Optimal. Leaf size=174 \[ -\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3} \]
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Rubi [A] time = 0.08, antiderivative size = 174, 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} \[ \frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 368
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^5 \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^5 \sqrt {a+b x}}{b^5}+\frac {5 a^4 (a+b x)^{3/2}}{b^5}-\frac {10 a^3 (a+b x)^{5/2}}{b^5}+\frac {10 a^2 (a+b x)^{7/2}}{b^5}-\frac {5 a (a+b x)^{9/2}}{b^5}+\frac {(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt {c x^2}\right )}{c^3}\\ &=-\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 103, normalized size = 0.59 \[ \frac {2 \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {c x^2}-480 a^3 b^2 c x^2+560 a^2 b^3 \left (c x^2\right )^{3/2}-630 a b^4 c^2 x^4+693 b^5 \left (c x^2\right )^{5/2}\right )}{9009 b^6 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 103, normalized size = 0.59 \[ \frac {2 \, {\left (693 \, b^{6} c^{3} x^{6} - 70 \, a^{2} b^{4} c^{2} x^{4} - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} + {\left (63 \, a b^{5} c^{2} x^{4} + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{9009 \, b^{6} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 231, normalized size = 1.33 \[ \frac {2 \, {\left (\frac {13 \, {\left (63 \, {\left (b \sqrt {c} x + a\right )}^{\frac {11}{2}} \sqrt {c} - 385 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} a \sqrt {c} + 990 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a^{2} \sqrt {c} - 1386 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{3} \sqrt {c} + 1155 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{4} \sqrt {c} - 693 \, \sqrt {b \sqrt {c} x + a} a^{5} \sqrt {c}\right )} a}{b^{5} c^{3}} + \frac {3 \, {\left (231 \, {\left (b \sqrt {c} x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b \sqrt {c} x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b \sqrt {c} x + a} a^{6}\right )}}{b^{5} c^{\frac {5}{2}}}\right )}}{9009 \, b \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 92, normalized size = 0.53 \[ \frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} \left (-630 a \,b^{4} c^{2} x^{4}-480 a^{3} b^{2} c \,x^{2}-256 a^{5}+384 \sqrt {c \,x^{2}}\, a^{4} b +560 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{2} b^{3}+693 \left (c \,x^{2}\right )^{\frac {5}{2}} b^{5}\right )}{9009 b^{6} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 127, normalized size = 0.73 \[ \frac {2 \, {\left (\frac {693 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {13}{2}}}{b^{6}} - \frac {4095 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {11}{2}} a}{b^{6}} + \frac {10010 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {9}{2}} a^{2}}{b^{6}} - \frac {12870 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {7}{2}} a^{3}}{b^{6}} + \frac {9009 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {3003 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \]
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 \[ \int x^5\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]
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 \[ \int x^{5} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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