Optimal. Leaf size=74 \[ \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (2 b c-a d)}{5 d^3}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^3} \]
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Rubi [A] time = 0.06, antiderivative size = 74, 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} \[ \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (2 b c-a d)}{5 d^3}-\frac {c \left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^5} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x (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 (b c-a d) \sqrt {c+d x}}{d^2}+\frac {(-2 b c+a d) (c+d x)^{3/2}}{d^2}+\frac {b (c+d x)^{5/2}}{d^2}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {c (b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3}+\frac {(2 b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 0.93 \[ \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (7 a d x^2 \left (2 c x^2-3 d\right )+b \left (-8 c^2 x^4+12 c d x^2-15 d^2\right )\right )}{105 d^3 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 85, normalized size = 1.15 \[ -\frac {{\left (2 \, {\left (4 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} - {\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x^{4} + 15 \, b d^{3} + 3 \, {\left (b c d^{2} + 7 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{105 \, d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 310, normalized size = 4.19 \[ \frac {4 \, {\left (105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 175 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {5}{2}} d \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {7}{2}} d \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {5}{2}} d^{2} \mathrm {sgn}\relax (x) + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {7}{2}} d^{2} \mathrm {sgn}\relax (x) - 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {5}{2}} d^{3} \mathrm {sgn}\relax (x) - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {7}{2}} d^{3} \mathrm {sgn}\relax (x) + 49 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {5}{2}} d^{4} \mathrm {sgn}\relax (x) + 4 \, b c^{\frac {7}{2}} d^{4} \mathrm {sgn}\relax (x) - 7 \, a c^{\frac {5}{2}} d^{5} \mathrm {sgn}\relax (x)\right )}}{105 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 0.95 \[ \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (14 a c d \,x^{4}-8 b \,c^{2} x^{4}-21 a \,d^{2} x^{2}+12 b c d \,x^{2}-15 b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{105 d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 84, normalized size = 1.14 \[ -\frac {1}{105} \, b {\left (\frac {15 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{3}} - \frac {42 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{3}} + \frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2}}{d^{3}}\right )} - \frac {1}{15} \, a {\left (\frac {3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{2}} - \frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.92, size = 126, normalized size = 1.70 \[ \frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{15\,d^2}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {a\,\sqrt {c+\frac {d}{x^2}}}{5\,x^4}-\frac {8\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{105\,d^3}-\frac {a\,c\,\sqrt {c+\frac {d}{x^2}}}{15\,d\,x^2}-\frac {b\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,x^4}+\frac {4\,b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.56, size = 78, normalized size = 1.05 \[ - \frac {a \left (- \frac {c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} - \frac {b \left (\frac {c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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