Optimal. Leaf size=99 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac {\left (c+d x^2\right )^{3/2} \left (35 b^2 c^2-4 a d (7 b c-2 a d)\right )}{105 c^3 x^3}-\frac {2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {462, 453, 264} \[ -\frac {\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-28 a b c d+35 b^2 c^2\right )}{105 c^3 x^3}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac {2 a \left (c+d x^2\right )^{3/2} (7 b c-2 a d)}{35 c^2 x^5} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^8} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}+\frac {\int \frac {\left (2 a (7 b c-2 a d)+7 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^6} \, dx}{7 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac {2 a (7 b c-2 a d) \left (c+d x^2\right )^{3/2}}{35 c^2 x^5}-\frac {1}{35} \left (-35 b^2+\frac {4 a d (7 b c-2 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{7 c x^7}-\frac {2 a (7 b c-2 a d) \left (c+d x^2\right )^{3/2}}{35 c^2 x^5}-\frac {\left (35 b^2-\frac {4 a d (7 b c-2 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 0.77 \[ -\frac {\left (c+d x^2\right )^{3/2} \left (a^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+14 a b c x^2 \left (3 c-2 d x^2\right )+35 b^2 c^2 x^4\right )}{105 c^3 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 107, normalized size = 1.08 \[ -\frac {{\left ({\left (35 \, b^{2} c^{2} d - 28 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{6} + 15 \, a^{2} c^{3} + {\left (35 \, b^{2} c^{3} + 14 \, a b c^{2} d - 4 \, a^{2} c d^{2}\right )} x^{4} + 3 \, {\left (14 \, a b c^{3} + a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{105 \, c^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 490, normalized size = 4.95 \[ \frac {2 \, {\left (105 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} d^{\frac {3}{2}} - 420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c d^{\frac {3}{2}} + 420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b d^{\frac {5}{2}} + 665 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{2} d^{\frac {3}{2}} - 700 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c d^{\frac {5}{2}} + 560 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} d^{\frac {7}{2}} - 560 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{3} d^{\frac {3}{2}} + 280 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac {5}{2}} + 280 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c d^{\frac {7}{2}} + 315 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{4} d^{\frac {3}{2}} - 168 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac {5}{2}} + 168 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {7}{2}} - 140 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{5} d^{\frac {3}{2}} + 196 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac {5}{2}} - 56 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac {7}{2}} + 35 \, b^{2} c^{6} d^{\frac {3}{2}} - 28 \, a b c^{5} d^{\frac {5}{2}} + 8 \, a^{2} c^{4} d^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 0.79 \[ -\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (8 a^{2} d^{2} x^{4}-28 a b c d \,x^{4}+35 b^{2} c^{2} x^{4}-12 a^{2} c d \,x^{2}+42 a b \,c^{2} x^{2}+15 a^{2} c^{2}\right )}{105 c^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 124, normalized size = 1.25 \[ -\frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{3 \, c x^{3}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{15 \, c^{2} x^{3}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{105 \, c^{3} x^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{5 \, c x^{5}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{35 \, c^{2} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{7 \, c x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 181, normalized size = 1.83 \[ \frac {4\,a^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {b^2\,\sqrt {d\,x^2+c}}{3\,x^3}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{5\,x^5}-\frac {a^2\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {8\,a^2\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{3\,c\,x}+\frac {4\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{15\,c^2\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{15\,c\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.51, size = 510, normalized size = 5.15 \[ - \frac {15 a^{2} c^{5} d^{\frac {9}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {33 a^{2} c^{4} d^{\frac {11}{2}} x^{2} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {17 a^{2} c^{3} d^{\frac {13}{2}} x^{4} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {3 a^{2} c^{2} d^{\frac {15}{2}} x^{6} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {12 a^{2} c d^{\frac {17}{2}} x^{8} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {8 a^{2} d^{\frac {19}{2}} x^{10} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {2 a b \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {2 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c x^{2}} + \frac {4 a b d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c^{2}} - \frac {b^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {b^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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