Optimal. Leaf size=104 \[ \frac {c^2 \left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^4} \]
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Rubi [A] time = 0.08, antiderivative size = 104, 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 \left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^4} \end {gather*}
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^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int 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) \sqrt {c+d x}}{d^3}+\frac {c (3 b c-2 a d) (c+d x)^{3/2}}{d^3}+\frac {(-3 b c+a d) (c+d x)^{5/2}}{d^3}+\frac {b (c+d x)^{7/2}}{d^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c^2 (b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^4}-\frac {c (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^4}+\frac {(3 b c-a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.76 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (x^2 \left (\frac {c x^2}{d}+1\right ) \left (8 c^2 x^4-12 c d x^2+15 d^2\right ) (6 b c-9 a d)-105 b d^2 \left (c x^2+d\right )\right )}{945 d^3 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 114, normalized size = 1.10 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-24 a c^3 d x^8+12 a c^2 d^2 x^6-9 a c d^3 x^4-45 a d^4 x^2+16 b c^4 x^8-8 b c^3 d x^6+6 b c^2 d^2 x^4-5 b c d^3 x^2-35 b d^4\right )}{315 d^4 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 109, normalized size = 1.05 \begin {gather*} \frac {{\left (8 \, {\left (2 \, b c^{4} - 3 \, a c^{3} d\right )} x^{8} - 4 \, {\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x^{6} - 35 \, b d^{4} + 3 \, {\left (2 \, b c^{2} d^{2} - 3 \, a c d^{3}\right )} x^{4} - 5 \, {\left (b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, d^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.06, size = 370, normalized size = 3.56 \begin {gather*} \frac {16 \, {\left (210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {9}{2}} \mathrm {sgn}\relax (x) - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {7}{2}} d \mathrm {sgn}\relax (x) + 378 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {9}{2}} d \mathrm {sgn}\relax (x) + 63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {7}{2}} d^{2} \mathrm {sgn}\relax (x) + 168 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {9}{2}} d^{2} \mathrm {sgn}\relax (x) - 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {7}{2}} d^{3} \mathrm {sgn}\relax (x) - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {9}{2}} d^{3} \mathrm {sgn}\relax (x) + 108 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {7}{2}} d^{4} \mathrm {sgn}\relax (x) + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {9}{2}} d^{4} \mathrm {sgn}\relax (x) - 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {7}{2}} d^{5} \mathrm {sgn}\relax (x) - 2 \, b c^{\frac {9}{2}} d^{5} \mathrm {sgn}\relax (x) + 3 \, a c^{\frac {7}{2}} d^{6} \mathrm {sgn}\relax (x)\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{9}} \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 = 0.90 \begin {gather*} -\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (24 a \,c^{2} d \,x^{6}-16 b \,c^{3} x^{6}-36 a c \,d^{2} x^{4}+24 b \,c^{2} d \,x^{4}+45 a \,d^{3} x^{2}-30 b c \,d^{2} x^{2}+35 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{315 d^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 118, normalized size = 1.13 \begin {gather*} -\frac {1}{315} \, b {\left (\frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}}}{d^{4}} - \frac {135 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c}{d^{4}} + \frac {189 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{2}}{d^{4}} - \frac {105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{3}}{d^{4}}\right )} - \frac {1}{105} \, a {\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 168, normalized size = 1.62 \begin {gather*} \frac {16\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^4}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {8\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{105\,d^3}-\frac {a\,\sqrt {c+\frac {d}{x^2}}}{7\,x^6}-\frac {a\,c\,\sqrt {c+\frac {d}{x^2}}}{35\,d\,x^4}-\frac {b\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,d\,x^6}+\frac {4\,a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d^2\,x^2}+\frac {2\,b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d^2\,x^4}-\frac {8\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.21, size = 112, normalized size = 1.08 \begin {gather*} - \frac {a \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}} - \frac {b \left (- \frac {c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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