Optimal. Leaf size=143 \[ \frac {a^3 c x^5 \sqrt {c \left (a+b x^2\right )^2}}{5 \left (a+b x^2\right )}+\frac {3 a^2 b c x^7 \sqrt {c \left (a+b x^2\right )^2}}{7 \left (a+b x^2\right )}+\frac {b^3 c x^{11} \sqrt {c \left (a+b x^2\right )^2}}{11 \left (a+b x^2\right )}+\frac {a b^2 c x^9 \sqrt {c \left (a+b x^2\right )^2}}{3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 187, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1989, 1112, 270} \[ \frac {b^3 c x^{11} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{11 \left (a+b x^2\right )}+\frac {a b^2 c x^9 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b c x^7 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{7 \left (a+b x^2\right )}+\frac {a^3 c x^5 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rule 1989
Rubi steps
\begin {align*} \int x^4 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\int x^4 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \, dx\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int x^4 \left (a b c+b^2 c x^2\right )^3 \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int \left (a^3 b^3 c^3 x^4+3 a^2 b^4 c^3 x^6+3 a b^5 c^3 x^8+b^6 c^3 x^{10}\right ) \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {a^3 c x^5 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{5 \left (a+b x^2\right )}+\frac {3 a^2 b c x^7 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{7 \left (a+b x^2\right )}+\frac {a b^2 c x^9 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{3 \left (a+b x^2\right )}+\frac {b^3 c x^{11} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{11 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.44 \[ \frac {x^5 \left (231 a^3+495 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{1155 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 74, normalized size = 0.52 \[ \frac {{\left (105 \, b^{3} c x^{11} + 385 \, a b^{2} c x^{9} + 495 \, a^{2} b c x^{7} + 231 \, a^{3} c x^{5}\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{1155 \, {\left (b x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 72, normalized size = 0.50 \[ \frac {1}{1155} \, {\left (105 \, b^{3} x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + 385 \, a b^{2} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + 495 \, a^{2} b x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 231 \, a^{3} x^{5} \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.42 \[ \frac {\left (105 b^{3} x^{6}+385 a \,b^{2} x^{4}+495 a^{2} b \,x^{2}+231 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} x^{5}}{1155 \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 47, normalized size = 0.33 \[ \frac {1}{11} \, b^{3} c^{\frac {3}{2}} x^{11} + \frac {1}{3} \, a b^{2} c^{\frac {3}{2}} x^{9} + \frac {3}{7} \, a^{2} b c^{\frac {3}{2}} x^{7} + \frac {1}{5} \, a^{3} c^{\frac {3}{2}} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (c\,{\left (b\,x^2+a\right )}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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