Optimal. Leaf size=143 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2 c^2-8 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{3/2}}{105 c^3 x^5}+\frac {2 d \left (21 b^2 c^2-8 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{3/2}}{315 c^4 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps
used = 4, 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 (8 a^2 d^2-24 a b c d+21 b^2 c^2\right )}{105 c^3 x^5}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac {2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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^{10}} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac {\int \frac {\left (6 a (3 b c-a d)+9 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^8} \, dx}{9 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {1}{21} \left (-21 b^2+\frac {8 a d (3 b c-a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^6} \, dx\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2-\frac {8 a d (3 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^5}-\frac {\left (2 d \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx}{105 c^3}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2-\frac {8 a d (3 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^5}+\frac {2 d \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{315 c^4 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 108, normalized size = 0.76 \begin {gather*} -\frac {\left (c+d x^2\right )^{3/2} \left (21 b^2 c^2 x^4 \left (3 c-2 d x^2\right )+6 a b c x^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+a^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )\right )}{315 c^4 x^9} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 194, normalized size = 1.36
method | result | size |
gosper | \(-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (-16 a^{2} d^{3} x^{6}+48 a b c \,d^{2} x^{6}-42 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-72 a b \,c^{2} d \,x^{4}+63 b^{2} c^{3} x^{4}-30 a^{2} c^{2} d \,x^{2}+90 a b \,c^{3} x^{2}+35 a^{2} c^{3}\right )}{315 x^{9} c^{4}}\) | \(117\) |
trager | \(-\frac {\left (-16 a^{2} d^{4} x^{8}+48 a b c \,d^{3} x^{8}-42 b^{2} c^{2} d^{2} x^{8}+8 a^{2} c \,d^{3} x^{6}-24 a b \,c^{2} d^{2} x^{6}+21 b^{2} c^{3} d \,x^{6}-6 a^{2} c^{2} d^{2} x^{4}+18 a b \,c^{3} d \,x^{4}+63 b^{2} c^{4} x^{4}+5 a^{2} c^{3} d \,x^{2}+90 a b \,c^{4} x^{2}+35 a^{2} c^{4}\right ) \sqrt {d \,x^{2}+c}}{315 x^{9} c^{4}}\) | \(158\) |
risch | \(-\frac {\left (-16 a^{2} d^{4} x^{8}+48 a b c \,d^{3} x^{8}-42 b^{2} c^{2} d^{2} x^{8}+8 a^{2} c \,d^{3} x^{6}-24 a b \,c^{2} d^{2} x^{6}+21 b^{2} c^{3} d \,x^{6}-6 a^{2} c^{2} d^{2} x^{4}+18 a b \,c^{3} d \,x^{4}+63 b^{2} c^{4} x^{4}+5 a^{2} c^{3} d \,x^{2}+90 a b \,c^{4} x^{2}+35 a^{2} c^{4}\right ) \sqrt {d \,x^{2}+c}}{315 x^{9} c^{4}}\) | \(158\) |
default | \(b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 190, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{15 \, c^{2} x^{3}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{3}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{315 \, c^{4} x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{5 \, c x^{5}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{35 \, c^{2} x^{5}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{105 \, c^{3} x^{5}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{7 \, c x^{7}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{21 \, c^{2} x^{7}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{9 \, c x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.01, size = 147, normalized size = 1.03 \begin {gather*} \frac {{\left (2 \, {\left (21 \, b^{2} c^{2} d^{2} - 24 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{8} - {\left (21 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{6} - 35 \, a^{2} c^{4} - 3 \, {\left (21 \, b^{2} c^{4} + 6 \, a b c^{3} d - 2 \, a^{2} c^{2} d^{2}\right )} x^{4} - 5 \, {\left (18 \, a b c^{4} + a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, c^{4} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1061 vs.
\(2 (134) = 268\).
time = 2.70, size = 1061, normalized size = 7.42 \begin {gather*} - \frac {35 a^{2} 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 {110 a^{2} 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 {114 a^{2} 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 {40 a^{2} 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 {5 a^{2} 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 {30 a^{2} 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 {40 a^{2} 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 {16 a^{2} 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 {30 a b 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 {66 a b 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 {34 a b 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 {6 a b 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 {24 a b 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 {16 a b 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}} - \frac {b^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {b^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c x^{2}} + \frac {2 b^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs.
\(2 (127) = 254\).
time = 1.34, size = 579, normalized size = 4.05 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} d^{\frac {5}{2}} - 1155 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c d^{\frac {5}{2}} + 1680 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b d^{\frac {7}{2}} + 1575 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{2} d^{\frac {5}{2}} - 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c d^{\frac {7}{2}} + 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} d^{\frac {9}{2}} - 1071 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{3} d^{\frac {5}{2}} + 504 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac {7}{2}} + 1512 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c d^{\frac {9}{2}} + 609 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{4} d^{\frac {5}{2}} - 336 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac {7}{2}} + 672 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac {9}{2}} - 441 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{5} d^{\frac {5}{2}} + 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac {7}{2}} - 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac {9}{2}} + 189 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{6} d^{\frac {5}{2}} - 216 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac {7}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac {9}{2}} - 21 \, b^{2} c^{7} d^{\frac {5}{2}} + 24 \, a b c^{6} d^{\frac {7}{2}} - 8 \, a^{2} c^{5} d^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.05, size = 249, normalized size = 1.74 \begin {gather*} \frac {2\,a^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {b^2\,\sqrt {d\,x^2+c}}{5\,x^5}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {a^2\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {8\,a^2\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {16\,a^2\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}+\frac {2\,b^2\,d^2\,\sqrt {d\,x^2+c}}{15\,c^2\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{15\,c\,x^3}+\frac {8\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________