Optimal. Leaf size=46 \[ \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2} \]
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Rubi [A] time = 0.04, antiderivative size = 46, 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 = {444, 43} \begin {gather*} \frac {\left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^3} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int (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 {(-b c+a d) (c+d x)^{3/2}}{d}+\frac {b (c+d x)^{5/2}}{d}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {(b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (7 a d x^2-2 b c x^2+5 b d\right )}{35 d^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 90, normalized size = 1.96 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-7 a c^2 d x^6-14 a c d^2 x^4-7 a d^3 x^2+2 b c^3 x^6-b c^2 d x^4-8 b c d^2 x^2-5 b d^3\right )}{35 d^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 84, normalized size = 1.83 \begin {gather*} \frac {{\left ({\left (2 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} - {\left (b c^{2} d + 14 \, a c d^{2}\right )} x^{4} - 5 \, b d^{3} - {\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{35 \, d^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.87, size = 370, normalized size = 8.04 \begin {gather*} \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {5}{2}} d \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {7}{2}} d \mathrm {sgn}\relax (x) + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {5}{2}} d^{2} \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {7}{2}} d^{2} \mathrm {sgn}\relax (x) - 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {5}{2}} d^{3} \mathrm {sgn}\relax (x) + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {7}{2}} d^{3} \mathrm {sgn}\relax (x) + 77 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {5}{2}} d^{4} \mathrm {sgn}\relax (x) + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {7}{2}} d^{4} \mathrm {sgn}\relax (x) - 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {5}{2}} d^{5} \mathrm {sgn}\relax (x) - 2 \, b c^{\frac {7}{2}} d^{5} \mathrm {sgn}\relax (x) + 7 \, a c^{\frac {5}{2}} d^{6} \mathrm {sgn}\relax (x)\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (7 a d \,x^{2}-2 b c \,x^{2}+5 b d \right ) \left (c \,x^{2}+d \right )}{35 d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 49, normalized size = 1.07 \begin {gather*} -\frac {a {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, d} - \frac {1}{35} \, {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{2}} - \frac {7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{2}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 122, normalized size = 2.65 \begin {gather*} \frac {2\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{35\,d^2}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{5\,d}-\frac {2\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{5\,x^2}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{5\,x^4}-\frac {8\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,x^4}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.16, size = 138, normalized size = 3.00 \begin {gather*} - \frac {a c \left (\begin {cases} \frac {\sqrt {c}}{x^{2}} & \text {for}\: d = 0 \\\frac {2 \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right )}{2} - \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} - \frac {b c \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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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