Optimal. Leaf size=101 \[ \frac {c^2 \sqrt {c+\frac {d}{x^2}} (b c-a d)}{d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (3 b c-a d)}{5 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{3 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4} \]
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Rubi [A] time = 0.07, antiderivative size = 101, 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}} (b c-a d)}{d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{5/2} (3 b c-a d)}{5 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{3 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4} \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}}{\sqrt {c+\frac {d}{x^2}} x^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (a+b x)}{\sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {c^2 (b c-a d)}{d^3 \sqrt {c+d x}}+\frac {c (3 b c-2 a d) \sqrt {c+d x}}{d^3}+\frac {(-3 b c+a d) (c+d x)^{3/2}}{d^3}+\frac {b (c+d x)^{5/2}}{d^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c^2 (b c-a d) \sqrt {c+\frac {d}{x^2}}}{d^4}-\frac {c (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^4}+\frac {(3 b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.90 \begin {gather*} \frac {\left (\frac {c x^2}{d}+1\right ) \left (8 c^2 x^4-4 c d x^2+3 d^2\right ) (6 b c-7 a d)}{105 d^3 x^6 \sqrt {c+\frac {d}{x^2}}}-\frac {b \left (c x^2+d\right )}{7 d x^8 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 90, normalized size = 0.89 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-56 a c^2 d x^6+28 a c d^2 x^4-21 a d^3 x^2+48 b c^3 x^6-24 b c^2 d x^4+18 b c d^2 x^2-15 b d^3\right )}{105 d^4 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 86, normalized size = 0.85 \begin {gather*} \frac {{\left (8 \, {\left (6 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} - 4 \, {\left (6 \, b c^{2} d - 7 \, a c d^{2}\right )} x^{4} - 15 \, b d^{3} + 3 \, {\left (6 \, b c d^{2} - 7 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{105 \, d^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 219, normalized size = 2.17 \begin {gather*} \frac {140 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{4} a c + 210 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3} b c^{\frac {3}{2}} + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3} a \sqrt {c} d + 252 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} b c d + 21 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} a d^{2} + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} b \sqrt {c} d^{2} + 15 \, b d^{3}}{105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 94, normalized size = 0.93 \begin {gather*} -\frac {\left (56 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}-28 a c \,d^{2} x^{4}+24 b \,c^{2} d \,x^{4}+21 a \,d^{3} x^{2}-18 b c \,d^{2} x^{2}+15 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{105 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, d^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 118, normalized size = 1.17 \begin {gather*} -\frac {1}{35} \, b {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{4}} - \frac {21 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{4}} + \frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2}}{d^{4}} - \frac {35 \, \sqrt {c + \frac {d}{x^{2}}} c^{3}}{d^{4}}\right )} - \frac {1}{15} \, a {\left (\frac {3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{3}} - \frac {10 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{3}} + \frac {15 \, \sqrt {c + \frac {d}{x^{2}}} c^{2}}{d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 102, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}}\,\left (48\,b\,c^3-56\,a\,c^2\,d\right )}{105\,d^4}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{7\,d\,x^6}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (24\,b\,c^2-28\,a\,c\,d\right )}{105\,d^3\,x^2}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (7\,a\,d-6\,b\,c\right )}{35\,d^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.00, size = 269, normalized size = 2.66 \begin {gather*} \frac {\begin {cases} \frac {- \frac {a}{3 x^{6}} - \frac {b}{4 x^{8}}}{\sqrt {c}} & \text {for}\: d = 0 \\\frac {\frac {2 a c \left (\frac {c^{2}}{\sqrt {c + \frac {d}{x^{2}}}} + 2 c \sqrt {c + \frac {d}{x^{2}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} + \frac {2 a \left (- \frac {c^{3}}{\sqrt {c + \frac {d}{x^{2}}}} - 3 c^{2} \sqrt {c + \frac {d}{x^{2}}} + c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} + \frac {2 b c \left (- \frac {c^{3}}{\sqrt {c + \frac {d}{x^{2}}}} - 3 c^{2} \sqrt {c + \frac {d}{x^{2}}} + c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{3}} + \frac {2 b \left (\frac {c^{4}}{\sqrt {c + \frac {d}{x^{2}}}} + 4 c^{3} \sqrt {c + \frac {d}{x^{2}}} - 2 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} + \frac {4 c \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}}}{d} & \text {otherwise} \end {cases}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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