Optimal. Leaf size=29 \[ \frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x} \]
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Rubi [A] time = 0.00, antiderivative size = 29, 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^2 \sqrt {c x^2} (a+b x)^3}{3 b 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 )^{5/2} (a+b x)^2}{x^5} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int (a+b x)^2 \, dx}{x}\\ &=\frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} \frac {\left (c x^2\right )^{5/2} (a+b x)^3}{3 b x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 1.17 \begin {gather*} \frac {\left (c x^2\right )^{5/2} \left (3 a^2+3 a b x+b^2 x^2\right )}{3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 36, normalized size = 1.24 \begin {gather*} \frac {1}{3} \, {\left (b^{2} c^{2} x^{2} + 3 \, a b c^{2} x + 3 \, a^{2} c^{2}\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 41, normalized size = 1.41 \begin {gather*} \frac {1}{3} \, {\left (b^{2} c^{2} x^{3} \mathrm {sgn}\relax (x) + 3 \, a b c^{2} x^{2} \mathrm {sgn}\relax (x) + 3 \, a^{2} c^{2} x \mathrm {sgn}\relax (x)\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.07 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{3 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = -0.03 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.03, size = 56, normalized size = 1.93 \begin {gather*} \frac {a^{2} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{x^{4}} + \frac {a b c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{x^{3}} + \frac {b^{2} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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