Integrand size = 24, antiderivative size = 111 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 65, 223, 212} \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac {2 B \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}}-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}} \]
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Rule 49
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {B \int \frac {\sqrt {d+e x}}{(a+b x)^{3/2}} \, dx}{b} \\ & = -\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(B e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b^2} \\ & = -\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 B e) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3} \\ & = -\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 B e) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^3} \\ & = -\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (A b^2 (d+e x)+B \left (-3 a^2 e+3 b^2 d x+2 a b (d-2 e x)\right )\right )}{3 b^2 (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{b^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(89)=178\).
Time = 1.12 (sec) , antiderivative size = 503, normalized size of antiderivative = 4.53
method | result | size |
default | \(\frac {\left (3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} e^{2} x^{2}-3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d e \,x^{2}+6 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{2} x -6 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d e x +2 A \,b^{2} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} e^{2}-3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b d e -8 B a b e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+6 B \,b^{2} d x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+2 A \,b^{2} d \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-6 B \,a^{2} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+4 B a b d \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right ) \sqrt {e x +d}}{3 \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} \left (b x +a \right )^{\frac {3}{2}}}\) | \(503\) |
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (89) = 178\).
Time = 0.71 (sec) , antiderivative size = 525, normalized size of antiderivative = 4.73 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (B a^{2} b d - B a^{3} e + {\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \, {\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt {\frac {e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {e}{b}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (3 \, B a^{2} e - {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} d - {\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + {\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \, {\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}, -\frac {3 \, {\left (B a^{2} b d - B a^{3} e + {\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \, {\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {e}{b}}}{2 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )}}\right ) - 2 \, {\left (3 \, B a^{2} e - {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} d - {\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + {\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \, {\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}\right ] \]
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\[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (89) = 178\).
Time = 0.41 (sec) , antiderivative size = 558, normalized size of antiderivative = 5.03 \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=-\frac {\sqrt {b e} B {\left | b \right |} \log \left ({\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{b^{4}} - \frac {4 \, {\left (3 \, \sqrt {b e} B b^{5} d^{3} {\left | b \right |} - 10 \, \sqrt {b e} B a b^{4} d^{2} e {\left | b \right |} + \sqrt {b e} A b^{5} d^{2} e {\left | b \right |} + 11 \, \sqrt {b e} B a^{2} b^{3} d e^{2} {\left | b \right |} - 2 \, \sqrt {b e} A a b^{4} d e^{2} {\left | b \right |} - 4 \, \sqrt {b e} B a^{3} b^{2} e^{3} {\left | b \right |} + \sqrt {b e} A a^{2} b^{3} e^{3} {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{3} d^{2} {\left | b \right |} + 12 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b^{2} d e {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b e^{2} {\left | b \right |} + 3 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B b d {\left | b \right |} - 6 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B a e {\left | b \right |} + 3 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A b e {\left | b \right |}\right )}}{3 \, {\left (b^{2} d - a b e - {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3} b^{3}} \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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