Integrand size = 21, antiderivative size = 324 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {378, 1412, 786} \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4} \]
[In]
[Out]
Rule 378
Rule 786
Rule 1412
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-c+x)^3}{\sqrt {a+b \sqrt {x}}} \, dx,x,c+d x\right )}{d^4} \\ & = \frac {2 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{\sqrt {a+b x}} \, 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}{b^7 \sqrt {a+b x}}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 \sqrt {a+b x}}{b^7}-\frac {3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{3/2}}{b^7}+\frac {\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{5/2}}{b^7}-\frac {5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{7/2}}{b^7}-\frac {3 \left (-7 a^2+b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac {7 a (a+b x)^{11/2}}{b^7}+\frac {(a+b x)^{13/2}}{b^7}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4} \\ & = -\frac {4 a \left (a^2-b^2 c\right )^3 \sqrt {a+b \sqrt {c+d x}}}{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 )^{3/2}}{3 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 )^{5/2}}{5 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 )^{7/2}}{7 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.72 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \sqrt {a+b \sqrt {c+d x}} \left (-14336 a^7+768 a^5 b^2 (58 c-7 d x)+7168 a^6 b \sqrt {c+d x}-640 a^4 b^3 (32 c-7 d x) \sqrt {c+d x}+24 a^2 b^5 \sqrt {c+d x} \left (784 c^2-356 c d x+147 d^2 x^2\right )-16 a^3 b^4 \left (2936 c^2-680 c d x+245 d^2 x^2\right )+6 a b^6 \left (2880 c^3-928 c^2 d x+658 c d^2 x^2-539 d^3 x^3\right )-39 b^7 \sqrt {c+d x} \left (128 c^3-96 c^2 d x+84 c d^2 x^2-77 d^3 x^3\right )\right )}{45045 b^8 d^4} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}+\left (-b^{2} c +a^{2}\right )^{3} a \sqrt {a +b \sqrt {d x +c}}\right )}{d^{4} b^{8}}\) | \(384\) |
default | \(-\frac {4 \left (-\frac {\left (a +b \sqrt {d x +c}\right )^{\frac {15}{2}}}{15}+\frac {7 a \left (a +b \sqrt {d x +c}\right )^{\frac {13}{2}}}{13}+\frac {\left (3 b^{2} c -21 a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {11}{2}}}{11}+\frac {\left (8 \left (-b^{2} c +a^{2}\right ) a +2 a \left (-2 b^{2} c +6 a^{2}\right )+\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {9}{2}}}{9}+\frac {\left (-\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )-8 a^{2} \left (-b^{2} c +a^{2}\right )-\left (-b^{2} c +a^{2}\right )^{2}+\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 a \left (-2 b^{2} c +6 a^{2}\right )\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {7}{2}}}{7}+\frac {\left (6 \left (-b^{2} c +a^{2}\right )^{2} a +\left (\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+8 a^{2} \left (-b^{2} c +a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +b \sqrt {d x +c}\right )^{\frac {5}{2}}}{5}+\frac {\left (-\left (-b^{2} c +a^{2}\right )^{3}-6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}\right ) \left (a +b \sqrt {d x +c}\right )^{\frac {3}{2}}}{3}+\left (-b^{2} c +a^{2}\right )^{3} a \sqrt {a +b \sqrt {d x +c}}\right )}{d^{4} b^{8}}\) | \(384\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 \, {\left (3234 \, a b^{6} d^{3} x^{3} - 17280 \, a b^{6} c^{3} + 46976 \, a^{3} b^{4} c^{2} - 44544 \, a^{5} b^{2} c + 14336 \, a^{7} - 28 \, {\left (141 \, a b^{6} c - 140 \, a^{3} b^{4}\right )} d^{2} x^{2} + 64 \, {\left (87 \, a b^{6} c^{2} - 170 \, a^{3} b^{4} c + 84 \, a^{5} b^{2}\right )} d x - {\left (3003 \, b^{7} d^{3} x^{3} - 4992 \, b^{7} c^{3} + 18816 \, a^{2} b^{5} c^{2} - 20480 \, a^{4} b^{3} c + 7168 \, a^{6} b - 252 \, {\left (13 \, b^{7} c - 14 \, a^{2} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (117 \, b^{7} c^{2} - 267 \, a^{2} b^{5} c + 140 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{45045 \, b^{8} d^{4}} \]
[In]
[Out]
Time = 0.80 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\begin {cases} \frac {2 \left (\begin {cases} \frac {2 \left (- \frac {7 a \left (a + b \sqrt {c + d x}\right )^{\frac {13}{2}}}{13 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {15}{2}}}{15 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {11}{2}} \cdot \left (21 a^{2} - 3 b^{2} c\right )}{11 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {9}{2}} \left (- 35 a^{3} + 15 a b^{2} c\right )}{9 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {7}{2}} \cdot \left (35 a^{4} - 30 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{7 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {5}{2}} \left (- 21 a^{5} + 30 a^{3} b^{2} c - 9 a b^{4} c^{2}\right )}{5 b^{6}} + \frac {\left (a + b \sqrt {c + d x}\right )^{\frac {3}{2}} \cdot \left (7 a^{6} - 15 a^{4} b^{2} c + 9 a^{2} b^{4} c^{2} - b^{6} c^{3}\right )}{3 b^{6}} + \frac {\sqrt {a + b \sqrt {c + d x}} \left (- a^{7} + 3 a^{5} b^{2} c - 3 a^{3} b^{4} c^{2} + a b^{6} c^{3}\right )}{b^{6}}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\frac {d^{4} x^{4}}{8 \sqrt {a}} & \text {otherwise} \end {cases}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \sqrt {a + b \sqrt {c}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\frac {4 \, {\left (3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} - 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 12285 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 25025 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} + 6435 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 27027 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} - 15015 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} + 45045 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt {\sqrt {d x + c} b + a}\right )}}{45045 \, b^{8} d^{4}} \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.26 \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=-\frac {4 \, {\left (15015 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} b^{6} c^{3} - 45045 \, \sqrt {\sqrt {d x + c} b + a} a b^{6} c^{3} - 19305 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} b^{4} c^{2} + 81081 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a b^{4} c^{2} - 135135 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{2} b^{4} c^{2} + 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{3} b^{4} c^{2} + 12285 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} b^{2} c - 75075 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a b^{2} c + 193050 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{2} b^{2} c - 270270 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{3} b^{2} c + 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{4} b^{2} c - 135135 \, \sqrt {\sqrt {d x + c} b + a} a^{5} b^{2} c - 3003 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} + 24255 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} a - 85995 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} a^{2} + 175175 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} a^{3} - 225225 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} a^{4} + 189189 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} a^{5} - 105105 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} a^{6} + 45045 \, \sqrt {\sqrt {d x + c} b + a} a^{7}\right )}}{45045 \, b^{8} d^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3}{\sqrt {a+b \sqrt {c+d x}}} \, dx=\int \frac {x^3}{\sqrt {a+b\,\sqrt {c+d\,x}}} \,d x \]
[In]
[Out]