Optimal. Leaf size=91 \[ \frac {2 \sqrt {a} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt {x}}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2028, 2031, 2029, 206} \begin {gather*} \frac {2 \sqrt {a} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt {x}}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^3+b x^n}}{(c x)^{5/2}} \, dx &=-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}+\frac {a \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx}{c^3}\\ &=-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}+\frac {\left (a \sqrt {c x}\right ) \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^n}} \, dx}{c^3 \sqrt {x}}\\ &=-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}+\frac {\left (2 a \sqrt {c x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt {x}}\\ &=-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}+\frac {2 \sqrt {a} \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 103, normalized size = 1.13 \begin {gather*} \frac {2 x \left (-\sqrt {a} \sqrt {b} x^{\frac {n+3}{2}} \sqrt {\frac {a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )+a x^3+b x^n\right )}{(n-3) (c x)^{5/2} \sqrt {a x^3+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^3+b x^n}}{(c x)^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x^{n}}}{\left (c x\right )^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a \,x^{3}+b \,x^{n}}}{\left (c x \right )^{\frac {5}{2}}}\, dx \end {gather*}
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*} \int \frac {\sqrt {a x^{3} + b x^{n}}}{\left (c x\right )^{\frac {5}{2}}}\,{d x} \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 {\sqrt {b\,x^n+a\,x^3}}{{\left (c\,x\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x^{n}}}{\left (c x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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