Integrand size = 19, antiderivative size = 240 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4}+\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4} \]
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Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786} \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {x \left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4} \]
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Rule 378
Rule 786
Rule 1412
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-c+x)^3}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d^4} \\ & = \frac {2 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d^4} \\ & = \frac {2 \text {Subst}\left (\int \left (-\frac {6 a \left (a^2-b^2 c\right )^2}{b^7}+\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6}-\frac {2 a \left (2 a^2-3 b^2 c\right ) x^2}{b^5}-\frac {3 \left (-a^2+b^2 c\right ) x^3}{b^4}-\frac {2 a x^4}{b^3}+\frac {x^5}{b^2}-\frac {a \left (a^2-b^2 c\right )^3}{b^7 (a+b x)^2}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4} \\ & = \frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4}+\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {60 a^7-360 a^6 b \sqrt {c+d x}-30 a^5 b^2 (13 c+7 d x)+10 a^4 b^3 \sqrt {c+d x} (79 c+7 d x)-3 a^2 b^5 \sqrt {c+d x} \left (163 c^2+36 c d x-7 d^2 x^2\right )+5 a^3 b^4 \left (119 c^2+76 c d x-7 d^2 x^2\right )+5 b^7 \sqrt {c+d x} \left (11 c^3+6 c^2 d x-3 c d^2 x^2+2 d^3 x^3\right )-a b^6 \left (269 c^3+162 c^2 d x-33 c d^2 x^2+14 d^3 x^3\right )+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )}{30 b^8 d^4 \left (a+b \sqrt {c+d x}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}}{d^{4}}\) | \(277\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}}{d^{4}}\) | \(277\) |
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Time = 0.32 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.63 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\frac {10 \, b^{8} d^{4} x^{4} + 55 \, b^{8} c^{4} - 220 \, a^{2} b^{6} c^{3} + 195 \, a^{4} b^{4} c^{2} + 30 \, a^{6} b^{2} c - 60 \, a^{8} - 5 \, {\left (b^{8} c - 7 \, a^{2} b^{6}\right )} d^{3} x^{3} + 15 \, {\left (b^{8} c^{2} - 8 \, a^{2} b^{6} c + 7 \, a^{4} b^{4}\right )} d^{2} x^{2} + 5 \, {\left (17 \, b^{8} c^{3} - 87 \, a^{2} b^{6} c^{2} + 96 \, a^{4} b^{4} c - 30 \, a^{6} b^{2}\right )} d x - 60 \, {\left (b^{8} c^{4} - 10 \, a^{2} b^{6} c^{3} + 24 \, a^{4} b^{4} c^{2} - 22 \, a^{6} b^{2} c + 7 \, a^{8} + {\left (b^{8} c^{3} - 9 \, a^{2} b^{6} c^{2} + 15 \, a^{4} b^{4} c - 7 \, a^{6} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (6 \, a b^{7} d^{3} x^{3} + 81 \, a b^{7} c^{3} - 271 \, a^{3} b^{5} c^{2} + 295 \, a^{5} b^{3} c - 105 \, a^{7} b - 2 \, {\left (6 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} x^{2} + 2 \, {\left (24 \, a b^{7} c^{2} - 61 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}}{30 \, {\left (b^{10} d^{5} x + {\left (b^{10} c - a^{2} b^{8}\right )} d^{4}\right )}} \]
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Time = 5.69 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.11 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\begin {cases} \frac {2 \left (- \frac {2 a \left (c + d x\right )^{\frac {5}{2}}}{5 b^{3}} - \frac {a \left (a^{2} - b^{2} c\right )^{3} \left (\begin {cases} \frac {\sqrt {c + d x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{b \left (a + b \sqrt {c + d x}\right )} & \text {otherwise} \end {cases}\right )}{b^{7}} + \frac {\left (c + d x\right )^{3}}{6 b^{2}} + \frac {\left (3 a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{2}}{4 b^{4}} + \frac {\left (- 4 a^{3} + 6 a b^{2} c\right ) \left (c + d x\right )^{\frac {3}{2}}}{3 b^{5}} + \frac {\left (c + d x\right ) \left (5 a^{4} - 9 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{2 b^{6}} + \frac {\left (a^{2} - b^{2} c\right )^{2} \cdot \left (7 a^{2} - b^{2} c\right ) \left (\begin {cases} \frac {\sqrt {c + d x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \sqrt {c + d x} \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{7}} + \frac {\sqrt {c + d x} \left (- 6 a^{5} + 12 a^{3} b^{2} c - 6 a b^{4} c^{2}\right )}{b^{7}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {x^{4}}{4 \left (a + b \sqrt {c}\right )^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.05 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=-\frac {\frac {60 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{\sqrt {d x + c} b^{9} + a b^{8}} - \frac {10 \, {\left (d x + c\right )}^{3} b^{5} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} - 45 \, {\left (b^{5} c - a^{2} b^{3}\right )} {\left (d x + c\right )}^{2} + 40 \, {\left (3 \, a b^{4} c - 2 \, a^{3} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 30 \, {\left (3 \, b^{5} c^{2} - 9 \, a^{2} b^{3} c + 5 \, a^{4} b\right )} {\left (d x + c\right )} - 360 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt {d x + c}}{b^{7}} + \frac {60 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{30 \, d^{4}} \]
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Time = 0.34 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=-\frac {2 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{8} d^{4}} + \frac {10 \, {\left (d x + c\right )}^{3} b^{10} d^{20} - 45 \, {\left (d x + c\right )}^{2} b^{10} c d^{20} + 90 \, {\left (d x + c\right )} b^{10} c^{2} d^{20} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{9} d^{20} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{9} c d^{20} - 360 \, \sqrt {d x + c} a b^{9} c^{2} d^{20} + 45 \, {\left (d x + c\right )}^{2} a^{2} b^{8} d^{20} - 270 \, {\left (d x + c\right )} a^{2} b^{8} c d^{20} - 80 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} d^{20} + 720 \, \sqrt {d x + c} a^{3} b^{7} c d^{20} + 150 \, {\left (d x + c\right )} a^{4} b^{6} d^{20} - 360 \, \sqrt {d x + c} a^{5} b^{5} d^{20}}{30 \, b^{12} d^{24}} \]
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Time = 0.09 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.92 \[ \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx=\left (\frac {4\,a^3}{3\,b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{3\,b}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {3\,c}{2\,b^2\,d^4}-\frac {3\,a^2}{2\,b^4\,d^4}\right )\,{\left (c+d\,x\right )}^2-\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {6\,c^2}{b^2\,d^4}\right )}{b}+\frac {a^2\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b^2}\right )\,\sqrt {c+d\,x}+\frac {{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}+\frac {2\,\left (a^7-3\,a^5\,b^2\,c+3\,a^3\,b^4\,c^2-a\,b^6\,c^3\right )}{b\,\left (b^8\,d^4\,\sqrt {c+d\,x}+a\,b^7\,d^4\right )}+d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{2\,b^2}-\frac {a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {3\,c^2}{b^2\,d^4}\right )+\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (14\,a^6-30\,a^4\,b^2\,c+18\,a^2\,b^4\,c^2-2\,b^6\,c^3\right )}{b^8\,d^4}-\frac {4\,a\,{\left (c+d\,x\right )}^{5/2}}{5\,b^3\,d^4} \]
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