Optimal. Leaf size=126 \[ \frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {c^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{d^5}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5} \]
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Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \begin {gather*} \frac {c^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{d^5}+\frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (a+b x)}{(c+d x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^3 (b c-a d)}{d^4 (c+d x)^{3/2}}-\frac {c^2 (4 b c-3 a d)}{d^4 \sqrt {c+d x}}+\frac {3 c (2 b c-a d) \sqrt {c+d x}}{d^4}+\frac {(-4 b c+a d) (c+d x)^{3/2}}{d^4}+\frac {b (c+d x)^{5/2}}{d^4}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c^3 (b c-a d)}{d^5 \sqrt {c+\frac {d}{x^2}}}+\frac {c^2 (4 b c-3 a d) \sqrt {c+\frac {d}{x^2}}}{d^5}-\frac {c (2 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{d^5}+\frac {(4 b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 104, normalized size = 0.83 \begin {gather*} \frac {b \left (128 c^4 x^8+64 c^3 d x^6-16 c^2 d^2 x^4+8 c d^3 x^2-5 d^4\right )-7 a d x^2 \left (16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right )}{35 d^5 x^8 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 123, normalized size = 0.98 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-112 a c^3 d x^8-56 a c^2 d^2 x^6+14 a c d^3 x^4-7 a d^4 x^2+128 b c^4 x^8+64 b c^3 d x^6-16 b c^2 d^2 x^4+8 b c d^3 x^2-5 b d^4\right )}{35 d^5 x^6 \left (c x^2+d\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 121, normalized size = 0.96 \begin {gather*} \frac {{\left (16 \, {\left (8 \, b c^{4} - 7 \, a c^{3} d\right )} x^{8} + 8 \, {\left (8 \, b c^{3} d - 7 \, a c^{2} d^{2}\right )} x^{6} - 5 \, b d^{4} - 2 \, {\left (8 \, b c^{2} d^{2} - 7 \, a c d^{3}\right )} x^{4} + {\left (8 \, b c d^{3} - 7 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{35 \, {\left (c d^{5} x^{8} + d^{6} x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 118, normalized size = 0.94 \begin {gather*} -\frac {\left (112 a \,c^{3} d \,x^{8}-128 b \,c^{4} x^{8}+56 a \,c^{2} d^{2} x^{6}-64 b \,c^{3} d \,x^{6}-14 a c \,d^{3} x^{4}+16 b \,c^{2} d^{2} x^{4}+7 a \,d^{4} x^{2}-8 b c \,d^{3} x^{2}+5 b \,d^{4}\right ) \left (c \,x^{2}+d \right )}{35 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{5} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 151, normalized size = 1.20 \begin {gather*} -\frac {1}{35} \, b {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{5}} - \frac {28 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{5}} + \frac {70 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2}}{d^{5}} - \frac {140 \, \sqrt {c + \frac {d}{x^{2}}} c^{3}}{d^{5}} - \frac {35 \, c^{4}}{\sqrt {c + \frac {d}{x^{2}}} d^{5}}\right )} - \frac {1}{5} \, a {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{4}} - \frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{4}} + \frac {15 \, \sqrt {c + \frac {d}{x^{2}}} c^{2}}{d^{4}} + \frac {5 \, c^{3}}{\sqrt {c + \frac {d}{x^{2}}} d^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.92, size = 154, normalized size = 1.22 \begin {gather*} \frac {c\,\sqrt {c+\frac {d}{x^2}}\,\left (21\,a\,d-29\,b\,c\right )}{35\,d^4\,x^2}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{7\,d^2\,x^6}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (7\,a\,d^2-13\,b\,c\,d\right )}{35\,d^4\,x^4}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (x^2\,\left (\frac {58\,b\,c^4-42\,a\,c^3\,d}{35\,d^5}+\frac {2\,c^3\,\left (77\,a\,d-93\,b\,c\right )}{35\,d^5}\right )+\frac {c^2\,\left (77\,a\,d-93\,b\,c\right )}{35\,d^4}\right )}{c\,x^2+d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.14, size = 122, normalized size = 0.97 \begin {gather*} - \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7 d^{5}} - \frac {c^{3} \left (a d - b c\right )}{d^{5} \sqrt {c + \frac {d}{x^{2}}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}} \left (a d - 4 b c\right )}{5 d^{5}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} \left (- 3 a c d + 6 b c^{2}\right )}{3 d^{5}} - \frac {\sqrt {c + \frac {d}{x^{2}}} \left (3 a c^{2} d - 4 b c^{3}\right )}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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