |}\right ) - 4 \, A b^{5} c \log \left ({\left | b \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{2048 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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