Integrand size = 22, antiderivative size = 248 \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}+\frac {\left (10 a c-\frac {35 b c^2}{d}+\frac {a^2 d}{b}\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^2 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{9/2}} \]
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Time = 0.17 (sec) , antiderivative size = 248, 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} \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=-\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {a^2 d}{b}+10 a c-\frac {35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{5/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {(a+b x)^{3/2} \left (\frac {1}{2} c (5 b c-a d)-\frac {1}{2} d (b c-a d) x\right )}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 b d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b d^3} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d^4} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-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^2 d^4} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-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^2 d^4} \\ & = \frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{9/2}} \\ \end{align*}
Time = 10.32 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x} \left (3 a^2 d^2 (c+d x)+2 a b d \left (-50 c^2-19 c d x+7 d^2 x^2\right )+b^2 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )-3 (b c-a d)^{3/2} \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{24 b^2 d^{9/2} \sqrt {c+d x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(691\) vs. \(2(212)=424\).
Time = 1.58 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.79
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (-16 b^{2} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b 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^{3} d^{4} 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^{2} b c \,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 \,b^{2} c^{2} d^{2} x +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 ) b^{3} c^{3} d x -28 a b \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+28 b^{2} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b 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^{3} c \,d^{3}+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 \,c^{2} d^{2}-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 \,b^{2} c^{3} 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 ) b^{3} c^{4}-6 a^{2} d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+76 a b c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-70 b^{2} c^{2} d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 a^{2} c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+200 a b \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-210 b^{2} c^{3} \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 {d x +c}\, b \,d^{4}}\) | \(692\) |
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Time = 0.37 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.42 \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 100 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 14 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 38 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{2} d^{6} x + b^{2} c d^{5}\right )}}, \frac {3 \, {\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 100 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 14 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 38 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{2} d^{6} x + b^{2} c d^{5}\right )}}\right ] \]
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\[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\int \frac {x^{2} \left (a + b x\right )^{\frac {3}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{d {\left | b \right |}} - \frac {7 \, b^{3} c d^{5} + 5 \, a b^{2} d^{6}}{b^{2} d^{7} {\left | b \right |}}\right )} + \frac {35 \, b^{4} c^{2} d^{4} - 10 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}}{b^{2} d^{7} {\left | b \right |}}\right )} + \frac {3 \, {\left (35 \, b^{5} c^{3} d^{3} - 45 \, a b^{4} c^{2} d^{4} + 9 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{2} d^{7} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt {b d} d^{4} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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