Optimal. Leaf size=188 \[ \frac{35 B-x \left (\frac{93 A b}{a}-16 C\right )}{35 a^4 \sqrt{a+b x^2}}+\frac{35 B-3 x \left (\frac{29 A b}{a}-8 C\right )}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{7 B-x \left (\frac{13 A b}{a}-6 C\right )}{35 a^2 \left (a+b x^2\right )^{5/2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{B-x \left (\frac{A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.381314, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1805, 807, 266, 63, 208} \[ \frac{35 B-x \left (\frac{93 A b}{a}-16 C\right )}{35 a^4 \sqrt{a+b x^2}}+\frac{35 B-3 x \left (\frac{29 A b}{a}-8 C\right )}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{7 B-x \left (\frac{13 A b}{a}-6 C\right )}{35 a^2 \left (a+b x^2\right )^{5/2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{B-x \left (\frac{A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{-7 A-7 B x+6 \left (\frac{A b}{a}-C\right ) x^2}{x^2 \left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{35 A+35 B x-4 \left (\frac{13 A b}{a}-6 C\right ) x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}-\frac{\int \frac{-105 A-105 B x+6 \left (\frac{29 A b}{a}-8 C\right ) x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{105 a^3}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{35 B-\left (\frac{93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt{a+b x^2}}+\frac{\int \frac{105 A+105 B x}{x^2 \sqrt{a+b x^2}} \, dx}{105 a^4}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{35 B-\left (\frac{93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}+\frac{B \int \frac{1}{x \sqrt{a+b x^2}} \, dx}{a^4}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{35 B-\left (\frac{93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}+\frac{B \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a^4}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{35 B-\left (\frac{93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}+\frac{B \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{a^4 b}\\ &=\frac{B-\left (\frac{A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac{7 B-\left (\frac{13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac{35 B-3 \left (\frac{29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac{35 B-\left (\frac{93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt{a+b x^2}}-\frac{A \sqrt{a+b x^2}}{a^5 x}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.15615, size = 158, normalized size = 0.84 \[ \frac{14 a^2 b^2 x^4 (x (25 B+12 C x)-120 A)+14 a^3 b x^2 (x (29 B+15 C x)-60 A)+a^4 (x (176 B+105 C x)-105 A)+3 a b^3 x^6 (x (35 B+16 C x)-448 A)-105 \sqrt{a} B x \left (a+b x^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-384 A b^4 x^8}{105 a^5 x \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 240, normalized size = 1.3 \begin{align*}{\frac{Cx}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{6\,Cx}{35\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,Cx}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,Cx}{35\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{B}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{B}{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{B}{3\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{B}{{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}-{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{8\,Abx}{7\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{48\,Abx}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{64\,Abx}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{128\,Abx}{35\,{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 = 1.88288, size = 1172, normalized size = 6.23 \begin{align*} \left [\frac{105 \,{\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \,{\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \,{\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \,{\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \,{\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \,{\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, \frac{105 \,{\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \,{\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \,{\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \,{\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \,{\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17356, size = 323, normalized size = 1.72 \begin{align*} \frac{{\left ({\left ({\left ({\left (3 \,{\left (x{\left (\frac{35 \, B b^{3}}{a^{4}} + \frac{{\left (16 \, C a^{20} b^{6} - 93 \, A a^{19} b^{7}\right )} x}{a^{24} b^{3}}\right )} + \frac{28 \,{\left (2 \, C a^{21} b^{5} - 11 \, A a^{20} b^{6}\right )}}{a^{24} b^{3}}\right )} x + \frac{350 \, B b^{2}}{a^{3}}\right )} x + \frac{210 \,{\left (C a^{22} b^{4} - 5 \, A a^{21} b^{5}\right )}}{a^{24} b^{3}}\right )} x + \frac{406 \, B b}{a^{2}}\right )} x + \frac{105 \,{\left (C a^{23} b^{3} - 4 \, A a^{22} b^{4}\right )}}{a^{24} b^{3}}\right )} x + \frac{176 \, B}{a}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \, B \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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