Optimal. Leaf size=137 \[ \frac {2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt {x} (3 a d+2 b c)+\frac {2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac {2}{17} b^2 d^3 x^{17/2} \]
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Rubi [A] time = 0.06, antiderivative size = 137, 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 = {448} \begin {gather*} \frac {2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt {x} (3 a d+2 b c)+\frac {2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac {2}{17} b^2 d^3 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^{5/2}} \, dx &=\int \left (\frac {a^2 c^3}{x^{5/2}}+\frac {a c^2 (2 b c+3 a d)}{\sqrt {x}}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{3/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7/2}+b d^2 (3 b c+2 a d) x^{11/2}+b^2 d^3 x^{15/2}\right ) \, dx\\ &=-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 (2 b c+3 a d) \sqrt {x}+\frac {2}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5/2}+\frac {2}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{9/2}+\frac {2}{13} b d^2 (3 b c+2 a d) x^{13/2}+\frac {2}{17} b^2 d^3 x^{17/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 137, normalized size = 1.00 \begin {gather*} \frac {2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt {x} (3 a d+2 b c)+\frac {2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac {2}{17} b^2 d^3 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 141, normalized size = 1.03 \begin {gather*} \frac {2 \left (-3315 a^2 c^3+29835 a^2 c^2 d x^2+5967 a^2 c d^2 x^4+1105 a^2 d^3 x^6+19890 a b c^3 x^2+11934 a b c^2 d x^4+6630 a b c d^2 x^6+1530 a b d^3 x^8+1989 b^2 c^3 x^4+3315 b^2 c^2 d x^6+2295 b^2 c d^2 x^8+585 b^2 d^3 x^{10}\right )}{9945 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 129, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (585 \, b^{2} d^{3} x^{10} + 765 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 1105 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 3315 \, a^{2} c^{3} + 1989 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 9945 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{9945 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 135, normalized size = 0.99 \begin {gather*} \frac {2}{17} \, b^{2} d^{3} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c d^{2} x^{\frac {13}{2}} + \frac {4}{13} \, a b d^{3} x^{\frac {13}{2}} + \frac {2}{3} \, b^{2} c^{2} d x^{\frac {9}{2}} + \frac {4}{3} \, a b c d^{2} x^{\frac {9}{2}} + \frac {2}{9} \, a^{2} d^{3} x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c^{3} x^{\frac {5}{2}} + \frac {12}{5} \, a b c^{2} d x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c d^{2} x^{\frac {5}{2}} + 4 \, a b c^{3} \sqrt {x} + 6 \, a^{2} c^{2} d \sqrt {x} - \frac {2 \, a^{2} c^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 1.01 \begin {gather*} -\frac {2 \left (-585 b^{2} d^{3} x^{10}-1530 a b \,d^{3} x^{8}-2295 b^{2} c \,d^{2} x^{8}-1105 a^{2} d^{3} x^{6}-6630 a b c \,d^{2} x^{6}-3315 b^{2} c^{2} d \,x^{6}-5967 a^{2} c \,d^{2} x^{4}-11934 a b \,c^{2} d \,x^{4}-1989 b^{2} c^{3} x^{4}-29835 a^{2} c^{2} d \,x^{2}-19890 a b \,c^{3} x^{2}+3315 a^{2} c^{3}\right )}{9945 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 127, normalized size = 0.93 \begin {gather*} \frac {2}{17} \, b^{2} d^{3} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {9}{2}} - \frac {2 \, a^{2} c^{3}}{3 \, x^{\frac {3}{2}}} + \frac {2}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {5}{2}} + 2 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 119, normalized size = 0.87 \begin {gather*} x^{5/2}\,\left (\frac {6\,a^2\,c\,d^2}{5}+\frac {12\,a\,b\,c^2\,d}{5}+\frac {2\,b^2\,c^3}{5}\right )+x^{9/2}\,\left (\frac {2\,a^2\,d^3}{9}+\frac {4\,a\,b\,c\,d^2}{3}+\frac {2\,b^2\,c^2\,d}{3}\right )-\frac {2\,a^2\,c^3}{3\,x^{3/2}}+\frac {2\,b^2\,d^3\,x^{17/2}}{17}+2\,a\,c^2\,\sqrt {x}\,\left (3\,a\,d+2\,b\,c\right )+\frac {2\,b\,d^2\,x^{13/2}\,\left (2\,a\,d+3\,b\,c\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.05, size = 189, normalized size = 1.38 \begin {gather*} - \frac {2 a^{2} c^{3}}{3 x^{\frac {3}{2}}} + 6 a^{2} c^{2} d \sqrt {x} + \frac {6 a^{2} c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 a^{2} d^{3} x^{\frac {9}{2}}}{9} + 4 a b c^{3} \sqrt {x} + \frac {12 a b c^{2} d x^{\frac {5}{2}}}{5} + \frac {4 a b c d^{2} x^{\frac {9}{2}}}{3} + \frac {4 a b d^{3} x^{\frac {13}{2}}}{13} + \frac {2 b^{2} c^{3} x^{\frac {5}{2}}}{5} + \frac {2 b^{2} c^{2} d x^{\frac {9}{2}}}{3} + \frac {6 b^{2} c d^{2} x^{\frac {13}{2}}}{13} + \frac {2 b^{2} d^{3} x^{\frac {17}{2}}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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