Optimal. Leaf size=117 \[ \frac {8 d^2 x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}-\frac {4 d x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac {x^7 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \begin {gather*} \frac {8 d^2 x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}+\frac {x^7 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}-\frac {4 d x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{3/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^8 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{9 c}+\frac {(9 b c-6 a d) \int \sqrt {c+\frac {d}{x^2}} x^6 \, dx}{9 c}\\ &=\frac {(3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{9 c}-\frac {(4 d (3 b c-2 a d)) \int \sqrt {c+\frac {d}{x^2}} x^4 \, dx}{21 c^2}\\ &=-\frac {4 d (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{105 c^3}+\frac {(3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{9 c}+\frac {\left (8 d^2 (3 b c-2 a d)\right ) \int \sqrt {c+\frac {d}{x^2}} x^2 \, dx}{105 c^3}\\ &=\frac {8 d^2 (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^3}{315 c^4}-\frac {4 d (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{105 c^3}+\frac {(3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{9 c}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 86, normalized size = 0.74 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (a \left (35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right )+3 b c \left (15 c^2 x^4-12 c d x^2+8 d^2\right )\right )}{315 c^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 88, normalized size = 0.75 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (35 a c^3 x^6-30 a c^2 d x^4+24 a c d^2 x^2-16 a d^3+45 b c^3 x^4-36 b c^2 d x^2+24 b c d^2\right )}{315 c^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 107, normalized size = 0.91 \begin {gather*} \frac {{\left (35 \, a c^{4} x^{9} + 5 \, {\left (9 \, b c^{4} + a c^{3} d\right )} x^{7} + 3 \, {\left (3 \, b c^{3} d - 2 \, a c^{2} d^{2}\right )} x^{5} - 4 \, {\left (3 \, b c^{2} d^{2} - 2 \, a c d^{3}\right )} x^{3} + 8 \, {\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 140, normalized size = 1.20 \begin {gather*} -\frac {8 \, {\left (3 \, b c d^{\frac {7}{2}} - 2 \, a d^{\frac {9}{2}}\right )} \mathrm {sgn}\relax (x)}{315 \, c^{4}} + \frac {35 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a \mathrm {sgn}\relax (x) + 45 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c \mathrm {sgn}\relax (x) - 135 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d \mathrm {sgn}\relax (x) - 126 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d \mathrm {sgn}\relax (x) + 189 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{2} \mathrm {sgn}\relax (x) + 105 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c d^{2} \mathrm {sgn}\relax (x) - 105 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d^{3} \mathrm {sgn}\relax (x)}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (35 a \,x^{6} c^{3}-30 a \,c^{2} d \,x^{4}+45 b \,c^{3} x^{4}+24 a c \,d^{2} x^{2}-36 b \,c^{2} d \,x^{2}-16 a \,d^{3}+24 b c \,d^{2}\right ) \left (c \,x^{2}+d \right ) x}{315 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 124, normalized size = 1.06 \begin {gather*} \frac {{\left (15 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 42 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5} + 35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} x^{3}\right )} b}{105 \, c^{3}} + \frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 135 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 189 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5} - 105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{3} x^{3}\right )} a}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 97, normalized size = 0.83 \begin {gather*} \sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,x^9}{9}-\frac {x\,\left (16\,a\,d^4-24\,b\,c\,d^3\right )}{315\,c^4}+\frac {x^7\,\left (45\,b\,c^4+5\,a\,d\,c^3\right )}{315\,c^4}-\frac {d\,x^5\,\left (2\,a\,d-3\,b\,c\right )}{105\,c^2}+\frac {4\,d^2\,x^3\,\left (2\,a\,d-3\,b\,c\right )}{315\,c^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.92, size = 910, normalized size = 7.78 \begin {gather*} \frac {35 a c^{7} d^{\frac {19}{2}} x^{14} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} + \frac {110 a c^{6} d^{\frac {21}{2}} x^{12} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} + \frac {114 a c^{5} d^{\frac {23}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} + \frac {40 a c^{4} d^{\frac {25}{2}} x^{8} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} - \frac {5 a c^{3} d^{\frac {27}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} - \frac {30 a c^{2} d^{\frac {29}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} - \frac {40 a c d^{\frac {31}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} - \frac {16 a d^{\frac {33}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{315 c^{7} d^{9} x^{6} + 945 c^{6} d^{10} x^{4} + 945 c^{5} d^{11} x^{2} + 315 c^{4} d^{12}} + \frac {15 b c^{5} d^{\frac {9}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {33 b c^{4} d^{\frac {11}{2}} x^{8} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {17 b c^{3} d^{\frac {13}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {3 b c^{2} d^{\frac {15}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {12 b c d^{\frac {17}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {8 b d^{\frac {19}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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