Optimal. Leaf size=189 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{231 c^3 x^7}+\frac {4 d \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 190, normalized size of antiderivative = 1.01, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {473, 464, 277,
270} \begin {gather*} -\frac {\left (c+d x^2\right )^{3/2} \left (16 a^2 d^2-44 a b c d+33 b^2 c^2\right )}{231 c^3 x^7}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac {4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rule 464
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}+\frac {\int \frac {\left (2 a (11 b c-4 a d)+11 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{10}} \, dx}{11 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {1}{33} \left (-33 b^2+\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^8} \, dx\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}-\frac {\left (4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^6} \, dx}{231 c^3}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}+\frac {\left (8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx}{1155 c^4}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 141, normalized size = 0.75 \begin {gather*} -\frac {\left (c+d x^2\right )^{3/2} \left (33 b^2 c^2 x^4 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+22 a b c x^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+a^2 \left (315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right )\right )}{3465 c^5 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 266, normalized size = 1.41
method | result | size |
gosper | \(-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{8}-352 a b c \,d^{3} x^{8}+264 b^{2} c^{2} d^{2} x^{8}-192 a^{2} c \,d^{3} x^{6}+528 a b \,c^{2} d^{2} x^{6}-396 b^{2} c^{3} d \,x^{6}+240 a^{2} c^{2} d^{2} x^{4}-660 a b \,c^{3} d \,x^{4}+495 b^{2} c^{4} x^{4}-280 a^{2} c^{3} d \,x^{2}+770 a b \,c^{4} x^{2}+315 a^{2} c^{4}\right )}{3465 x^{11} c^{5}}\) | \(158\) |
trager | \(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 x^{11} c^{5}}\) | \(199\) |
risch | \(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 x^{11} c^{5}}\) | \(199\) |
default | \(a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{11 c \,x^{11}}-\frac {8 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )+b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 258, normalized size = 1.37 \begin {gather*} -\frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2}}{105 \, c^{3} x^{3}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{3}}{315 \, c^{4} x^{3}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{4}}{3465 \, c^{5} x^{3}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{35 \, c^{2} x^{5}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{5}} + \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{1155 \, c^{4} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{7 \, c x^{7}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{21 \, c^{2} x^{7}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{231 \, c^{3} x^{7}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{9 \, c x^{9}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{99 \, c^{2} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{11 \, c x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 185, normalized size = 0.98 \begin {gather*} -\frac {{\left (8 \, {\left (33 \, b^{2} c^{2} d^{3} - 44 \, a b c d^{4} + 16 \, a^{2} d^{5}\right )} x^{10} - 4 \, {\left (33 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 16 \, a^{2} c d^{4}\right )} x^{8} + 315 \, a^{2} c^{5} + 3 \, {\left (33 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 16 \, a^{2} c^{2} d^{3}\right )} x^{6} + 5 \, {\left (99 \, b^{2} c^{5} + 22 \, a b c^{4} d - 8 \, a^{2} c^{3} d^{2}\right )} x^{4} + 35 \, {\left (22 \, a b c^{5} + a^{2} c^{4} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, c^{5} x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1856 vs.
\(2 (187) = 374\).
