Optimal. Leaf size=159 \[ -\frac {c^3 (3 b c-8 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{128 d^{5/2}}+\frac {c^2 \sqrt {c+\frac {d}{x^2}} (3 b c-8 a d)}{128 d^2 x}+\frac {c \sqrt {c+\frac {d}{x^2}} (3 b c-8 a d)}{64 d x^3}+\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-8 a d)}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {459, 335, 279, 321, 217, 206} \begin {gather*} \frac {c^2 \sqrt {c+\frac {d}{x^2}} (3 b c-8 a d)}{128 d^2 x}-\frac {c^3 (3 b c-8 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{128 d^{5/2}}+\frac {c \sqrt {c+\frac {d}{x^2}} (3 b c-8 a d)}{64 d x^3}+\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-8 a d)}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 335
Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^4} \, dx &=-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {(-3 b c+8 a d) \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{x^4} \, dx}{8 d}\\ &=-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}-\frac {(-3 b c+8 a d) \operatorname {Subst}\left (\int x^2 \left (c+d x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )}{8 d}\\ &=\frac {(3 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {(c (3 b c-8 a d)) \operatorname {Subst}\left (\int x^2 \sqrt {c+d x^2} \, dx,x,\frac {1}{x}\right )}{16 d}\\ &=\frac {c (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{64 d x^3}+\frac {(3 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {\left (c^2 (3 b c-8 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{64 d}\\ &=\frac {c (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{64 d x^3}+\frac {(3 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {c^2 (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{128 d^2 x}-\frac {\left (c^3 (3 b c-8 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{128 d^2}\\ &=\frac {c (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{64 d x^3}+\frac {(3 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {c^2 (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{128 d^2 x}-\frac {\left (c^3 (3 b c-8 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )}{128 d^2}\\ &=\frac {c (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{64 d x^3}+\frac {(3 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{48 d x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{8 d x^3}+\frac {c^2 (3 b c-8 a d) \sqrt {c+\frac {d}{x^2}}}{128 d^2 x}-\frac {c^3 (3 b c-8 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{128 d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 71, normalized size = 0.45 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (c^3 x^8 (8 a d-3 b c) \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {c x^2}{d}+1\right )-5 b d^4\right )}{40 d^5 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 152, normalized size = 0.96 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {\left (8 a c^3 d-3 b c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{128 d^{5/2}}+\frac {\sqrt {c x^2+d} \left (-24 a c^2 d x^6-112 a c d^2 x^4-64 a d^3 x^2+9 b c^3 x^6-6 b c^2 d x^4-72 b c d^2 x^2-48 b d^3\right )}{384 d^2 x^8}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 298, normalized size = 1.87 \begin {gather*} \left [-\frac {3 \, {\left (3 \, b c^{4} - 8 \, a c^{3} d\right )} \sqrt {d} x^{7} \log \left (-\frac {c x^{2} + 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (3 \, b c^{3} d - 8 \, a c^{2} d^{2}\right )} x^{6} - 48 \, b d^{4} - 2 \, {\left (3 \, b c^{2} d^{2} + 56 \, a c d^{3}\right )} x^{4} - 8 \, {\left (9 \, b c d^{3} + 8 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{768 \, d^{3} x^{7}}, \frac {3 \, {\left (3 \, b c^{4} - 8 \, a c^{3} d\right )} \sqrt {-d} x^{7} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (3 \, {\left (3 \, b c^{3} d - 8 \, a c^{2} d^{2}\right )} x^{6} - 48 \, b d^{4} - 2 \, {\left (3 \, b c^{2} d^{2} + 56 \, a c d^{3}\right )} x^{4} - 8 \, {\left (9 \, b c d^{3} + 8 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{384 \, d^{3} x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 214, normalized size = 1.35 \begin {gather*} \frac {\frac {3 \, {\left (3 \, b c^{5} \mathrm {sgn}\relax (x) - 8 \, a c^{4} d \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d} d^{2}} + \frac {9 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c^{5} \mathrm {sgn}\relax (x) - 24 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a c^{4} d \mathrm {sgn}\relax (x) - 33 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c^{5} d \mathrm {sgn}\relax (x) - 40 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a c^{4} d^{2} \mathrm {sgn}\relax (x) - 33 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{5} d^{2} \mathrm {sgn}\relax (x) + 88 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{4} d^{3} \mathrm {sgn}\relax (x) + 9 \, \sqrt {c x^{2} + d} b c^{5} d^{3} \mathrm {sgn}\relax (x) - 24 \, \sqrt {c x^{2} + d} a c^{4} d^{4} \mathrm {sgn}\relax (x)}{c^{4} d^{2} x^{8}}}{384 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 302, normalized size = 1.90 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (24 a \,c^{3} d^{\frac {5}{2}} x^{8} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-9 b \,c^{4} d^{\frac {3}{2}} x^{8} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-24 \sqrt {c \,x^{2}+d}\, a \,c^{3} d^{2} x^{8}+9 \sqrt {c \,x^{2}+d}\, b \,c^{4} d \,x^{8}-8 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{3} d \,x^{8}+3 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{4} x^{8}+8 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,c^{2} d \,x^{6}-3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{3} x^{6}+16 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a c \,d^{2} x^{4}-6 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{2} d \,x^{4}-64 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,d^{3} x^{2}+24 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b c \,d^{2} x^{2}-48 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,d^{3}\right )}{384 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.57, size = 354, normalized size = 2.23 \begin {gather*} -\frac {1}{96} \, {\left (\frac {3 \, c^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3} x^{5} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{3} d x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{3} d^{2} x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} d x^{6} - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} x^{4} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} d^{3} x^{2} - d^{4}}\right )} a + \frac {1}{256} \, {\left (\frac {3 \, c^{4} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c^{4} x^{7} - 11 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{4} d x^{5} - 11 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{4} d^{2} x^{3} + 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{4} d^{3} x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{4} d^{2} x^{8} - 4 \, {\left (c + \frac {d}{x^{2}}\right )}^{3} d^{3} x^{6} + 6 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} d^{4} x^{4} - 4 \, {\left (c + \frac {d}{x^{2}}\right )} d^{5} x^{2} + d^{6}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+\frac {b}{x^2}\right )\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 29.16, size = 287, normalized size = 1.81 \begin {gather*} - \frac {a c^{\frac {5}{2}}}{16 d x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {17 a c^{\frac {3}{2}}}{48 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {11 a \sqrt {c} d}{24 x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {a c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{16 d^{\frac {3}{2}}} - \frac {a d^{2}}{6 \sqrt {c} x^{7} \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {3 b c^{\frac {7}{2}}}{128 d^{2} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{\frac {5}{2}}}{128 d x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {13 b c^{\frac {3}{2}}}{64 x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {5 b \sqrt {c} d}{16 x^{7} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{128 d^{\frac {5}{2}}} - \frac {b d^{2}}{8 \sqrt {c} x^{9} \sqrt {1 + \frac {d}{c x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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