Optimal. Leaf size=98 \[ \frac {2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{a^{3/2} n} \]
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Rubi [A] time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1357, 740, 12, 724, 206} \[ \frac {2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{a^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 740
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^n\right )}{n}\\ &=\frac {2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {b^2}{2}+2 a c}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{a \left (b^2-4 a c\right ) n}\\ &=\frac {2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{a n}\\ &=\frac {2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{a n}\\ &=\frac {2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{a^{3/2} n}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 94, normalized size = 0.96 \[ \frac {\frac {2 \left (-2 a c+b^2+b c x^n\right )}{a \left (b^2-4 a c\right ) \sqrt {a+x^n \left (b+c x^n\right )}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+x^n \left (b+c x^n\right )}}\right )}{a^{3/2}}}{n} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 449, normalized size = 4.58 \[ \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {a} x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} \sqrt {a} x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {a}\right )} \log \left (-\frac {8 \, a b x^{n} + 8 \, a^{2} + {\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \, {\left (\sqrt {a} b x^{n} + 2 \, a^{\frac {3}{2}}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 4 \, {\left (a b c x^{n} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} n x^{2 \, n} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} n x^{n} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {-a} x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} \sqrt {-a} x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {-a}\right )} \arctan \left (\frac {{\left (\sqrt {-a} b x^{n} + 2 \, \sqrt {-a} a\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) + 2 \, {\left (a b c x^{n} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} n x^{2 \, n} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} n x^{n} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}} x}\, dx \]
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 \[ \int \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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