time = 3.38, size = 1856, normalized size = 9.82 \begin {gather*} - \frac {315 a^{2} c^{9} d^{\frac {33}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {1295 a^{2} c^{8} d^{\frac {35}{2}} x^{2} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {1990 a^{2} c^{7} d^{\frac {37}{2}} x^{4} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {1358 a^{2} c^{6} d^{\frac {39}{2}} x^{6} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {343 a^{2} c^{5} d^{\frac {41}{2}} x^{8} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {35 a^{2} c^{4} d^{\frac {43}{2}} x^{10} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {280 a^{2} c^{3} d^{\frac {45}{2}} x^{12} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {560 a^{2} c^{2} d^{\frac {47}{2}} x^{14} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {448 a^{2} c d^{\frac {49}{2}} x^{16} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {128 a^{2} d^{\frac {51}{2}} x^{18} \sqrt {\frac {c}{d x^{2}} + 1}}{3465 c^{9} d^{16} x^{10} + 13860 c^{8} d^{17} x^{12} + 20790 c^{7} d^{18} x^{14} + 13860 c^{6} d^{19} x^{16} + 3465 c^{5} d^{20} x^{18}} - \frac {70 a b c^{7} d^{\frac {19}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} - \frac {220 a b c^{6} d^{\frac {21}{2}} x^{2} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} - \frac {228 a b c^{5} d^{\frac {23}{2}} x^{4} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} - \frac {80 a b c^{4} d^{\frac {25}{2}} x^{6} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} + \frac {10 a b c^{3} d^{\frac {27}{2}} x^{8} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} + \frac {60 a b c^{2} d^{\frac {29}{2}} x^{10} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} + \frac {80 a b c d^{\frac {31}{2}} x^{12} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} + \frac {32 a b d^{\frac {33}{2}} x^{14} \sqrt {\frac {c}{d x^{2}} + 1}}{315 c^{7} d^{9} x^{8} + 945 c^{6} d^{10} x^{10} + 945 c^{5} d^{11} x^{12} + 315 c^{4} d^{12} x^{14}} - \frac {15 b^{2} c^{5} d^{\frac {9}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {33 b^{2} c^{4} d^{\frac {11}{2}} x^{2} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {17 b^{2} c^{3} d^{\frac {13}{2}} x^{4} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {3 b^{2} c^{2} d^{\frac {15}{2}} x^{6} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {12 b^{2} c d^{\frac {17}{2}} x^{8} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} - \frac {8 b^{2} d^{\frac {19}{2}} x^{10} \sqrt {\frac {c}{d x^{2}} + 1}}{105 c^{5} d^{4} x^{6} + 210 c^{4} d^{5} x^{8} + 105 c^{3} d^{6} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 668 vs.
\(2 (169) = 338\).
time = 1.32, size = 668, normalized size = 3.53 \begin {gather*} \frac {16 \, {\left (2310 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{16} b^{2} d^{\frac {7}{2}} - 8085 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} c d^{\frac {7}{2}} + 13860 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} a b d^{\frac {9}{2}} + 9933 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c^{2} d^{\frac {7}{2}} - 19404 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b c d^{\frac {9}{2}} + 22176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a^{2} d^{\frac {11}{2}} - 5313 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{3} d^{\frac {7}{2}} + 924 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c^{2} d^{\frac {9}{2}} + 14784 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} c d^{\frac {11}{2}} + 2805 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{4} d^{\frac {7}{2}} - 660 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{3} d^{\frac {9}{2}} + 5280 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c^{2} d^{\frac {11}{2}} - 3135 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{5} d^{\frac {7}{2}} + 7260 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{4} d^{\frac {9}{2}} - 2640 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{3} d^{\frac {11}{2}} + 1815 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{6} d^{\frac {7}{2}} - 2420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{5} d^{\frac {9}{2}} + 880 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{4} d^{\frac {11}{2}} - 363 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{7} d^{\frac {7}{2}} + 484 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{6} d^{\frac {9}{2}} - 176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{5} d^{\frac {11}{2}} + 33 \, b^{2} c^{8} d^{\frac {7}{2}} - 44 \, a b c^{7} d^{\frac {9}{2}} + 16 \, a^{2} c^{6} d^{\frac {11}{2}}\right )}}{3465 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 317, normalized size = 1.68 \begin {gather*} \frac {8\,a^2\,d^2\,\sqrt {d\,x^2+c}}{693\,c^2\,x^7}-\frac {b^2\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {a^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {16\,a^2\,d^3\,\sqrt {d\,x^2+c}}{1155\,c^3\,x^5}+\frac {64\,a^2\,d^4\,\sqrt {d\,x^2+c}}{3465\,c^4\,x^3}-\frac {128\,a^2\,d^5\,\sqrt {d\,x^2+c}}{3465\,c^5\,x}+\frac {4\,b^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {8\,b^2\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{99\,c\,x^9}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5}+\frac {4\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {32\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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