Optimal. Leaf size=145 \[ -\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )-\frac {3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{8 \sqrt {c}} \]
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Rubi [A]
time = 0.09, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1356, 746, 828,
857, 635, 212, 738} \begin {gather*} -\frac {3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right )}{8 \sqrt {c}}+x \left (a+\frac {b}{x}+\frac {c}{x^2}\right )^{3/2}-\frac {3}{4} \left (3 b+\frac {2 c}{x}\right ) \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 828
Rule 857
Rule 1356
Rubi steps
\begin {align*} \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} \, dx &=-\text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x-\frac {3}{2} \text {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+\frac {3 \text {Subst}\left (\int \frac {-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )}{8 c}\\ &=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x-\frac {1}{2} (3 a b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 \left (b^2+4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+(3 a b) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+\frac {b}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )-\frac {1}{4} \left (3 \left (b^2+4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+\frac {2 c}{x}}{\sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )\\ &=-\frac {3}{4} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} \left (3 b+\frac {2 c}{x}\right )+\left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{3/2} x+\frac {3}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+\frac {b}{x}}{2 \sqrt {a} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )-\frac {3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+\frac {2 c}{x}}{2 \sqrt {c} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 158, normalized size = 1.09 \begin {gather*} \frac {\sqrt {a+\frac {c+b x}{x^2}} \left (3 \left (b^2+4 a c\right ) x^2 \tanh ^{-1}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )-\sqrt {c} \left ((2 c+x (5 b-4 a x)) \sqrt {c+x (b+a x)}+6 \sqrt {a} b x^2 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{4 \sqrt {c} x \sqrt {c+x (b+a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(333\) vs.
\(2(119)=238\).
time = 0.04, size = 334, normalized size = 2.30
method | result | size |
risch | \(-\frac {\left (5 b x +2 c \right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{4 x}+\frac {\left (a \sqrt {a \,x^{2}+b x +c}+\frac {3 \sqrt {a}\, b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2}-\frac {3 \sqrt {c}\, \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a}{2}-\frac {3 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2}}{8 \sqrt {c}}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b x +c}}\) | \(179\) |
default | \(-\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {3}{2}} x \left (12 a^{\frac {5}{2}} c^{\frac {5}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) x^{2}-2 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,x^{3}-4 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c \,x^{2}-6 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b c \,x^{3}+3 a^{\frac {3}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{2} x^{2}-12 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{2} x^{2}+2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b x -2 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} x^{2}+4 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c \,a^{\frac {3}{2}}-6 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c \,x^{2}-12 a^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b \,c^{2} x^{2}\right )}{8 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{2} a^{\frac {3}{2}}}\) | \(334\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 709, normalized size = 4.89 \begin {gather*} \left [\frac {12 \, \sqrt {a} b c x \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) + 4 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{16 \, c x}, -\frac {24 \, \sqrt {-a} b c x \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) - 4 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{16 \, c x}, \frac {6 \, \sqrt {a} b c x \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) + 2 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{8 \, c x}, -\frac {12 \, \sqrt {-a} b c x \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) - 2 \, {\left (4 \, a c x^{2} - 5 \, b c x - 2 \, c^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{8 \, c x}\right ] \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 \left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+\frac {b}{x}+\frac {c}{x^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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