Optimal. Leaf size=135 \[ \frac {a^3 c x \sqrt {c \left (a+b x^2\right )^2}}{a+b x^2}+\frac {a^2 b c x^3 \sqrt {c \left (a+b x^2\right )^2}}{a+b x^2}+\frac {b^3 c x^7 \sqrt {c \left (a+b x^2\right )^2}}{7 \left (a+b x^2\right )}+\frac {3 a b^2 c x^5 \sqrt {c \left (a+b x^2\right )^2}}{5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 175, normalized size of antiderivative = 1.30, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1988, 1088, 194} \[ \frac {b^3 x^7 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}+\frac {3 a b^2 x^5 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac {a^2 b x^3 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {a^3 x \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{\left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 194
Rule 1088
Rule 1988
Rubi steps
\begin {align*} \int \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\int \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \, dx\\ &=\frac {\left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \int \left (2 a b c+2 b^2 c x^2\right )^3 \, dx}{\left (2 a b c+2 b^2 c x^2\right )^3}\\ &=\frac {\left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \int \left (8 a^3 b^3 c^3+24 a^2 b^4 c^3 x^2+24 a b^5 c^3 x^4+8 b^6 c^3 x^6\right ) \, dx}{\left (2 a b c+2 b^2 c x^2\right )^3}\\ &=\frac {a^3 x \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {a^2 b x^3 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {3 a b^2 x^5 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac {b^3 x^7 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.45 \[ \frac {\left (35 a^3 x+35 a^2 b x^3+21 a b^2 x^5+5 b^3 x^7\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{35 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 72, normalized size = 0.53 \[ \frac {{\left (5 \, b^{3} c x^{7} + 21 \, a b^{2} c x^{5} + 35 \, a^{2} b c x^{3} + 35 \, a^{3} c x\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{35 \, {\left (b x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 46, normalized size = 0.34 \[ \frac {1}{35} \, {\left (5 \, b^{3} x^{7} + 21 \, a b^{2} x^{5} + 35 \, a^{2} b x^{3} + 35 \, a^{3} x\right )} c^{\frac {3}{2}} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 0.43 \[ \frac {\left (5 b^{3} x^{6}+21 a \,b^{2} x^{4}+35 a^{2} b \,x^{2}+35 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} x}{35 \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 43, normalized size = 0.32 \[ \frac {1}{7} \, b^{3} c^{\frac {3}{2}} x^{7} + \frac {3}{5} \, a b^{2} c^{\frac {3}{2}} x^{5} + a^{2} b c^{\frac {3}{2}} x^{3} + a^{3} c^{\frac {3}{2}} x \]
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 \[ \int {\left (c\,{\left (b\,x^2+a\right )}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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