Optimal. Leaf size=170 \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^{10}}-\frac{8 c^2 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^{12}}+\frac{2 c \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^{14}}-\frac{\left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^{16}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]
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Rubi [A] time = 0.321332, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^{10}}-\frac{8 c^2 \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^{12}}+\frac{2 c \left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^{14}}-\frac{\left (b x^2+c x^4\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^{16}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{17}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^9} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}+\frac{\left (-9 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^8} \, dx,x,x^2\right )}{13 b}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac{(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}-\frac{(3 c (13 b B-8 A c)) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )}{143 b^2}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac{(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac{2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}+\frac{\left (4 c^2 (13 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{429 b^3}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac{(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac{2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}-\frac{8 c^2 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}-\frac{\left (8 c^3 (13 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{3003 b^4}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}}-\frac{(13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}+\frac{2 c (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}-\frac{8 c^2 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}+\frac{16 c^3 (13 b B-8 A c) \left (b x^2+c x^4\right )^{5/2}}{15015 b^5 x^{10}}\\ \end{align*}
Mathematica [A] time = 0.0704421, size = 95, normalized size = 0.56 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\left (\frac{c x^3}{b}+x\right )^2 \left (-70 b^2 c x^2+105 b^3+40 b c^2 x^4-16 c^3 x^6\right ) (8 A c-13 b B)-1155 A b^2 \left (b+c x^2\right )^2\right )}{15015 b^3 x^{14}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 118, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,A{c}^{4}{x}^{8}-208\,Bb{c}^{3}{x}^{8}-320\,Ab{c}^{3}{x}^{6}+520\,B{b}^{2}{c}^{2}{x}^{6}+560\,A{b}^{2}{c}^{2}{x}^{4}-910\,B{b}^{3}c{x}^{4}-840\,A{b}^{3}c{x}^{2}+1365\,B{b}^{4}{x}^{2}+1155\,A{b}^{4} \right ) }{15015\,{x}^{16}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{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 = 1.81635, size = 356, normalized size = 2.09 \begin{align*} \frac{{\left (16 \,{\left (13 \, B b c^{5} - 8 \, A c^{6}\right )} x^{12} - 8 \,{\left (13 \, B b^{2} c^{4} - 8 \, A b c^{5}\right )} x^{10} + 6 \,{\left (13 \, B b^{3} c^{3} - 8 \, A b^{2} c^{4}\right )} x^{8} - 1155 \, A b^{6} - 5 \,{\left (13 \, B b^{4} c^{2} - 8 \, A b^{3} c^{3}\right )} x^{6} - 35 \,{\left (52 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} - 105 \,{\left (13 \, B b^{6} + 14 \, A b^{5} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15015 \, b^{5} x^{14}} \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 \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{17}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 5.15756, size = 743, normalized size = 4.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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