Optimal. Leaf size=134 \[ -\frac {c^3 \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^5}+\frac {c^2 \left (c+\frac {d}{x^2}\right )^{7/2} (4 b c-3 a d)}{7 d^5}+\frac {\left (c+\frac {d}{x^2}\right )^{11/2} (4 b c-a d)}{11 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{9/2} (2 b c-a d)}{3 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{13/2}}{13 d^5} \]
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Rubi [A] time = 0.10, antiderivative size = 134, 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 )^{7/2} (4 b c-3 a d)}{7 d^5}-\frac {c^3 \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^5}+\frac {\left (c+\frac {d}{x^2}\right )^{11/2} (4 b c-a d)}{11 d^5}-\frac {c \left (c+\frac {d}{x^2}\right )^{9/2} (2 b c-a d)}{3 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{13/2}}{13 d^5} \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 ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^9} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x^3 (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 {c^3 (b c-a d) (c+d x)^{3/2}}{d^4}-\frac {c^2 (4 b c-3 a d) (c+d x)^{5/2}}{d^4}+\frac {3 c (2 b c-a d) (c+d x)^{7/2}}{d^4}+\frac {(-4 b c+a d) (c+d x)^{9/2}}{d^4}+\frac {b (c+d x)^{11/2}}{d^4}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {c^3 (b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^5}+\frac {c^2 (4 b c-3 a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5}-\frac {c (2 b c-a d) \left (c+\frac {d}{x^2}\right )^{9/2}}{3 d^5}+\frac {(4 b c-a d) \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{13/2}}{13 d^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 115, normalized size = 0.86 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (13 a d x^2 \left (16 c^3 x^6-40 c^2 d x^4+70 c d^2 x^2-105 d^3\right )+b \left (-128 c^4 x^8+320 c^3 d x^6-560 c^2 d^2 x^4+840 c d^3 x^2-1155 d^4\right )\right )}{15015 d^5 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 162, normalized size = 1.21 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (208 a c^5 d x^{12}-104 a c^4 d^2 x^{10}+78 a c^3 d^3 x^8-65 a c^2 d^4 x^6-1820 a c d^5 x^4-1365 a d^6 x^2-128 b c^6 x^{12}+64 b c^5 d x^{10}-48 b c^4 d^2 x^8+40 b c^3 d^3 x^6-35 b c^2 d^4 x^4-1470 b c d^5 x^2-1155 b d^6\right )}{15015 d^5 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 157, normalized size = 1.17 \begin {gather*} -\frac {{\left (16 \, {\left (8 \, b c^{6} - 13 \, a c^{5} d\right )} x^{12} - 8 \, {\left (8 \, b c^{5} d - 13 \, a c^{4} d^{2}\right )} x^{10} + 6 \, {\left (8 \, b c^{4} d^{2} - 13 \, a c^{3} d^{3}\right )} x^{8} + 1155 \, b d^{6} - 5 \, {\left (8 \, b c^{3} d^{3} - 13 \, a c^{2} d^{4}\right )} x^{6} + 35 \, {\left (b c^{2} d^{4} + 52 \, a c d^{5}\right )} x^{4} + 105 \, {\left (14 \, b c d^{5} + 13 \, a d^{6}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15015 \, d^{5} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.63, size = 550, normalized size = 4.10 \begin {gather*} \frac {32 \, {\left (15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{18} a c^{\frac {11}{2}} \mathrm {sgn}\relax (x) + 48048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{16} b c^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 3003 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{16} a c^{\frac {11}{2}} d \mathrm {sgn}\relax (x) + 96096 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} b c^{\frac {13}{2}} d \mathrm {sgn}\relax (x) - 6006 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {11}{2}} d^{2} \mathrm {sgn}\relax (x) + 109824 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {13}{2}} d^{2} \mathrm {sgn}\relax (x) - 28314 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {11}{2}} d^{3} \mathrm {sgn}\relax (x) + 37752 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {13}{2}} d^{3} \mathrm {sgn}\relax (x) + 13728 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {11}{2}} d^{4} \mathrm {sgn}\relax (x) + 5720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {13}{2}} d^{4} \mathrm {sgn}\relax (x) + 5720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {11}{2}} d^{5} \mathrm {sgn}\relax (x) - 2288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {13}{2}} d^{5} \mathrm {sgn}\relax (x) + 3718 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {11}{2}} d^{6} \mathrm {sgn}\relax (x) + 624 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {13}{2}} d^{6} \mathrm {sgn}\relax (x) - 1014 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {11}{2}} d^{7} \mathrm {sgn}\relax (x) - 104 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {13}{2}} d^{7} \mathrm {sgn}\relax (x) + 169 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {11}{2}} d^{8} \mathrm {sgn}\relax (x) + 8 \, b c^{\frac {13}{2}} d^{8} \mathrm {sgn}\relax (x) - 13 \, a c^{\frac {11}{2}} d^{9} \mathrm {sgn}\relax (x)\right )}}{15015 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 118, normalized size = 0.88 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (208 a \,c^{3} d \,x^{8}-128 b \,c^{4} x^{8}-520 a \,c^{2} d^{2} x^{6}+320 b \,c^{3} d \,x^{6}+910 a c \,d^{3} x^{4}-560 b \,c^{2} d^{2} x^{4}-1365 a \,d^{4} x^{2}+840 b c \,d^{3} x^{2}-1155 b \,d^{4}\right ) \left (c \,x^{2}+d \right )}{15015 d^{5} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 152, normalized size = 1.13 \begin {gather*} -\frac {1}{1155} \, {\left (\frac {105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}}}{d^{4}} - \frac {385 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} c}{d^{4}} + \frac {495 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c^{2}}{d^{4}} - \frac {231 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3}}{d^{4}}\right )} a - \frac {1}{15015} \, {\left (\frac {1155 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {13}{2}}}{d^{5}} - \frac {5460 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} c}{d^{5}} + \frac {10010 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} c^{2}}{d^{5}} - \frac {8580 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c^{3}}{d^{5}} + \frac {3003 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{4}}{d^{5}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.81, size = 248, normalized size = 1.85 \begin {gather*} \frac {16\,a\,c^5\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^4}-\frac {128\,b\,c^6\,\sqrt {c+\frac {d}{x^2}}}{15015\,d^5}-\frac {4\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{33\,x^8}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{11\,x^{10}}-\frac {14\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{143\,x^{10}}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{13\,x^{12}}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{231\,d\,x^6}+\frac {2\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{385\,d^2\,x^4}-\frac {8\,a\,c^4\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^3\,x^2}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{429\,d\,x^8}+\frac {8\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{3003\,d^2\,x^6}-\frac {16\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{5005\,d^3\,x^4}+\frac {64\,b\,c^5\,\sqrt {c+\frac {d}{x^2}}}{15015\,d^4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 19.45, size = 326, normalized size = 2.43 \begin {gather*} - \frac {a c \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}} - \frac {a \left (\frac {c^{4} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} - \frac {b c \left (\frac {c^{4} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} - \frac {b \left (- \frac {c^{5} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + \frac {d}{x^{2}}\right )^{\frac {11}{2}}}{11} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {13}{2}}}{13}\right )}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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