Optimal. Leaf size=326 \[ -\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 326, 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 {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int x^3 \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+b \sqrt {x}} (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int x \sqrt {a+b x} \left (-c+x^2\right )^3 \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^3 \sqrt {a+b x}}{b^7}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 (a+b x)^{3/2}}{b^7}-\frac {3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{5/2}}{b^7}+\frac {\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{7/2}}{b^7}-\frac {5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac {3 \left (-7 a^2+b^2 c\right ) (a+b x)^{11/2}}{b^7}-\frac {7 a (a+b x)^{13/2}}{b^7}+\frac {(a+b x)^{15/2}}{b^7}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 232, normalized size = 0.71 \begin {gather*} \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-14336 a^7+3840 a^5 b^2 (10 c-7 d x)+21504 a^6 b \sqrt {c+d x}-640 a^4 b^3 (104 c-49 d x) \sqrt {c+d x}-48 a^3 b^4 \left (616 c^2-1080 c d x+735 d^2 x^2\right )+24 a^2 b^5 \sqrt {c+d x} \left (2960 c^2-2716 c d x+1617 d^2 x^2\right )+6 a b^6 \left (320 c^3-3936 c^2 d x+5754 c d^2 x^2-7007 d^3 x^3\right )-231 b^7 \sqrt {c+d x} \left (128 c^3-160 c^2 d x+180 c d^2 x^2-195 d^3 x^3\right )\right )}{765765 b^8 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 383, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {17}{2}}}{17}-\frac {28 a \left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (-3 b^{2} c +21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {4 \left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )-\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {4 \left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}-\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-6 \left (-b^{2} c +a^{2}\right )^{2} a -\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (\left (-b^{2} c +a^{2}\right )^{3}+6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{3} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{4} b^{8}}\) | \(383\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {d x +c}\right )^{\frac {17}{2}}}{17}-\frac {28 a \left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {4 \left (-3 b^{2} c +21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {4 \left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )-\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {4 \left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}-\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-6 \left (-b^{2} c +a^{2}\right )^{2} a -\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {4 \left (\left (-b^{2} c +a^{2}\right )^{3}+6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (-b^{2} c +a^{2}\right )^{3} a \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}}{d^{4} b^{8}}\) | \(383\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 268, normalized size = 0.82 \begin {gather*} \frac {4 \, {\left (45045 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {17}{2}} - 357357 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} a - 176715 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} + 348075 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 85085 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} - 328185 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 153153 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 255255 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{765765 \, b^{8} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 286, normalized size = 0.88 \begin {gather*} \frac {4 \, {\left (45045 \, b^{8} d^{4} x^{4} - 29568 \, b^{8} c^{4} + 72960 \, a^{2} b^{6} c^{3} - 96128 \, a^{4} b^{4} c^{2} + 59904 \, a^{6} b^{2} c - 14336 \, a^{8} + 231 \, {\left (15 \, b^{8} c - 14 \, a^{2} b^{6}\right )} d^{3} x^{3} - 28 \, {\left (165 \, b^{8} c^{2} - 291 \, a^{2} b^{6} c + 140 \, a^{4} b^{4}\right )} d^{2} x^{2} + 32 \, {\left (231 \, b^{8} c^{3} - 555 \, a^{2} b^{6} c^{2} + 520 \, a^{4} b^{4} c - 168 \, a^{6} b^{2}\right )} d x + {\left (3003 \, a b^{7} d^{3} x^{3} - 27648 \, a b^{7} c^{3} + 41472 \, a^{3} b^{5} c^{2} - 28160 \, a^{5} b^{3} c + 7168 \, a^{7} b - 3528 \, {\left (2 \, a b^{7} c - a^{3} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (417 \, a b^{7} c^{2} - 417 \, a^{3} b^{5} c + 140 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{765765 \, b^{8} d^{4}} \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^{3} \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. 915 vs.
\(2 (279) = 558\).
time = 3.24, size = 915, normalized size = 2.81 \begin {gather*} -\frac {4 \, {\left (\frac {17 \, {\left (15015 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{6} c^{3} - 45045 \, \sqrt {\sqrt {d x + c} b + a} a b^{6} c^{3} - 19305 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{4} c^{2} + 81081 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{4} c^{2} - 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{4} c^{2} + 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{4} c^{2} + 12285 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} b^{2} c - 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a b^{2} c + 193050 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} b^{2} c - 270270 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} b^{2} c + 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} b^{2} c - 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{5} b^{2} c - 3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} + 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 85995 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a^{2} + 175175 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{3} - 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{4} + 189189 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{5} - 105105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{6} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{7}\right )} a}{b^{7} d^{3}} + \frac {153153 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} b^{6} c^{3} - 510510 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a b^{6} c^{3} + 765765 \, \sqrt {\sqrt {d x + c} b + a} a^{2} b^{6} c^{3} - 255255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} b^{4} c^{2} + 1312740 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a b^{4} c^{2} - 2756754 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{2} b^{4} c^{2} + 3063060 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{3} b^{4} c^{2} - 2297295 \, \sqrt {\sqrt {d x + c} b + a} a^{4} b^{4} c^{2} + 176715 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} b^{2} c - 1253070 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a b^{2} c + 3828825 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{2} b^{2} c - 6563700 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{3} b^{2} c + 6891885 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{4} b^{2} c - 4594590 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{5} b^{2} c + 2297295 \, \sqrt {\sqrt {d x + c} b + a} a^{6} b^{2} c - 45045 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {17}{2}} + 408408 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} a - 1649340 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a^{2} + 3898440 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a^{3} - 5955950 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{4} + 6126120 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{5} - 4288284 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{6} + 2042040 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{7} - 765765 \, \sqrt {\sqrt {d x + c} b + a} a^{8}}{b^{7} d^{3}}\right )}}{765765 \, 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^3\,\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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