Optimal. Leaf size=141 \[ -\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {2 a^{3/2} x^{3 j/2} (c x)^{-3 j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)} \]
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Rubi [A]
time = 0.17, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2056, 2053,
2054, 212} \begin {gather*} \frac {2 a^{3/2} 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 )}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}-\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2053
Rule 2054
Rule 2056
Rubi steps
\begin {align*} \int (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 x^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx}{c}\\ &=-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {\left (a x^{3 j/2} (c x)^{-3 j/2}\right ) \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx}{c}\\ &=-\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {\left (a^2 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}{c}\\ &=-\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {\left (2 a^2 x^{3 j/2} (c x)^{-3 j/2}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}\\ &=-\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {2 a^{3/2} 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 )}{c (j-n)}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 131, normalized size = 0.93 \begin {gather*} -\frac {2 (c x)^{-3 j/2} \left (4 a^2 x^{2 j}+b^2 x^{2 n}+5 a b x^{j+n}-3 a^{3/2} \sqrt {b} x^{\frac {1}{2} (3 j+n)} \sqrt {1+\frac {a x^{j-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{3 c (j-n) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{-1-\frac {3 j}{2}} \left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 (a\,x^j+b\,x^n\right )}^{3/2}}{{\left (c\,x\right )}^{\frac {3\,j}{2}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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