Optimal. Leaf size=281 \[ -\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {16 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^7 \sqrt {a+b x^2}}-\frac {8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {2 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 275, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1803, 12, 192, 191} \begin {gather*} -\frac {16 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{105 a^7 \sqrt {a+b x^2}}-\frac {8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {2 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 1803
Rubi steps
\begin {align*} \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {12 A b-5 a \left (B+C x^2+D x^4\right )}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{5 a}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {10 b (12 A b-5 a B)-3 a \left (-5 a C-5 a D x^2\right )}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^2}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {8 b \left (120 A b^2-50 a b B+15 a^2 C\right )-15 a^3 D}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^3}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {\left (2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^4}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {\left (8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^5}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^6}\\ &=-\frac {A}{5 a x^5 \left (a+b x^2\right )^{7/2}}+\frac {12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac {24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}-\frac {\left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac {2 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac {8 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac {16 \left (192 A b^3-8 a b (10 b B-3 a C)-3 a^3 D\right ) x}{105 a^7 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 202, normalized size = 0.72 \begin {gather*} \frac {-7 a^6 \left (3 A+5 x^2 \left (B+3 C x^2-3 D x^4\right )\right )+14 a^5 b x^2 \left (6 A+25 B x^2-60 C x^4+15 D x^6\right )+56 a^4 b^2 x^4 \left (-15 A+50 B x^2-30 C x^4+3 D x^6\right )+16 a^3 b^3 x^6 \left (-420 A+350 B x^2-84 C x^4+3 D x^6\right )-128 a^2 b^4 x^8 \left (105 A-35 B x^2+3 C x^4\right )+256 a b^5 x^{10} \left (5 B x^2-42 A\right )-3072 A b^6 x^{12}}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 256, normalized size = 0.91 \begin {gather*} \frac {-21 a^6 A-35 a^6 B x^2-105 a^6 C x^4+105 a^6 D x^6+84 a^5 A b x^2+350 a^5 b B x^4-840 a^5 b C x^6+210 a^5 b D x^8-840 a^4 A b^2 x^4+2800 a^4 b^2 B x^6-1680 a^4 b^2 C x^8+168 a^4 b^2 D x^{10}-6720 a^3 A b^3 x^6+5600 a^3 b^3 B x^8-1344 a^3 b^3 C x^{10}+48 a^3 b^3 D x^{12}-13440 a^2 A b^4 x^8+4480 a^2 b^4 B x^{10}-384 a^2 b^4 C x^{12}-10752 a A b^5 x^{10}+1280 a b^5 B x^{12}-3072 A b^6 x^{12}}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.83, size = 270, normalized size = 0.96 \begin {gather*} \frac {{\left (16 \, {\left (3 \, D a^{3} b^{3} - 24 \, C a^{2} b^{4} + 80 \, B a b^{5} - 192 \, A b^{6}\right )} x^{12} + 56 \, {\left (3 \, D a^{4} b^{2} - 24 \, C a^{3} b^{3} + 80 \, B a^{2} b^{4} - 192 \, A a b^{5}\right )} x^{10} + 70 \, {\left (3 \, D a^{5} b - 24 \, C a^{4} b^{2} + 80 \, B a^{3} b^{3} - 192 \, A a^{2} b^{4}\right )} x^{8} - 21 \, A a^{6} + 35 \, {\left (3 \, D a^{6} - 24 \, C a^{5} b + 80 \, B a^{4} b^{2} - 192 \, A a^{3} b^{3}\right )} x^{6} - 35 \, {\left (3 \, C a^{6} - 10 \, B a^{5} b + 24 \, A a^{4} b^{2}\right )} x^{4} - 7 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.63, size = 592, normalized size = 2.11 \begin {gather*} \frac {{\left ({\left (x^{2} {\left (\frac {{\left (48 \, D a^{18} b^{6} - 279 \, C a^{17} b^{7} + 790 \, B a^{16} b^{8} - 1686 \, A a^{15} b^{9}\right )} x^{2}}{a^{22} b^{3}} + \frac {7 \, {\left (24 \, D a^{19} b^{5} - 132 \, C a^{18} b^{6} + 365 \, B a^{17} b^{7} - 768 \, A a^{16} b^{8}\right )}}{a^{22} b^{3}}\right )} + \frac {35 \, {\left (6 \, D a^{20} b^{4} - 30 \, C a^{19} b^{5} + 80 \, B a^{18} b^{6} - 165 \, A a^{17} b^{7}\right )}}{a^{22} b^{3}}\right )} x^{2} + \frac {105 \, {\left (D a^{21} b^{3} - 4 \, C a^{20} b^{4} + 10 \, B a^{19} b^{5} - 20 \, A a^{18} b^{6}\right )}}{a^{22} b^{3}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} C a^{2} \sqrt {b} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{3} \sqrt {b} + 270 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} - 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{4} \sqrt {b} - 430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} + 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{5} \sqrt {b} + 290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} - 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 15 \, C a^{6} \sqrt {b} - 70 \, B a^{5} b^{\frac {3}{2}} + 198 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 253, normalized size = 0.90 \begin {gather*} -\frac {3072 A \,b^{6} x^{12}-1280 B a \,b^{5} x^{12}+384 C \,a^{2} b^{4} x^{12}-48 D a^{3} b^{3} x^{12}+10752 A a \,b^{5} x^{10}-4480 B \,a^{2} b^{4} x^{10}+1344 C \,a^{3} b^{3} x^{10}-168 D a^{4} b^{2} x^{10}+13440 A \,a^{2} b^{4} x^{8}-5600 B \,a^{3} b^{3} x^{8}+1680 C \,a^{4} b^{2} x^{8}-210 D a^{5} b \,x^{8}+6720 A \,a^{3} b^{3} x^{6}-2800 B \,a^{4} b^{2} x^{6}+840 C \,a^{5} b \,x^{6}-105 D a^{6} x^{6}+840 A \,a^{4} b^{2} x^{4}-350 B \,a^{5} b \,x^{4}+105 C \,a^{6} x^{4}-84 A \,a^{5} b \,x^{2}+35 B \,a^{6} x^{2}+21 A \,a^{6}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{7} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 398, normalized size = 1.42 \begin {gather*} \frac {16 \, D x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, D x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {D x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, C b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, C b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, C b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} + \frac {256 \, B b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, B b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, B b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, A b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, A b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, A b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} - \frac {C}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} + \frac {10 \, B b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, A b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, A b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 405, normalized size = 1.44 \begin {gather*} \frac {\frac {61\,A\,b}{35\,a^3}+\frac {78\,A\,b^2\,x^2}{35\,a^4}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {\frac {128\,B\,b}{21\,a^5}+\frac {256\,B\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}+\frac {x\,D}{{\left (b\,x^2+a\right )}^{9/2}}-\frac {\frac {B}{3\,a^2}+\frac {19\,B\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {C}{a^4}+\frac {128\,C\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {512\,A\,b^2}{35\,a^6}+\frac {1024\,A\,b^3\,x^2}{35\,a^7}}{x\,\sqrt {b\,x^2+a}}-\frac {A\,\sqrt {b\,x^2+a}}{5\,a^5\,x^5}+\frac {18\,b^2\,x^5\,D}{5\,a^2\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {72\,b^3\,x^7\,D}{35\,a^3\,{\left (b\,x^2+a\right )}^{9/2}}+\frac {16\,b^4\,x^9\,D}{35\,a^4\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {A\,b}{7\,a^2\,x^3\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {32\,B\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {B\,b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {27\,A\,b^2}{7\,a^5\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {3\,b\,x^3\,D}{a\,{\left (b\,x^2+a\right )}^{9/2}}-\frac {29\,C\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,C\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {C\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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