Integrand size = 22, antiderivative size = 165 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {91, 79, 52, 65, 223, 212} \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=-\frac {(5 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}}+\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{d^3 (b c-a d)}-\frac {4 c (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)} \]
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Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {3}{2} c (b c-a d)-\frac {3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \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 d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \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 d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (a d \left (13 c^2+18 c d x+3 d^2 x^2\right )-b c \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (-b c+a d) (c+d x)^{3/2}}+\frac {(-5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(139)=278\).
Time = 0.54 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.99
method | result | size |
default | \(\frac {\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} d^{4} x^{2}-18 \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 c \,d^{3} x^{2}+15 \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^{2} c^{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^{2} c \,d^{3} x -36 \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 \,c^{2} d^{2} x +30 \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^{2} c^{3} d x +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{3} x^{2}-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b c \,d^{2} 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 ) a^{2} c^{2} d^{2}-18 \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 \,c^{3} d +15 \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^{2} c^{4}+36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c \,d^{2} x -40 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2} d x +26 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,c^{2} d -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{3}\right ) \sqrt {b x +a}}{6 \left (a d -b c \right ) \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\) | \(659\) |
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (139) = 278\).
Time = 0.36 (sec) , antiderivative size = 640, normalized size of antiderivative = 3.88 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a b c d^{6}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a b c d^{6}\right )} x\right )}}\right ] \]
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\[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{2} \sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (139) = 278\).
Time = 0.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.72 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} {\left (b x + a\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}} + \frac {2 \, {\left (10 \, b^{7} c^{2} d^{3} - 12 \, a b^{6} c d^{4} + 3 \, a^{2} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{3} d^{2} - 11 \, a b^{7} c^{2} d^{3} + 7 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^2\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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