Optimal. Leaf size=64 \[ \frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 64, 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, 43} \begin {gather*} \frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x}\\ &=\frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 36, normalized size = 0.56 \begin {gather*} \frac {1}{12} c^2 x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 35, normalized size = 0.55 \begin {gather*} \frac {\left (c x^2\right )^{5/2} \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 40, normalized size = 0.62 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{2} c^{2} x^{3} + 8 \, a b c^{2} x^{2} + 6 \, a^{2} c^{2} x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 44, normalized size = 0.69 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{2} c^{2} x^{4} \mathrm {sgn}\relax (x) + 8 \, a b c^{2} x^{3} \mathrm {sgn}\relax (x) + 6 \, a^{2} c^{2} x^{2} \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 = 32, normalized size = 0.50 \begin {gather*} \frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{3}} \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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.01, size = 60, normalized size = 0.94 \begin {gather*} \frac {a^{2} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} + \frac {2 a b c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b^{2} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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