Optimal. Leaf size=31 \[ \frac {x (a+b x)^{n+1}}{b c (n+1) \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, 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} \[ \frac {x (a+b x)^{n+1}}{b c (n+1) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int (a+b x)^n \, dx}{c \sqrt {c x^2}}\\ &=\frac {x (a+b x)^{1+n}}{b c (1+n) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.97 \[ \frac {x^3 (a+b x)^{n+1}}{b (n+1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 37, normalized size = 1.19 \[ \frac {\sqrt {c x^{2}} {\left (b x + a\right )} {\left (b x + a\right )}^{n}}{{\left (b c^{2} n + b c^{2}\right )} x} \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.94 \[ \frac {x^{3} \left (b x +a \right )^{n +1}}{\left (n +1\right ) \left (c \,x^{2}\right )^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 31, normalized size = 1.00 \[ \frac {{\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{b c^{2} {\left (n + 1\right )}} \]
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 \[ \frac {\left (\frac {x^2}{c\,\left (n+1\right )}+\frac {a\,x}{b\,c\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \]
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 \[ \begin {cases} \frac {x^{4}}{a c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} x^{4}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a x^{3} \left (a + b x\right )^{n}}{b c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{4} \left (a + b x\right )^{n}}{b c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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