Optimal. Leaf size=114 \[ -\frac {c (b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{9/2}}{9 d^4}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 78}
\begin {gather*} -\frac {b \left (c+d x^2\right )^{9/2} (3 b c-2 a d)}{9 d^4}+\frac {\left (c+d x^2\right )^{7/2} (b c-a d) (3 b c-a d)}{7 d^4}-\frac {c \left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^4}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^2 (c+d x)^{3/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {c (b c-a d)^2 (c+d x)^{3/2}}{d^3}+\frac {(b c-a d) (3 b c-a d) (c+d x)^{5/2}}{d^3}-\frac {b (3 b c-2 a d) (c+d x)^{7/2}}{d^3}+\frac {b^2 (c+d x)^{9/2}}{d^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {c (b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{9/2}}{9 d^4}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 100, normalized size = 0.88 \begin {gather*} \frac {\left (c+d x^2\right )^{5/2} \left (99 a^2 d^2 \left (-2 c+5 d x^2\right )+22 a b d \left (8 c^2-20 c d x^2+35 d^2 x^4\right )-3 b^2 \left (16 c^3-40 c^2 d x^2+70 c d^2 x^4-105 d^3 x^6\right )\right )}{3465 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 185, normalized size = 1.62
method | result | size |
gosper | \(-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} \left (-315 b^{2} x^{6} d^{3}-770 a b \,d^{3} x^{4}+210 b^{2} c \,d^{2} x^{4}-495 a^{2} d^{3} x^{2}+440 a b c \,d^{2} x^{2}-120 b^{2} c^{2} d \,x^{2}+198 a^{2} c \,d^{2}-176 a b \,c^{2} d +48 b^{2} c^{3}\right )}{3465 d^{4}}\) | \(108\) |
default | \(b^{2} \left (\frac {x^{6} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{11 d}-\frac {6 c \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )}{11 d}\right )+2 a b \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )+a^{2} \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{35 d^{2}}\right )\) | \(185\) |
trager | \(-\frac {\left (-315 b^{2} d^{5} x^{10}-770 a b \,d^{5} x^{8}-420 b^{2} c \,d^{4} x^{8}-495 a^{2} d^{5} x^{6}-1100 a b c \,d^{4} x^{6}-15 b^{2} c^{2} d^{3} x^{6}-792 a^{2} c \,d^{4} x^{4}-66 a b \,c^{2} d^{3} x^{4}+18 b^{2} c^{3} d^{2} x^{4}-99 a^{2} c^{2} d^{3} x^{2}+88 a b \,c^{3} d^{2} x^{2}-24 b^{2} c^{4} d \,x^{2}+198 a^{2} c^{3} d^{2}-176 a b \,c^{4} d +48 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 d^{4}}\) | \(190\) |
risch | \(-\frac {\left (-315 b^{2} d^{5} x^{10}-770 a b \,d^{5} x^{8}-420 b^{2} c \,d^{4} x^{8}-495 a^{2} d^{5} x^{6}-1100 a b c \,d^{4} x^{6}-15 b^{2} c^{2} d^{3} x^{6}-792 a^{2} c \,d^{4} x^{4}-66 a b \,c^{2} d^{3} x^{4}+18 b^{2} c^{3} d^{2} x^{4}-99 a^{2} c^{2} d^{3} x^{2}+88 a b \,c^{3} d^{2} x^{2}-24 b^{2} c^{4} d \,x^{2}+198 a^{2} c^{3} d^{2}-176 a b \,c^{4} d +48 b^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 d^{4}}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 181, normalized size = 1.59 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{6}}{11 \, d} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{4}}{33 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{4}}{9 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} x^{2}}{231 \, d^{3}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c x^{2}}{63 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} x^{2}}{7 \, d} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3}}{1155 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2}}{315 \, d^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c}{35 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 179, normalized size = 1.57 \begin {gather*} \frac {{\left (315 \, b^{2} d^{5} x^{10} + 70 \, {\left (6 \, b^{2} c d^{4} + 11 \, a b d^{5}\right )} x^{8} - 48 \, b^{2} c^{5} + 176 \, a b c^{4} d - 198 \, a^{2} c^{3} d^{2} + 5 \, {\left (3 \, b^{2} c^{2} d^{3} + 220 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{6} - 6 \, {\left (3 \, b^{2} c^{3} d^{2} - 11 \, a b c^{2} d^{3} - 132 \, a^{2} c d^{4}\right )} x^{4} + {\left (24 \, b^{2} c^{4} d - 88 \, a b c^{3} d^{2} + 99 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, d^{4}} \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. 384 vs.
\(2 (102) = 204\).
time = 0.43, size = 384, normalized size = 3.37 \begin {gather*} \begin {cases} - \frac {2 a^{2} c^{3} \sqrt {c + d x^{2}}}{35 d^{2}} + \frac {a^{2} c^{2} x^{2} \sqrt {c + d x^{2}}}{35 d} + \frac {8 a^{2} c x^{4} \sqrt {c + d x^{2}}}{35} + \frac {a^{2} d x^{6} \sqrt {c + d x^{2}}}{7} + \frac {16 a b c^{4} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {8 a b c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{2}} + \frac {2 a b c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d} + \frac {20 a b c x^{6} \sqrt {c + d x^{2}}}{63} + \frac {2 a b d x^{8} \sqrt {c + d x^{2}}}{9} - \frac {16 b^{2} c^{5} \sqrt {c + d x^{2}}}{1155 d^{4}} + \frac {8 b^{2} c^{4} x^{2} \sqrt {c + d x^{2}}}{1155 d^{3}} - \frac {2 b^{2} c^{3} x^{4} \sqrt {c + d x^{2}}}{385 d^{2}} + \frac {b^{2} c^{2} x^{6} \sqrt {c + d x^{2}}}{231 d} + \frac {4 b^{2} c x^{8} \sqrt {c + d x^{2}}}{33} + \frac {b^{2} d x^{10} \sqrt {c + d x^{2}}}{11} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 150, normalized size = 1.32 \begin {gather*} \frac {315 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} - 1155 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c + 1485 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} - 693 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} + 770 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b d - 1980 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c d + 1386 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} d + 495 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2} - 693 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c d^{2}}{3465 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 170, normalized size = 1.49 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {x^6\,\left (495\,a^2\,d^5+1100\,a\,b\,c\,d^4+15\,b^2\,c^2\,d^3\right )}{3465\,d^4}-\frac {198\,a^2\,c^3\,d^2-176\,a\,b\,c^4\,d+48\,b^2\,c^5}{3465\,d^4}+\frac {2\,b\,x^8\,\left (11\,a\,d+6\,b\,c\right )}{99}+\frac {b^2\,d\,x^{10}}{11}+\frac {2\,c\,x^4\,\left (132\,a^2\,d^2+11\,a\,b\,c\,d-3\,b^2\,c^2\right )}{1155\,d^2}+\frac {c^2\,x^2\,\left (99\,a^2\,d^2-88\,a\,b\,c\,d+24\,b^2\,c^2\right )}{3465\,d^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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