Optimal. Leaf size=177 \[ \frac {3 \sqrt {x} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{256 c^{5/2} \sqrt {a x+b x^3+c x^5}}-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1918, 1914, 1107, 621, 206} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {3 \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 )}{256 c^{5/2} \sqrt {a x+b x^3+c x^5}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 1107
Rule 1914
Rule 1918
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3+c x^5\right )^{3/2}}{\sqrt {x}} \, dx &=\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx}{16 c}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int \frac {x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{128 c^2}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac {\left (3 \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}{128 c^2 \sqrt {a x+b x^3+c x^5}}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac {\left (3 \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 )}{256 c^2 \sqrt {a x+b x^3+c x^5}}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac {\left (3 \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 )}{128 c^2 \sqrt {a x+b x^3+c x^5}}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{128 c^2 \sqrt {x}}+\frac {\left (b+2 c x^2\right ) \left (a x+b x^3+c x^5\right )^{3/2}}{16 c x^{3/2}}+\frac {3 \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 )}{256 c^{5/2} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 152, normalized size = 0.86 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{256 c^{5/2} \sqrt {x \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.52, size = 181, normalized size = 1.02 \begin {gather*} -\frac {3 \left (16 a^2 c^2-8 a b^2 c+b^4\right ) \log \left (-2 \sqrt {c} \sqrt {a x+b x^3+c x^5}+b \sqrt {x}+2 c x^{5/2}\right )}{256 c^{5/2}}+\frac {3 \log \left (\sqrt {x}\right ) \left (16 a^2 c^2-8 a b^2 c+b^4\right )}{256 c^{5/2}}+\frac {\sqrt {a x+b x^3+c x^5} \left (20 a b c+40 a c^2 x^2-3 b^3+2 b^2 c x^2+24 b c^2 x^4+16 c^3 x^6\right )}{128 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 332, normalized size = 1.88 \begin {gather*} \left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} 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 (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{512 \, c^{3} x}, -\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} 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 (16 \, c^{4} x^{6} + 24 \, b c^{3} x^{4} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \, {\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{256 \, c^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.58, size = 518, normalized size = 2.93 \begin {gather*} \frac {1}{16} \, {\left (2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, x^{2} + \frac {b}{c}\right )} + \frac {{\left (b^{2} - 4 \, a 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 {3}{2}}} - \frac {b^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2 \, \sqrt {a} b \sqrt {c}}{c^{\frac {3}{2}}}\right )} a + \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 )} b + \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 )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 295, normalized size = 1.67 \begin {gather*} \frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (32 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {7}{2}} x^{6}+48 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{\frac {5}{2}} x^{4}+80 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,c^{\frac {5}{2}} x^{2}+4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c^{\frac {3}{2}} x^{2}+48 a^{2} 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 )-24 a \,b^{2} c \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3 b^{4} \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )+40 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,c^{\frac {3}{2}}-6 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} \sqrt {c}\right )}{256 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {5}{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 \frac {{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}}}{\sqrt {x}}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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