Optimal. Leaf size=66 \[ \frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \left (c x^2\right )^{5/2} (a+b x)^2 \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx}{x}\\ &=\frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.50 \[ \frac {1}{168} x \left (c x^2\right )^{5/2} \left (28 a^2+48 a b x+21 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 42, normalized size = 0.64 \[ \frac {1}{168} \, {\left (21 \, b^{2} c^{2} x^{7} + 48 \, a b c^{2} x^{6} + 28 \, a^{2} c^{2} x^{5}\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 44, normalized size = 0.67 \[ \frac {1}{168} \, {\left (21 \, b^{2} c^{2} x^{8} \mathrm {sgn}\relax (x) + 48 \, a b c^{2} x^{7} \mathrm {sgn}\relax (x) + 28 \, a^{2} c^{2} x^{6} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.45 \[ \frac {\left (21 b^{2} x^{2}+48 a b x +28 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}} x}{168} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 44, normalized size = 0.67 \[ \frac {1}{6} \, \left (c x^{2}\right )^{\frac {5}{2}} a^{2} x + \frac {\left (c x^{2}\right )^{\frac {7}{2}} b^{2} x}{8 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {7}{2}} a b}{7 \, c} \]
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 \[ \int {\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.80, size = 60, normalized size = 0.91 \[ \frac {a^{2} c^{\frac {5}{2}} x \left (x^{2}\right )^{\frac {5}{2}}}{6} + \frac {2 a b c^{\frac {5}{2}} x^{2} \left (x^{2}\right )^{\frac {5}{2}}}{7} + \frac {b^{2} c^{\frac {5}{2}} x^{3} \left (x^{2}\right )^{\frac {5}{2}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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