Integrand size = 22, antiderivative size = 246 \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4 d}-\frac {\left (10 a c+\frac {b c^2}{d}-\frac {35 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} d^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {91, 81, 52, 65, 223, 212} \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=-\frac {(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} d^{3/2}}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-\frac {35 a^2 d}{b}+10 a c+\frac {b c^2}{d}\right )}{12 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 \sqrt {a+b x} (b c-a d)}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{8 b^4 d}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {2 \int \frac {(c+d x)^{3/2} \left (-\frac {1}{2} a (b c-5 a d)+\frac {1}{2} b (b c-a d) x\right )}{\sqrt {a+b x}} \, dx}{b^2 (b c-a d)} \\ & = -\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^2 d (b c-a d)} \\ & = -\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{8 b^3 d} \\ & = -\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^4 d} \\ & = -\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^5 d} \\ & = -\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b^5 d} \\ & = -\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4 d}-\frac {\left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3 d (b c-a d)}-\frac {2 a^2 (c+d x)^{5/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 d}-\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} d^{3/2}} \\ \end{align*}
Time = 10.33 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {\sqrt {d} \left (105 a^3 d^2+5 a^2 b d (-20 c+7 d x)+a b^2 \left (3 c^2-38 c d x-14 d^2 x^2\right )+b^3 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{\sqrt {a+b x}}-\frac {3 \sqrt {b c-a d} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^4 d^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(691\) vs. \(2(210)=420\).
Time = 0.62 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.81
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (-16 b^{3} d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b \,d^{3} x -135 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c \,d^{2} x +27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{3} x +28 a \,b^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-28 b^{3} c d \,x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{3}-135 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{2}+27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3}-70 a^{2} b \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+76 a \,b^{2} c d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 b^{3} c^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-210 a^{3} d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+200 a^{2} b c d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 a \,b^{2} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{4} d}\) | \(692\) |
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Time = 0.33 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.44 \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\left [\frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (8 \, b^{4} d^{3} x^{3} + 3 \, a b^{3} c^{2} d - 100 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 14 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + {\left (3 \, b^{4} c^{2} d - 38 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}}, \frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{3} x^{3} + 3 \, a b^{3} c^{2} d - 100 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 14 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + {\left (3 \, b^{4} c^{2} d - 38 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{6} d^{2} x + a b^{5} d^{2}\right )}}\right ] \]
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\[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.45 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{24} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{6}} + \frac {7 \, b^{18} c d^{4} {\left | b \right |} - 19 \, a b^{17} d^{5} {\left | b \right |}}{b^{23} d^{4}}\right )} + \frac {3 \, {\left (b^{19} c^{2} d^{3} {\left | b \right |} - 22 \, a b^{18} c d^{4} {\left | b \right |} + 29 \, a^{2} b^{17} d^{5} {\left | b \right |}\right )}}{b^{23} d^{4}}\right )} - \frac {4 \, {\left (a^{2} b^{2} c^{2} d {\left | b \right |} - 2 \, a^{3} b c d^{2} {\left | b \right |} + a^{4} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{4}} + \frac {{\left (b^{3} c^{3} {\left | b \right |} + 9 \, a b^{2} c^{2} d {\left | b \right |} - 45 \, a^{2} b c d^{2} {\left | b \right |} + 35 \, a^{3} d^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, \sqrt {b d} b^{5} d} \]
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Timed out. \[ \int \frac {x^2 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
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