Integrand size = 22, antiderivative size = 189 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (6 a c-\frac {15 b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}} \]
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Time = 0.13 (sec) , antiderivative size = 189, 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, 81, 52, 65, 223, 212} \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+6 a c-\frac {15 b c^2}{d}\right )}{4 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2} \]
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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)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {1}{2} c (3 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)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 b d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\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 )}{4 b^2 d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\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 )}{4 b^2 d^3} \\ & = \frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 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 )}{4 b^{3/2} d^{7/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (a d (c+d x)+b \left (-15 c^2-5 c d x+2 d^2 x^2\right )\right )}{4 b d^3 \sqrt {c+d x}}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(159)=318\).
Time = 0.54 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.41
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x +6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x -4 b \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c \,d^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3}-2 a \,d^{2} x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+10 b c d x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-2 a c d \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+30 b \,c^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 \sqrt {b d}\, b \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} \sqrt {d x +c}}\) | \(456\) |
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Time = 0.30 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.35 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\left [-\frac {{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2} + {\left (15 \, b^{2} c^{2} d - 6 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (2 \, b^{2} d^{3} x^{2} - 15 \, b^{2} c^{2} d + a b c d^{2} - {\left (5 \, b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}, -\frac {{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2} + {\left (15 \, b^{2} c^{2} d - 6 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{3} x^{2} - 15 \, b^{2} c^{2} d + a b c d^{2} - {\left (5 \, b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}\right ] \]
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\[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x^{2} \sqrt {a + b x}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )}}{d {\left | b \right |}} - \frac {5 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}}{b^{2} d^{5} {\left | b \right |}}\right )} - \frac {15 \, b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}}{b^{2} d^{5} {\left | b \right |}}\right )}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} d^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2 \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x^2\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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