Optimal. Leaf size=32 \[ \frac {c \left (a+b x^2\right )^4 \sqrt {c \left (a+b x^2\right )^3}}{11 b} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ \frac {c \left (a+b x^2\right )^4 \sqrt {c \left (a+b x^2\right )^3}}{11 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 1591
Rubi steps
\begin {align*} \int x \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (c x^3\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int x^{9/2} \, dx,x,a+b x^2\right )}{2 b \left (a+b x^2\right )^{3/2}}\\ &=\frac {c \left (a+b x^2\right )^4 \sqrt {c \left (a+b x^2\right )^3}}{11 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.91 \[ \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^3\right )^{3/2}}{11 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 87, normalized size = 2.72 \[ \frac {{\left (b^{4} c x^{8} + 4 \, a b^{3} c x^{6} + 6 \, a^{2} b^{2} c x^{4} + 4 \, a^{3} b c x^{2} + a^{4} c\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{11 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 28, normalized size = 0.88 \[ \frac {{\left (b c x^{2} + a c\right )}^{\frac {11}{2}} \mathrm {sgn}\left (b x^{2} + a\right )}{11 \, b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 0.81 \[ \frac {\left (b \,x^{2}+a \right ) \left (\left (b \,x^{2}+a \right )^{3} c \right )^{\frac {3}{2}}}{11 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.08, size = 70, normalized size = 2.19 \[ \frac {{\left (b^{4} c^{\frac {3}{2}} x^{8} + 4 \, a b^{3} c^{\frac {3}{2}} x^{6} + 6 \, a^{2} b^{2} c^{\frac {3}{2}} x^{4} + 4 \, a^{3} b c^{\frac {3}{2}} x^{2} + a^{4} c^{\frac {3}{2}}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{11 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.71, size = 62, normalized size = 1.94 \[ \sqrt {c\,{\left (b\,x^2+a\right )}^3}\,\left (\frac {a^4\,c}{11\,b}+\frac {4\,a^3\,c\,x^2}{11}+\frac {b^3\,c\,x^8}{11}+\frac {6\,a^2\,b\,c\,x^4}{11}+\frac {4\,a\,b^2\,c\,x^6}{11}\right ) \]
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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