Optimal. Leaf size=100 \[ -\frac {2 a^{3/2} c^5 \tanh ^{-1}\left (\frac {\sqrt {a}}{x^2 \sqrt {\frac {a}{x^4}+b x^n}}\right )}{n+4}+\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (n+4)}+\frac {2 a c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n}}{n+4} \]
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Rubi [A] time = 0.21, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2028, 2029, 206} \[ -\frac {2 a^{3/2} c^5 \tanh ^{-1}\left (\frac {\sqrt {a}}{x^2 \sqrt {\frac {a}{x^4}+b x^n}}\right )}{n+4}+\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (n+4)}+\frac {2 a c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n}}{n+4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2028
Rule 2029
Rubi steps
\begin {align*} \int c^5 x^5 \left (\frac {a}{x^4}+b x^n\right )^{3/2} \, dx &=c^5 \int x^5 \left (\frac {a}{x^4}+b x^n\right )^{3/2} \, dx\\ &=\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (4+n)}+\left (a c^5\right ) \int x \sqrt {\frac {a}{x^4}+b x^n} \, dx\\ &=\frac {2 a c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n}}{4+n}+\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (4+n)}+\left (a^2 c^5\right ) \int \frac {1}{x^3 \sqrt {\frac {a}{x^4}+b x^n}} \, dx\\ &=\frac {2 a c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n}}{4+n}+\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (4+n)}-\frac {\left (2 a^2 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x^2 \sqrt {\frac {a}{x^4}+b x^n}}\right )}{4+n}\\ &=\frac {2 a c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n}}{4+n}+\frac {2 c^5 x^6 \left (\frac {a}{x^4}+b x^n\right )^{3/2}}{3 (4+n)}-\frac {2 a^{3/2} c^5 \tanh ^{-1}\left (\frac {\sqrt {a}}{x^2 \sqrt {\frac {a}{x^4}+b x^n}}\right )}{4+n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 96, normalized size = 0.96 \[ \frac {2 c^5 x^2 \sqrt {\frac {a}{x^4}+b x^n} \left (\sqrt {a+b x^{n+4}} \left (4 a+b x^{n+4}\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+4}}}{\sqrt {a}}\right )\right )}{3 (n+4) \sqrt {a+b x^{n+4}}} \]
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 \[ \int {\left (b x^{n} + \frac {a}{x^{4}}\right )}^{\frac {3}{2}} c^{5} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.73, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+\frac {a}{x^{4}}\right )^{\frac {3}{2}} c^{5} x^{5}\, dx \]
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 \[ c^{5} \int {\left (b x^{n} + \frac {a}{x^{4}}\right )}^{\frac {3}{2}} x^{5}\,{d x} \]
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 \[ \int c^5\,x^5\,{\left (b\,x^n+\frac {a}{x^4}\right )}^{3/2} \,d x \]
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 \[ c^{5} \left (\int a x \sqrt {\frac {a}{x^{4}} + b x^{n}}\, dx + \int b x^{5} x^{n} \sqrt {\frac {a}{x^{4}} + b x^{n}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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