Optimal. Leaf size=104 \[ \frac {c^2 \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^4} \]
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Rubi [A] time = 0.07, 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 )^{5/2} (b c-a d)}{5 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 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 ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x^2 (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^2 (b c-a d) (c+d x)^{3/2}}{d^3}+\frac {c (3 b c-2 a d) (c+d x)^{5/2}}{d^3}+\frac {(-3 b c+a d) (c+d x)^{7/2}}{d^3}+\frac {b (c+d x)^{9/2}}{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 )^{5/2}}{5 d^4}-\frac {c (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4}+\frac {(3 b c-a d) \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 94, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (-11 a d x^2 \left (8 c^2 x^4-20 c d x^2+35 d^2\right )-3 b \left (-16 c^3 x^6+40 c^2 d x^4-70 c d^2 x^2+105 d^3\right )\right )}{3465 d^4 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 138, normalized size = 1.33 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-88 a c^4 d x^{10}+44 a c^3 d^2 x^8-33 a c^2 d^3 x^6-550 a c d^4 x^4-385 a d^5 x^2+48 b c^5 x^{10}-24 b c^4 d x^8+18 b c^3 d^2 x^6-15 b c^2 d^3 x^4-420 b c d^4 x^2-315 b d^5\right )}{3465 d^4 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 134, normalized size = 1.29 \begin {gather*} \frac {{\left (8 \, {\left (6 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 4 \, {\left (6 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 3 \, {\left (6 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} - 315 \, b d^{5} - 5 \, {\left (3 \, b c^{2} d^{3} + 110 \, a c d^{4}\right )} x^{4} - 35 \, {\left (12 \, b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, d^{4} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.28, size = 490, normalized size = 4.71 \begin {gather*} \frac {16 \, {\left (2310 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{16} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} b c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 1155 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {9}{2}} d \mathrm {sgn}\relax (x) + 12474 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {11}{2}} d \mathrm {sgn}\relax (x) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {9}{2}} d^{2} \mathrm {sgn}\relax (x) + 15246 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {11}{2}} d^{2} \mathrm {sgn}\relax (x) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {9}{2}} d^{3} \mathrm {sgn}\relax (x) + 4950 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {11}{2}} d^{3} \mathrm {sgn}\relax (x) + 2475 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {9}{2}} d^{4} \mathrm {sgn}\relax (x) + 990 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {11}{2}} d^{4} \mathrm {sgn}\relax (x) + 495 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {9}{2}} d^{5} \mathrm {sgn}\relax (x) - 330 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {11}{2}} d^{5} \mathrm {sgn}\relax (x) + 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {9}{2}} d^{6} \mathrm {sgn}\relax (x) + 66 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {11}{2}} d^{6} \mathrm {sgn}\relax (x) - 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {9}{2}} d^{7} \mathrm {sgn}\relax (x) - 6 \, b c^{\frac {11}{2}} d^{7} \mathrm {sgn}\relax (x) + 11 \, a c^{\frac {9}{2}} d^{8} \mathrm {sgn}\relax (x)\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{11}} \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 {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (88 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}-220 a c \,d^{2} x^{4}+120 b \,c^{2} d \,x^{4}+385 a \,d^{3} x^{2}-210 b c \,d^{2} x^{2}+315 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{3465 d^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 118, normalized size = 1.13 \begin {gather*} -\frac {1}{315} \, {\left (\frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}}}{d^{3}} - \frac {90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c}{d^{3}} + \frac {63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{2}}{d^{3}}\right )} a - \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 )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.31, size = 206, normalized size = 1.98 \begin {gather*} \frac {16\,b\,c^5\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^4}-\frac {8\,a\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3}-\frac {10\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,x^6}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {4\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{33\,x^8}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{11\,x^{10}}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d\,x^4}+\frac {4\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^2\,x^2}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{231\,d\,x^6}+\frac {2\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{385\,d^2\,x^4}-\frac {8\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.96, size = 262, normalized size = 2.52 \begin {gather*} - \frac {a c \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 {a \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^{3}} - \frac {b 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 {b \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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