Optimal. Leaf size=224 \[ -\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 224, normalized size of antiderivative = 1.00, 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 = {378, 1412, 786}
\begin {gather*} \frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}-\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+b \sqrt {x}} (-c+x)^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \text {Subst}\left (\int x \sqrt {a+b x} \left (-c+x^2\right )^2 \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^2 \sqrt {a+b x}}{b^5}+\frac {\left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) (a+b x)^{3/2}}{b^5}-\frac {2 \left (5 a^3-3 a b^2 c\right ) (a+b x)^{5/2}}{b^5}-\frac {2 \left (-5 a^2+b^2 c\right ) (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+d x}\right )}{d^3}\\ &=-\frac {4 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^6 d^3}+\frac {4 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^6 d^3}-\frac {8 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^6 d^3}+\frac {8 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^6 d^3}-\frac {20 a \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^6 d^3}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^6 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 147, normalized size = 0.66 \begin {gather*} \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-1280 a^5+32 a^3 b^2 (68 c-75 d x)+1920 a^4 b \sqrt {c+d x}+16 a^2 b^3 \sqrt {c+d x} (-254 c+175 d x)+77 b^5 \sqrt {c+d x} \left (32 c^2-40 c d x+45 d^2 x^2\right )-6 a b^4 \left (96 c^2-380 c d x+525 d^2 x^2\right )\right )}{45045 b^6 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 183, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {4 \left (-2 b^{2} c +10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-4 \left (-b^{2} c +a^{2}\right ) a -a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (\left (-b^{2} c +a^{2}\right )^{2}+4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{2} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{3} b^{6}}\) | \(183\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {4 \left (-2 b^{2} c +10 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-4 \left (-b^{2} c +a^{2}\right ) a -a \left (-2 b^{2} c +6 a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (\left (-b^{2} c +a^{2}\right )^{2}+4 a^{2} \left (-b^{2} c +a^{2}\right )\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{2} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{3} b^{6}}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 167, normalized size = 0.75 \begin {gather*} \frac {4 \, {\left (3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 20475 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a - 10010 \, {\left (b^{2} c - 5 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 12870 \, {\left (3 \, a b^{2} c - 5 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{45045 \, b^{6} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 184, normalized size = 0.82 \begin {gather*} \frac {4 \, {\left (3465 \, b^{6} d^{3} x^{3} + 2464 \, b^{6} c^{3} - 4640 \, a^{2} b^{4} c^{2} + 4096 \, a^{4} b^{2} c - 1280 \, a^{6} + 35 \, {\left (11 \, b^{6} c - 10 \, a^{2} b^{4}\right )} d^{2} x^{2} - 8 \, {\left (77 \, b^{6} c^{2} - 127 \, a^{2} b^{4} c + 60 \, a^{4} b^{2}\right )} d x + {\left (315 \, a b^{5} d^{2} x^{2} + 1888 \, a b^{5} c^{2} - 1888 \, a^{3} b^{3} c + 640 \, a^{5} b - 400 \, {\left (2 \, a b^{5} c - a^{3} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{6} d^{3}} \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^{2} \sqrt {a + b \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 549 vs.
\(2 (188) = 376\).
time = 3.13, size = 549, normalized size = 2.45 \begin {gather*} \frac {4 \, {\left (\frac {13 \, {\left (1155 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{4} c^{2} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a b^{4} c^{2} - 990 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{2} c + 4158 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{2} c - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{2} c + 6930 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{2} c + 315 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} - 1925 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a + 4950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} - 6930 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} + 5775 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} - 3465 \, \sqrt {\sqrt {d x + c} b + a} a^{5}\right )} a}{b^{5} d^{2}} + \frac {9009 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{4} c^{2} - 30030 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{4} c^{2} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{4} c^{2} - 10010 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} b^{2} c + 51480 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a b^{2} c - 108108 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} b^{2} c + 120120 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} b^{2} c - 90090 \, \sqrt {\sqrt {d x + c} b + a} a^{4} b^{2} c + 3465 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} - 24570 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a + 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{2} - 128700 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{3} + 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{4} - 90090 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{5} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{6}}{b^{5} d^{2}}\right )}}{45045 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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