Integrand size = 20, antiderivative size = 209 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \]
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Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {90, 52, 65, 214} \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 a^3 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {2 a^3 \sqrt {c+d x} (b c-a d)^2}{b^6}-\frac {2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3} \]
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Rule 52
Rule 65
Rule 90
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{b^3 d^2}-\frac {a^3 (c+d x)^{5/2}}{b^3 (a+b x)}+\frac {(-2 b c-a d) (c+d x)^{7/2}}{b^2 d^2}+\frac {(c+d x)^{9/2}}{b d^2}\right ) \, dx \\ & = \frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {a^3 \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b^3} \\ & = -\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^4} \\ & = -\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^5} \\ & = -\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^6} \\ & = -\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (2 a^3 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^6 d} \\ & = -\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {c+d x} \left (-3465 a^5 d^5+495 a^2 b^3 d^2 (c+d x)^3+55 a b^4 d (2 c-7 d x) (c+d x)^3+1155 a^4 b d^4 (7 c+d x)-231 a^3 b^2 d^3 \left (23 c^2+11 c d x+3 d^2 x^2\right )+5 b^5 (c+d x)^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{3465 b^6 d^3}+\frac {2 a^3 (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \]
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Time = 2.01 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {8 \left (\frac {63}{8} d^{2} x^{2}-\frac {7}{2} c d x +c^{2}\right ) \left (d x +c \right )^{3} b^{5}}{693}-\frac {2 \left (-\frac {7 d x}{2}+c \right ) d \left (d x +c \right )^{3} a \,b^{4}}{63}-\frac {a^{2} d^{2} \left (d x +c \right )^{3} b^{3}}{7}+\frac {23 d^{3} \left (\frac {3}{23} d^{2} x^{2}+\frac {11}{23} c d x +c^{2}\right ) a^{3} b^{2}}{15}-\frac {7 \left (\frac {d x}{7}+c \right ) d^{4} a^{4} b}{3}+a^{5} d^{5}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}-a^{3} d^{3} \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{3} b^{6}}\) | \(200\) |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a d \,b^{4}+2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(348\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a d \,b^{4}+2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(348\) |
risch | \(-\frac {2 \left (-315 b^{5} d^{5} x^{5}+385 x^{4} a \,b^{4} d^{5}-805 x^{4} b^{5} c \,d^{4}-495 x^{3} a^{2} b^{3} d^{5}+1045 a \,b^{4} c \,d^{4} x^{3}-565 x^{3} b^{5} c^{2} d^{3}+693 x^{2} a^{3} b^{2} d^{5}-1485 a^{2} b^{3} c \,d^{4} x^{2}+825 a \,b^{4} c^{2} d^{3} x^{2}-15 x^{2} b^{5} c^{3} d^{2}-1155 x \,a^{4} b \,d^{5}+2541 a^{3} b^{2} c \,d^{4} x -1485 a^{2} b^{3} c^{2} d^{3} x +55 a \,b^{4} c^{3} d^{2} x +20 x \,b^{5} c^{4} d +3465 a^{5} d^{5}-8085 a^{4} b c \,d^{4}+5313 a^{3} b^{2} c^{2} d^{3}-495 a^{2} b^{3} c^{3} d^{2}-110 a \,b^{4} c^{4} d -40 b^{5} c^{5}\right ) \sqrt {d x +c}}{3465 d^{3} b^{6}}+\frac {2 a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}\) | \(355\) |
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Time = 0.23 (sec) , antiderivative size = 713, normalized size of antiderivative = 3.41 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\left [\frac {3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, b^{6} d^{3}}, \frac {2 \, {\left (3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}\right )}}{3465 \, b^{6} d^{3}}\right ] \]
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Time = 2.86 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (- \frac {a^{3} d^{4} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{4}} + \frac {a^{3} d^{4} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{7} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x\right )^{\frac {11}{2}}}{11 b} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- a d^{2} - 2 b c d\right )}{9 b^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a^{2} d^{3} + a b c d^{2} + b^{2} c^{2} d\right )}{7 b^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} d^{5} - a^{3} b c d^{4}\right )}{3 b^{5}} + \frac {\sqrt {c + d x} \left (- a^{5} d^{6} + 2 a^{4} b c d^{5} - a^{3} b^{2} c^{2} d^{4}\right )}{b^{6}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (- \frac {a^{3} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.46 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{10} d^{30} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{10} c d^{30} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{10} c^{2} d^{30} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} a b^{9} d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{9} c d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a^{2} b^{8} d^{32} - 693 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{3} b^{7} d^{33} - 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} c d^{33} - 3465 \, \sqrt {d x + c} a^{3} b^{7} c^{2} d^{33} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{6} d^{34} + 6930 \, \sqrt {d x + c} a^{4} b^{6} c d^{34} - 3465 \, \sqrt {d x + c} a^{5} b^{5} d^{35}\right )}}{3465 \, b^{11} d^{33}} \]
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Time = 0.11 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.71 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\left (\frac {6\,c^2}{7\,b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{7\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{7/2}-\left (\frac {2\,c^3}{5\,b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{5\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {2\,c}{3\,b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{9\,b^2\,d^6}\right )\,{\left (c+d\,x\right )}^{9/2}+\frac {2\,{\left (c+d\,x\right )}^{11/2}}{11\,b\,d^3}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {a^3\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{13/2}}-\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,{\left (a\,d^4-b\,c\,d^3\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^6}+\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^3} \]
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