Optimal. Leaf size=244 \[ \frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {3 b \sqrt {x} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt {a x+b x^3+c x^5}}-\frac {x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \]
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Rubi [A] time = 0.36, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1919, 1945, 1949, 12, 1914, 1107, 621, 206} \begin {gather*} \frac {\left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (4 c x^2 \left (5 b^2-16 a c\right )+b \left (5 b^2-4 a c\right )\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}-\frac {3 b \sqrt {x} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1107
Rule 1914
Rule 1919
Rule 1945
Rule 1949
Rubi steps
\begin {align*} \int x^{3/2} \left (a x+b x^3+c x^5\right )^{3/2} \, dx &=\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac {3 \int \sqrt {x} \left (-2 a b-\left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5} \, dx}{80 c}\\ &=-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}+\frac {\int \frac {x^{3/2} \left (2 a b \left (5 b^2-28 a c\right )+\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{640 c^2}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\int \frac {15 b \left (b^2-4 a c\right )^2 x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{1280 c^3}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \int \frac {x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{256 c^3}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx}{256 c^3 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 c^3 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt {a x+b x^3+c x^5}}{1280 c^3 \sqrt {x}}-\frac {x^{3/2} \left (b \left (5 b^2-4 a c\right )+4 c \left (5 b^2-16 a c\right ) x^2\right ) \sqrt {a x+b x^3+c x^5}}{640 c^2}+\frac {\sqrt {x} \left (3 b+8 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{80 c}-\frac {3 b \left (b^2-4 a c\right )^2 \sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 192, normalized size = 0.79 \begin {gather*} \frac {\left (x \left (a+b x^2+c x^4\right )\right )^{3/2} \left (-\frac {3 b \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )-2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}\right )}{256 c^{7/2}}-\frac {b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 c^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )}{2 x^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.92, size = 214, normalized size = 0.88 \begin {gather*} \frac {3 \left (16 a^2 b c^2-8 a b^3 c+b^5\right ) \log \left (-2 \sqrt {c} \sqrt {a x+b x^3+c x^5}+b \sqrt {x}+2 c x^{5/2}\right )}{512 c^{7/2}}-\frac {3 \log \left (\sqrt {x}\right ) \left (16 a^2 b c^2-8 a b^3 c+b^5\right )}{512 c^{7/2}}+\frac {\sqrt {a x+b x^3+c x^5} \left (128 a^2 c^2-100 a b^2 c+56 a b c^2 x^2+256 a c^3 x^4+15 b^4-10 b^3 c x^2+8 b^2 c^2 x^4+176 b c^3 x^6+128 c^4 x^8\right )}{1280 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 396, normalized size = 1.62 \begin {gather*} \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{5120 \, c^{4} x}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{8} + 176 \, b c^{4} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{4} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{2560 \, c^{4} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.11, size = 662, normalized size = 2.71 \begin {gather*} \frac {1}{96} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {3 \, b^{2} - 8 \, a c}{c^{2}}\right )} - \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {5}{2}}} + \frac {3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}}{c^{\frac {5}{2}}}\right )} a + \frac {1}{768} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {5 \, b^{2} c - 12 \, a c^{2}}{c^{3}}\right )} x^{2} + \frac {15 \, b^{3} - 52 \, a b c}{c^{3}}\right )} + \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {7}{2}}} - \frac {15 \, b^{4} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{3} \sqrt {c} - 104 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}}{c^{\frac {7}{2}}}\right )} b + \frac {1}{7680} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {9}{2}}} + \frac {105 \, b^{5} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{4} \sqrt {c} - 920 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 512 \, a^{\frac {5}{2}} c^{\frac {5}{2}}}{c^{\frac {9}{2}}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 369, normalized size = 1.51 \begin {gather*} -\frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (-256 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {9}{2}} x^{8}-352 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{\frac {7}{2}} x^{6}-512 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{\frac {7}{2}} x^{4}-16 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c^{\frac {5}{2}} x^{4}-112 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,c^{\frac {5}{2}} x^{2}+20 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} c^{\frac {3}{2}} x^{2}+240 a^{2} b \,c^{2} \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )-120 a \,b^{3} c \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )+15 b^{5} \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )-256 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} c^{\frac {5}{2}}+200 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,b^{2} c^{\frac {3}{2}}-30 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4} \sqrt {c}\right )}{2560 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {7}{2}} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} x^{\frac {3}{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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