Optimal. Leaf size=102 \[ \frac {2 a^2 \left (a+b x^2\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{7 b^3}+\frac {2 \left (a+b x^2\right )^3 \left (c \sqrt {a+b x^2}\right )^{3/2}}{15 b^3}-\frac {4 a \left (a+b x^2\right )^2 \left (c \sqrt {a+b x^2}\right )^{3/2}}{11 b^3} \]
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Rubi [A] time = 0.16, antiderivative size = 113, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6720, 266, 43} \[ \frac {2 a^2 c \left (a+b x^2\right )^{3/2} \sqrt {c \sqrt {a+b x^2}}}{7 b^3}+\frac {2 c \left (a+b x^2\right )^{7/2} \sqrt {c \sqrt {a+b x^2}}}{15 b^3}-\frac {4 a c \left (a+b x^2\right )^{5/2} \sqrt {c \sqrt {a+b x^2}}}{11 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6720
Rubi steps
\begin {align*} \int x^5 \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int x^5 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int x^2 (a+b x)^{3/4} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/4}}{b^2}-\frac {2 a (a+b x)^{7/4}}{b^2}+\frac {(a+b x)^{11/4}}{b^2}\right ) \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=\frac {2 a^2 c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b^3}-\frac {4 a c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{5/2}}{11 b^3}+\frac {2 c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{7/2}}{15 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.51 \[ \frac {2 \left (a+b x^2\right ) \left (32 a^2-56 a b x^2+77 b^2 x^4\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{1155 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 63, normalized size = 0.62 \[ \frac {2 \, {\left (77 \, b^{3} c x^{6} + 21 \, a b^{2} c x^{4} - 24 \, a^{2} b c x^{2} + 32 \, a^{3} c\right )} \sqrt {b x^{2} + a} \sqrt {\sqrt {b x^{2} + a} c}}{1155 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 109, normalized size = 1.07 \[ \frac {2 \, c^{\frac {3}{2}} {\left (\frac {5 \, {\left (21 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} - 66 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a + 77 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{2}\right )} a}{b^{2}} + \frac {77 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}} - 315 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{2} - 385 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{3}}{b^{2}}\right )}}{1155 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.46 \[ \frac {2 \left (b \,x^{2}+a \right ) \left (77 x^{4} b^{2}-56 a b \,x^{2}+32 a^{2}\right ) \left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}}}{1155 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 64, normalized size = 0.63 \[ \frac {2 \, {\left (165 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {7}{2}} a^{2} c^{4} - 210 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {11}{2}} a c^{2} + 77 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {15}{2}}\right )}}{1155 \, b^{3} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 88, normalized size = 0.86 \[ \sqrt {c\,\sqrt {b\,x^2+a}}\,\left (\frac {2\,c\,x^6\,\sqrt {b\,x^2+a}}{15}+\frac {64\,a^3\,c\,\sqrt {b\,x^2+a}}{1155\,b^3}+\frac {2\,a\,c\,x^4\,\sqrt {b\,x^2+a}}{55\,b}-\frac {16\,a^2\,c\,x^2\,\sqrt {b\,x^2+a}}{385\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 88.61, size = 116, normalized size = 1.14 \[ \begin {cases} \frac {64 a^{3} c^{\frac {3}{2}} \left (a + b x^{2}\right )^{\frac {3}{4}}}{1155 b^{3}} - \frac {16 a^{2} c^{\frac {3}{2}} x^{2} \left (a + b x^{2}\right )^{\frac {3}{4}}}{385 b^{2}} + \frac {2 a c^{\frac {3}{2}} x^{4} \left (a + b x^{2}\right )^{\frac {3}{4}}}{55 b} + \frac {2 c^{\frac {3}{2}} x^{6} \left (a + b x^{2}\right )^{\frac {3}{4}}}{15} & \text {for}\: b \neq 0 \\\frac {x^{6} \left (\sqrt {a} c\right )^{\frac {3}{2}}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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