Optimal. Leaf size=31 \[ \frac {x (a+b x)^{1+n}}{b c^2 (1+n) \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32}
\begin {gather*} \frac {x (a+b x)^{n+1}}{b c^2 (n+1) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x^5 (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int (a+b x)^n \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x (a+b x)^{1+n}}{b c^2 (1+n) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {x (a+b x)^{1+n}}{b c^2 (1+n) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 29, normalized size = 0.94
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1+n} x^{5}}{b \left (1+n \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(29\) |
risch | \(\frac {x \left (b x +a \right ) \left (b x +a \right )^{n}}{c^{2} \sqrt {c \,x^{2}}\, b \left (1+n \right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 31, normalized size = 1.00 \begin {gather*} \frac {{\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{b c^{3} {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 37, normalized size = 1.19 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a\right )} {\left (b x + a\right )}^{n}}{{\left (b c^{3} n + b c^{3}\right )} 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*} \begin {cases} \frac {x^{6}}{a \left (c x^{2}\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} x^{6}}{\left (c x^{2}\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{5}}{\left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a x^{5} \left (a + b x\right )^{n}}{b n \left (c x^{2}\right )^{\frac {5}{2}} + b \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b x^{6} \left (a + b x\right )^{n}}{b n \left (c x^{2}\right )^{\frac {5}{2}} + b \left (c x^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [B]
time = 0.23, size = 42, normalized size = 1.35 \begin {gather*} \frac {\left (\frac {x^2}{c^2\,\left (n+1\right )}+\frac {a\,x}{b\,c^2\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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