Optimal. Leaf size=173 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}+\frac {\left (8 a c+b^2+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n} \]
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Rubi [A] time = 0.16, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1357, 734, 814, 843, 621, 206, 724} \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}+\frac {\left (8 a c+b^2+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 734
Rule 814
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {\operatorname {Subst}\left (\int \frac {(-2 a-b x) \sqrt {a+b x+c x^2}}{x} \, dx,x,x^n\right )}{2 n}\\ &=\frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}+\frac {\operatorname {Subst}\left (\int \frac {8 a^2 c-\frac {1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{8 c n}\\ &=\frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{n}-\frac {\left (b \left (b^2-12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^n\right )}{16 c n}\\ &=\frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {\left (2 a^2\right ) \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 )}{n}-\frac {\left (b \left (b^2-12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^n}{\sqrt {a+b x^n+c x^{2 n}}}\right )}{8 c n}\\ &=\frac {\left (b^2+8 a c+2 b c x^n\right ) \sqrt {a+b x^n+c x^{2 n}}}{8 c n}+\frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{3 n}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+b x^n+c x^{2 n}}}\right )}{n}-\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+b x^n+c x^{2 n}}}\right )}{16 c^{3/2} n}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 158, normalized size = 0.91 \[ \frac {-48 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {2 a+b x^n}{2 \sqrt {a} \sqrt {a+x^n \left (b+c x^n\right )}}\right )-3 b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )}}\right )+2 \sqrt {c} \sqrt {a+x^n \left (b+c x^n\right )} \left (8 c \left (4 a+c x^{2 n}\right )+3 b^2+14 b c x^n\right )}{48 c^{3/2} n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 827, normalized size = 4.78 \[ \left [\frac {48 \, a^{\frac {3}{2}} c^{2} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{96 \, c^{2} n}, \frac {24 \, a^{\frac {3}{2}} c^{2} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{48 \, c^{2} n}, \frac {96 \, \sqrt {-a} a c^{2} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \, {\left (2 \, c^{\frac {3}{2}} x^{n} + b \sqrt {c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{96 \, c^{2} n}, \frac {48 \, \sqrt {-a} a c^{2} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-c} \arctan \left (\frac {{\left (2 \, \sqrt {-c} c x^{n} + b \sqrt {-c}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{2 \, {\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, {\left (8 \, c^{3} x^{2 \, n} + 14 \, b c^{2} x^{n} + 3 \, b^{2} c + 32 \, a c^{2}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{48 \, c^{2} 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 \[ \int \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 209, normalized size = 1.21 \[ -\frac {a^{\frac {3}{2}} \ln \left (\left (b \,{\mathrm e}^{n \ln \relax (x )}+2 a +2 \sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}\, \sqrt {a}\right ) {\mathrm e}^{-n \ln \relax (x )}\right )}{n}+\frac {3 a b \ln \left (\frac {c \,{\mathrm e}^{n \ln \relax (x )}+\frac {b}{2}}{\sqrt {c}}+\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}\right )}{4 \sqrt {c}\, n}-\frac {b^{3} \ln \left (\frac {c \,{\mathrm e}^{n \ln \relax (x )}+\frac {b}{2}}{\sqrt {c}}+\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}\right )}{16 c^{\frac {3}{2}} n}+\frac {\left (14 b c \,{\mathrm e}^{n \ln \relax (x )}+8 c^{2} {\mathrm e}^{2 n \ln \relax (x )}+32 a c +3 b^{2}\right ) \sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+c \,{\mathrm e}^{2 n \ln \relax (x )}+a}}{24 c n} \]
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 \[ \int \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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