Integrand size = 22, antiderivative size = 126 \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}} \]
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Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {91, 79, 65, 223, 212} \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}}+\frac {4 a \sqrt {c+d x} (3 b c-2 a d)}{3 b^2 \sqrt {a+b x} (b c-a d)^2} \]
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Rule 65
Rule 79
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {2 \int \frac {-\frac {1}{2} a (3 b c-a d)+\frac {3}{2} b (b c-a d) x}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 b^2 (b c-a d)} \\ & = -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^2} \\ & = -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3} \\ & = -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3} \\ & = -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=-\frac {2 a \sqrt {c+d x} \left (-6 b c+3 a d+\frac {a b (c+d x)}{a+b x}\right )}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(102)=204\).
Time = 1.74 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.79
method | result | size |
default | \(\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} d^{2} x^{2}-6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c d \,x^{2}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b \,d^{2} x -12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c d x +6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{2} x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{2}-6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2}-8 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b d x +12 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d +10 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \right )}{3 \sqrt {b d}\, \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} \left (b x +a \right )^{\frac {3}{2}}}\) | \(604\) |
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (102) = 204\).
Time = 0.31 (sec) , antiderivative size = 670, normalized size of antiderivative = 5.32 \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}, -\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=\int \frac {x^{2}}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (102) = 204\).
Time = 0.38 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.32 \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=-\frac {\log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d} b {\left | b \right |}} + \frac {8 \, {\left (3 \, a b^{4} c^{2} d - 5 \, a^{2} b^{3} c d^{2} + 2 \, a^{3} b^{2} d^{3} - 6 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c d + 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b d^{2} + 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a d\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} \sqrt {b d} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx=\int \frac {x^2}{{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]
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