Optimal. Leaf size=77 \[ \frac {(b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^3}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, 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 = {455, 45}
\begin {gather*} -\frac {2 b \left (c+d x^2\right )^{7/2} (b c-a d)}{7 d^3}+\frac {\left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (c+d x)^{3/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {(-b c+a d)^2 (c+d x)^{3/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{5/2}}{d^2}+\frac {b^2 (c+d x)^{7/2}}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^3}-\frac {2 b (b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^3}+\frac {b^2 \left (c+d x^2\right )^{9/2}}{9 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 0.87 \begin {gather*} \frac {\left (c+d x^2\right )^{5/2} \left (63 a^2 d^2+18 a b d \left (-2 c+5 d x^2\right )+b^2 \left (8 c^2-20 c d x^2+35 d^2 x^4\right )\right )}{315 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 117, normalized size = 1.52
method | result | size |
gosper | \(\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} \left (35 b^{2} x^{4} d^{2}+90 a b \,d^{2} x^{2}-20 b^{2} c d \,x^{2}+63 a^{2} d^{2}-36 a b c d +8 b^{2} c^{2}\right )}{315 d^{3}}\) | \(69\) |
default | \(b^{2} \left (\frac {x^{4} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )+2 a b \left (\frac {x^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{35 d^{2}}\right )+\frac {a^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5 d}\) | \(117\) |
trager | \(\frac {\left (35 b^{2} x^{8} d^{4}+90 a b \,d^{4} x^{6}+50 b^{2} c \,d^{3} x^{6}+63 a^{2} x^{4} d^{4}+144 x^{4} a b c \,d^{3}+3 b^{2} c^{2} x^{4} d^{2}+126 a^{2} c \,d^{3} x^{2}+18 a b \,c^{2} d^{2} x^{2}-4 b^{2} c^{3} d \,x^{2}+63 a^{2} c^{2} d^{2}-36 a b \,c^{3} d +8 b^{2} c^{4}\right ) \sqrt {d \,x^{2}+c}}{315 d^{3}}\) | \(149\) |
risch | \(\frac {\left (35 b^{2} x^{8} d^{4}+90 a b \,d^{4} x^{6}+50 b^{2} c \,d^{3} x^{6}+63 a^{2} x^{4} d^{4}+144 x^{4} a b c \,d^{3}+3 b^{2} c^{2} x^{4} d^{2}+126 a^{2} c \,d^{3} x^{2}+18 a b \,c^{2} d^{2} x^{2}-4 b^{2} c^{3} d \,x^{2}+63 a^{2} c^{2} d^{2}-36 a b \,c^{3} d +8 b^{2} c^{4}\right ) \sqrt {d \,x^{2}+c}}{315 d^{3}}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 115, normalized size = 1.49 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{4}}{9 \, d} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{2}}{63 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{2}}{7 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2}}{315 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c}{35 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (65) = 130\).
time = 1.09, size = 141, normalized size = 1.83 \begin {gather*} \frac {{\left (35 \, b^{2} d^{4} x^{8} + 10 \, {\left (5 \, b^{2} c d^{3} + 9 \, a b d^{4}\right )} x^{6} + 8 \, b^{2} c^{4} - 36 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} + 3 \, {\left (b^{2} c^{2} d^{2} + 48 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{4} - 2 \, {\left (2 \, b^{2} c^{3} d - 9 \, a b c^{2} d^{2} - 63 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, d^{3}} \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. 303 vs.
\(2 (66) = 132\).
time = 0.30, size = 303, normalized size = 3.94 \begin {gather*} \begin {cases} \frac {a^{2} c^{2} \sqrt {c + d x^{2}}}{5 d} + \frac {2 a^{2} c x^{2} \sqrt {c + d x^{2}}}{5} + \frac {a^{2} d x^{4} \sqrt {c + d x^{2}}}{5} - \frac {4 a b c^{3} \sqrt {c + d x^{2}}}{35 d^{2}} + \frac {2 a b c^{2} x^{2} \sqrt {c + d x^{2}}}{35 d} + \frac {16 a b c x^{4} \sqrt {c + d x^{2}}}{35} + \frac {2 a b d x^{6} \sqrt {c + d x^{2}}}{7} + \frac {8 b^{2} c^{4} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {4 b^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{2}} + \frac {b^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d} + \frac {10 b^{2} c x^{6} \sqrt {c + d x^{2}}}{63} + \frac {b^{2} d x^{8} \sqrt {c + d x^{2}}}{9} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.47, size = 98, normalized size = 1.27 \begin {gather*} \frac {35 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} - 90 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c + 63 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} + 90 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d - 126 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d + 63 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{315 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 136, normalized size = 1.77 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {63\,a^2\,c^2\,d^2-36\,a\,b\,c^3\,d+8\,b^2\,c^4}{315\,d^3}+\frac {x^4\,\left (63\,a^2\,d^4+144\,a\,b\,c\,d^3+3\,b^2\,c^2\,d^2\right )}{315\,d^3}+\frac {2\,b\,x^6\,\left (9\,a\,d+5\,b\,c\right )}{63}+\frac {b^2\,d\,x^8}{9}+\frac {2\,c\,x^2\,\left (63\,a^2\,d^2+9\,a\,b\,c\,d-2\,b^2\,c^2\right )}{315\,d^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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