Optimal. Leaf size=107 \[ \frac {2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}} \]
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Rubi [A] time = 0.19, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2031, 2030, 2029, 206} \[ \frac {2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rule 2030
Rule 2031
Rubi steps
\begin {align*} \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx &=\frac {\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac {x^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx}{c}\\ &=-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx}{a c}\\ &=-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {\left (2 x^{-3 j/2} (c x)^{3 j/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a c (j-n)}\\ &=-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 117, normalized size = 1.09 \[ -\frac {2 x^{-3 j/2} (c x)^{3 j/2} \left (\sqrt {a} x^{j/2}-\sqrt {b} x^{n/2} \sqrt {\frac {a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{a^{3/2} c (j-n) \sqrt {a x^j+b x^n}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {3 j}{2}-1}}{\left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{\frac {3\,j}{2}-1}}{{\left (a\,x^j+b\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (c x\right )^{\frac {3 j}{2} - 1}}{\left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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