Optimal. Leaf size=116 \[ -\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2} \]
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Rubi [A] time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {368, 43} \[ \frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}-\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int x^3 \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\sqrt {c x^2}\right )}{c^2}\\ &=-\frac {2 a^3 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^4 c^2}+\frac {6 a^2 \left (a+b \sqrt {c x^2}\right )^{5/2}}{5 b^4 c^2}-\frac {6 a \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^4 c^2}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 0.62 \[ \frac {2 \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-16 a^3+24 a^2 b \sqrt {c x^2}-30 a b^2 c x^2+35 b^3 \left (c x^2\right )^{3/2}\right )}{315 b^4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 75, normalized size = 0.65 \[ \frac {2 \, {\left (35 \, b^{4} c^{2} x^{4} - 6 \, a^{2} b^{2} c x^{2} - 16 \, a^{4} + {\left (5 \, a b^{3} c x^{2} + 8 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{315 \, b^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 164, normalized size = 1.41 \[ \frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} \sqrt {c} - 21 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a \sqrt {c} + 35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{2} \sqrt {c} - 35 \, \sqrt {b \sqrt {c} x + a} a^{3} \sqrt {c}\right )} a}{b^{3} c^{2}} + \frac {35 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sqrt {c} x + a} a^{4}}{b^{3} c^{\frac {3}{2}}}\right )}}{315 \, b \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 63, normalized size = 0.54 \[ \frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} \left (-30 a \,b^{2} c \,x^{2}-16 a^{3}+24 \sqrt {c \,x^{2}}\, a^{2} b +35 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{3}\right )}{315 b^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 85, normalized size = 0.73 \[ \frac {2 \, {\left (\frac {35 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {9}{2}}}{b^{4}} - \frac {135 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {7}{2}} a}{b^{4}} + \frac {189 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {5}{2}} a^{2}}{b^{4}} - \frac {105 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \]
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 \[ \int x^3\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]
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 \[ \int x^{3} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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