Optimal. Leaf size=134 \[ \frac {128 b^4 \left (b x^2+c x^4\right )^{5/2}}{15015 c^5 x^5}-\frac {64 b^3 \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {16 b^2 \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c} \]
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Rubi [A] time = 0.25, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \begin {gather*} \frac {128 b^4 \left (b x^2+c x^4\right )^{5/2}}{15015 c^5 x^5}-\frac {64 b^3 \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {16 b^2 \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x^6 \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}-\frac {(8 b) \int x^4 \left (b x^2+c x^4\right )^{3/2} \, dx}{13 c}\\ &=-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}+\frac {\left (48 b^2\right ) \int x^2 \left (b x^2+c x^4\right )^{3/2} \, dx}{143 c^2}\\ &=\frac {16 b^2 \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}-\frac {\left (64 b^3\right ) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{429 c^3}\\ &=-\frac {64 b^3 \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {16 b^2 \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}+\frac {\left (128 b^4\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{3003 c^4}\\ &=\frac {128 b^4 \left (b x^2+c x^4\right )^{5/2}}{15015 c^5 x^5}-\frac {64 b^3 \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {16 b^2 \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {8 b x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 0.56 \begin {gather*} \frac {x \left (b+c x^2\right )^3 \left (128 b^4-320 b^3 c x^2+560 b^2 c^2 x^4-840 b c^3 x^6+1155 c^4 x^8\right )}{15015 c^5 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 68, normalized size = 0.51 \begin {gather*} \frac {\left (b x^2+c x^4\right )^{5/2} \left (128 b^4-320 b^3 c x^2+560 b^2 c^2 x^4-840 b c^3 x^6+1155 c^4 x^8\right )}{15015 c^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 86, normalized size = 0.64 \begin {gather*} \frac {{\left (1155 \, c^{6} x^{12} + 1470 \, b c^{5} x^{10} + 35 \, b^{2} c^{4} x^{8} - 40 \, b^{3} c^{3} x^{6} + 48 \, b^{4} c^{2} x^{4} - 64 \, b^{5} c x^{2} + 128 \, b^{6}\right )} \sqrt {c x^{4} + b x^{2}}}{15015 \, c^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 92, normalized size = 0.69 \begin {gather*} -\frac {128 \, b^{\frac {13}{2}} \mathrm {sgn}\relax (x)}{15015 \, c^{5}} + \frac {1155 \, {\left (c x^{2} + b\right )}^{\frac {13}{2}} \mathrm {sgn}\relax (x) - 5460 \, {\left (c x^{2} + b\right )}^{\frac {11}{2}} b \mathrm {sgn}\relax (x) + 10010 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} b^{2} \mathrm {sgn}\relax (x) - 8580 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} b^{3} \mathrm {sgn}\relax (x) + 3003 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b^{4} \mathrm {sgn}\relax (x)}{15015 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.54 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) \left (1155 c^{4} x^{8}-840 c^{3} x^{6} b +560 c^{2} x^{4} b^{2}-320 c \,x^{2} b^{3}+128 b^{4}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{15015 c^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 79, normalized size = 0.59 \begin {gather*} \frac {{\left (1155 \, c^{6} x^{12} + 1470 \, b c^{5} x^{10} + 35 \, b^{2} c^{4} x^{8} - 40 \, b^{3} c^{3} x^{6} + 48 \, b^{4} c^{2} x^{4} - 64 \, b^{5} c x^{2} + 128 \, b^{6}\right )} \sqrt {c x^{2} + b}}{15015 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.48, size = 73, normalized size = 0.54 \begin {gather*} \frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}\,\left (128\,b^4-320\,b^3\,c\,x^2+560\,b^2\,c^2\,x^4-840\,b\,c^3\,x^6+1155\,c^4\,x^8\right )}{15015\,c^5\,x} \end {gather*}
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 \begin {gather*} \int x^{6} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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