Optimal. Leaf size=87 \[ -\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}}+\frac {2 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c (1-n) \sqrt {c x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2053, 2056,
2054, 212} \begin {gather*} \frac {2 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c (1-n) \sqrt {c x}}-\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2053
Rule 2054
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt {a x+b x^n}}{(c x)^{3/2}} \, dx &=-\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}}+\frac {a \int \frac {1}{\sqrt {c x} \sqrt {a x+b x^n}} \, dx}{c}\\ &=-\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}}+\frac {\left (a \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a x+b x^n}} \, dx}{c \sqrt {c x}}\\ &=-\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}}+\frac {\left (2 a \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c (1-n) \sqrt {c x}}\\ &=-\frac {2 \sqrt {a x+b x^n}}{c (1-n) \sqrt {c x}}+\frac {2 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c (1-n) \sqrt {c x}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 100, normalized size = 1.15 \begin {gather*} \frac {x \left (2 a x+2 b x^n-2 \sqrt {a} \sqrt {b} x^{\frac {1+n}{2}} \sqrt {1+\frac {a x^{1-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{(-1+n) (c x)^{3/2} \sqrt {a x+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a x +b \,x^{n}}}{\left (c x \right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + b x^{n}}}{\left (c x\right )^{\frac {3}{2}}}\, dx \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 {\sqrt {b\,x^n+a\,x}}{{\left (c\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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