Optimal. Leaf size=31 \[ \frac {c \sqrt {c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
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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} \begin {gather*} \frac {c \sqrt {c x^2} (a+b x)^{n+1}}{b (n+1) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^3} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int (a+b x)^n \, dx}{x}\\ &=\frac {c \sqrt {c x^2} (a+b x)^{1+n}}{b (1+n) x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.97 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (a+b x)^{n+1}}{b (n+1) x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.01, size = 33, normalized size = 1.06 \begin {gather*} \frac {{\left (b c x + a c\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 42, normalized size = 1.35 \begin {gather*} -c^{\frac {3}{2}} {\left (\frac {a^{n + 1} \mathrm {sgn}\relax (x)}{b n + b} - \frac {{\left (b x + a\right )}^{n + 1} \mathrm {sgn}\relax (x)}{b {\left (n + 1\right )}}\right )} \end {gather*}
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 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{n +1}}{\left (n +1\right ) b \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 28, normalized size = 0.90 \begin {gather*} \frac {{\left (b c^{\frac {3}{2}} x + a c^{\frac {3}{2}}\right )} {\left (b x + a\right )}^{n}}{b {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 45, normalized size = 1.45 \begin {gather*} \frac {\left (\frac {c\,x\,\sqrt {c\,x^2}}{n+1}+\frac {a\,c\,\sqrt {c\,x^2}}{b\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{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 {c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{a x^{2}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{x^{2}} & \text {for}\: b = 0 \\\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a c^{\frac {3}{2}} \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {3}{2}}}{b n x^{3} + b x^{3}} + \frac {b c^{\frac {3}{2}} x \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {3}{2}}}{b n x^{3} + b x^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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