Optimal. Leaf size=139 \[ \frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 (2 b c+3 a d) x^{11/2}+\frac {2}{15} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+\frac {2}{19} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+\frac {2}{23} b d^2 (3 b c+2 a d) x^{23/2}+\frac {2}{27} b^2 d^3 x^{27/2} \]
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Rubi [A]
time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459}
\begin {gather*} \frac {2}{19} d x^{19/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{15} c x^{15/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac {2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac {2}{27} b^2 d^3 x^{27/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^{5/2}+a c^2 (2 b c+3 a d) x^{9/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{13/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{17/2}+b d^2 (3 b c+2 a d) x^{21/2}+b^2 d^3 x^{25/2}\right ) \, dx\\ &=\frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 (2 b c+3 a d) x^{11/2}+\frac {2}{15} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+\frac {2}{19} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+\frac {2}{23} b d^2 (3 b c+2 a d) x^{23/2}+\frac {2}{27} b^2 d^3 x^{27/2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 126, normalized size = 0.91 \begin {gather*} \frac {2 x^{7/2} \left (621 a^2 \left (1045 c^3+1995 c^2 d x^2+1463 c d^2 x^4+385 d^3 x^6\right )+378 a b x^2 \left (2185 c^3+4807 c^2 d x^2+3795 c d^2 x^4+1045 d^3 x^6\right )+77 b^2 x^4 \left (3933 c^3+9315 c^2 d x^2+7695 c d^2 x^4+2185 d^3 x^6\right )\right )}{4542615} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 128, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7}\) | \(128\) |
default | \(\frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7}\) | \(128\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 a b c \,d^{2} x^{6}+717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\) | \(138\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 a b c \,d^{2} x^{6}+717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\) | \(138\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 a b c \,d^{2} x^{6}+717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 127, normalized size = 0.91 \begin {gather*} \frac {2}{27} \, b^{2} d^{3} x^{\frac {27}{2}} + \frac {2}{23} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {19}{2}} + \frac {2}{7} \, a^{2} c^{3} x^{\frac {7}{2}} + \frac {2}{15} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.78, size = 132, normalized size = 0.95 \begin {gather*} \frac {2}{4542615} \, {\left (168245 \, b^{2} d^{3} x^{13} + 197505 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + 239085 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + 648945 \, a^{2} c^{3} x^{3} + 302841 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + 412965 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.67, size = 192, normalized size = 1.38 \begin {gather*} \frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7} + \frac {6 a^{2} c^{2} d x^{\frac {11}{2}}}{11} + \frac {2 a^{2} c d^{2} x^{\frac {15}{2}}}{5} + \frac {2 a^{2} d^{3} x^{\frac {19}{2}}}{19} + \frac {4 a b c^{3} x^{\frac {11}{2}}}{11} + \frac {4 a b c^{2} d x^{\frac {15}{2}}}{5} + \frac {12 a b c d^{2} x^{\frac {19}{2}}}{19} + \frac {4 a b d^{3} x^{\frac {23}{2}}}{23} + \frac {2 b^{2} c^{3} x^{\frac {15}{2}}}{15} + \frac {6 b^{2} c^{2} d x^{\frac {19}{2}}}{19} + \frac {6 b^{2} c d^{2} x^{\frac {23}{2}}}{23} + \frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 135, normalized size = 0.97 \begin {gather*} \frac {2}{27} \, b^{2} d^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b^{2} c d^{2} x^{\frac {23}{2}} + \frac {4}{23} \, a b d^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c^{2} d x^{\frac {19}{2}} + \frac {12}{19} \, a b c d^{2} x^{\frac {19}{2}} + \frac {2}{19} \, a^{2} d^{3} x^{\frac {19}{2}} + \frac {2}{15} \, b^{2} c^{3} x^{\frac {15}{2}} + \frac {4}{5} \, a b c^{2} d x^{\frac {15}{2}} + \frac {2}{5} \, a^{2} c d^{2} x^{\frac {15}{2}} + \frac {4}{11} \, a b c^{3} x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c^{2} d x^{\frac {11}{2}} + \frac {2}{7} \, a^{2} c^{3} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 119, normalized size = 0.86 \begin {gather*} x^{15/2}\,\left (\frac {2\,a^2\,c\,d^2}{5}+\frac {4\,a\,b\,c^2\,d}{5}+\frac {2\,b^2\,c^3}{15}\right )+x^{19/2}\,\left (\frac {2\,a^2\,d^3}{19}+\frac {12\,a\,b\,c\,d^2}{19}+\frac {6\,b^2\,c^2\,d}{19}\right )+\frac {2\,a^2\,c^3\,x^{7/2}}{7}+\frac {2\,b^2\,d^3\,x^{27/2}}{27}+\frac {2\,a\,c^2\,x^{11/2}\,\left (3\,a\,d+2\,b\,c\right )}{11}+\frac {2\,b\,d^2\,x^{23/2}\,\left (2\,a\,d+3\,b\,c\right )}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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