Optimal. Leaf size=242 \[ \frac{8 b^2 x^7 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac{x \left (80 A b^2-3 a (8 b B-a C)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.321803, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1803, 1813, 12, 271, 264} \[ \frac{8 b^2 x^7 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{7/2}}+\frac{4 b x^5 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (160 A b^3-a \left (a^2 (-D)-6 a b C+48 b^2 B\right )\right )}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac{x \left (80 A b^2-3 a (8 b B-a C)\right )}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 1803
Rule 1813
Rule 12
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{10 A b-3 a \left (B+C x^2+D x^4\right )}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{8 b (10 A b-3 a B)-a \left (-3 a C-3 a D x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^2}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{\int \frac{\left (6 b \left (80 A b^2-24 a b B+3 a^2 C\right )+3 a^3 D\right ) x^2}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) \int \frac{x^2}{\left (a+b x^2\right )^{9/2}} \, dx}{a^3}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac{\left (4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right )\right ) \int \frac{x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^4}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac{4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^5}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac{\left (8 b^2 \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right )\right ) \int \frac{x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^5}\\ &=-\frac{A}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac{10 A b-3 a B}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac{\left (80 A b^2-3 a (8 b B-a C)\right ) x}{3 a^3 \left (a+b x^2\right )^{7/2}}+\frac{\left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^3}{3 a^4 \left (a+b x^2\right )^{7/2}}+\frac{4 b \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^5}{15 a^5 \left (a+b x^2\right )^{7/2}}+\frac{8 b^2 \left (160 A b^3-a \left (48 b^2 B-6 a b C-a^2 D\right )\right ) x^7}{105 a^6 \left (a+b x^2\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.129439, size = 165, normalized size = 0.68 \[ \frac{8 a^3 b^2 x^4 \left (350 A-210 B x^2+21 C x^4+D x^6\right )+16 a^2 b^3 x^6 \left (350 A-84 B x^2+3 C x^4\right )+14 a^4 b x^2 \left (25 A-60 B x^2+15 C x^4+2 D x^6\right )-35 a^5 \left (A+3 B x^2-3 C x^4-D x^6\right )+128 a b^4 x^8 \left (35 A-3 B x^2\right )+1280 A b^5 x^{10}}{105 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 205, normalized size = 0.9 \begin{align*} -{\frac{-1280\,A{b}^{5}{x}^{10}+384\,Ba{b}^{4}{x}^{10}-48\,C{a}^{2}{b}^{3}{x}^{10}-8\,D{a}^{3}{b}^{2}{x}^{10}-4480\,Aa{b}^{4}{x}^{8}+1344\,B{a}^{2}{b}^{3}{x}^{8}-168\,C{a}^{3}{b}^{2}{x}^{8}-28\,D{a}^{4}b{x}^{8}-5600\,A{a}^{2}{b}^{3}{x}^{6}+1680\,B{a}^{3}{b}^{2}{x}^{6}-210\,C{a}^{4}b{x}^{6}-35\,D{a}^{5}{x}^{6}-2800\,A{a}^{3}{b}^{2}{x}^{4}+840\,B{a}^{4}b{x}^{4}-105\,C{a}^{5}{x}^{4}-350\,A{a}^{4}b{x}^{2}+105\,B{a}^{5}{x}^{2}+35\,A{a}^{5}}{105\,{x}^{3}{a}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 = 1.21572, size = 471, normalized size = 1.95 \begin{align*} \frac{{\left ({\left (x^{2}{\left (\frac{{\left (8 \, D a^{15} b^{5} + 48 \, C a^{14} b^{6} - 279 \, B a^{13} b^{7} + 790 \, A a^{12} b^{8}\right )} x^{2}}{a^{18} b^{3}} + \frac{7 \,{\left (4 \, D a^{16} b^{4} + 24 \, C a^{15} b^{5} - 132 \, B a^{14} b^{6} + 365 \, A a^{13} b^{7}\right )}}{a^{18} b^{3}}\right )} + \frac{35 \,{\left (D a^{17} b^{3} + 6 \, C a^{16} b^{4} - 30 \, B a^{15} b^{5} + 80 \, A a^{14} b^{6}\right )}}{a^{18} b^{3}}\right )} x^{2} + \frac{105 \,{\left (C a^{17} b^{3} - 4 \, B a^{16} b^{4} + 10 \, A a^{15} b^{5}\right )}}{a^{18} b^{3}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} \sqrt{b} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{3}{2}} + 3 \, B a^{3} \sqrt{b} - 14 \, A a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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