Optimal. Leaf size=65 \[ -\frac {a c \sqrt {c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac {c \sqrt {c x^2} (a+b x)^{2+n}}{b^2 (2+n) x} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45}
\begin {gather*} \frac {c \sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a c \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x (a+b x)^n \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx}{x}\\ &=-\frac {a c \sqrt {c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac {c \sqrt {c x^2} (a+b x)^{2+n}}{b^2 (2+n) x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 0.71 \begin {gather*} \frac {c^2 x (a+b x)^{1+n} (-a+b (1+n) x)}{b^2 (1+n) (2+n) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 46, normalized size = 0.71
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (c \,x^{2}\right )^{\frac {3}{2}} \left (-b n x -b x +a \right )}{x^{3} b^{2} \left (n^{2}+3 n +2\right )}\) | \(46\) |
risch | \(-\frac {c \sqrt {c \,x^{2}}\, \left (-b^{2} n \,x^{2}-a b n x -x^{2} b^{2}+a^{2}\right ) \left (b x +a \right )^{n}}{x \,b^{2} \left (2+n \right ) \left (1+n \right )}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 51, normalized size = 0.78 \begin {gather*} \frac {{\left (b^{2} c^{\frac {3}{2}} {\left (n + 1\right )} x^{2} + a b c^{\frac {3}{2}} n x - a^{2} c^{\frac {3}{2}}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 68, normalized size = 1.05 \begin {gather*} \frac {{\left (a b c n x - a^{2} c + {\left (b^{2} c n + b^{2} c\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\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 {a^{n} \left (c x^{2}\right )^{\frac {3}{2}}}{2 x} & \text {for}\: b = 0 \\\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{b^{2} n^{2} x^{3} + 3 b^{2} n x^{3} + 2 b^{2} x^{3}} + \frac {a b n x \left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{b^{2} n^{2} x^{3} + 3 b^{2} n x^{3} + 2 b^{2} x^{3}} + \frac {b^{2} n x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{b^{2} n^{2} x^{3} + 3 b^{2} n x^{3} + 2 b^{2} x^{3}} + \frac {b^{2} x^{2} \left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{b^{2} n^{2} x^{3} + 3 b^{2} n x^{3} + 2 b^{2} x^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 119, normalized size = 1.83 \begin {gather*} {\left (\frac {a^{2} a^{n} \mathrm {sgn}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} + \frac {{\left (b x + a\right )}^{n} b^{2} n x^{2} \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} a b n x \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} b^{2} x^{2} \mathrm {sgn}\left (x\right ) - {\left (b x + a\right )}^{n} a^{2} \mathrm {sgn}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}}\right )} c^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 88, normalized size = 1.35 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {c\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{n^2+3\,n+2}-\frac {a^2\,c\,\sqrt {c\,x^2}}{b^2\,\left (n^2+3\,n+2\right )}+\frac {a\,c\,n\,x\,\sqrt {c\,x^2}}{b\,\left (n^2+3\,n+2\right )}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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