Integrand size = 24, antiderivative size = 202 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {3 (b d-a e) (5 b B d-4 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}} \]
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Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {3 (b d-a e) (-a B e-4 A b e+5 b B d) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \]
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Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(5 b B d-4 A b e-a B e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}-\frac {(3 (5 b B d-4 A b e-a B e)) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{4 e^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 e^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a 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 )}{4 b e^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a 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 )}{4 b e^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {3 (b d-a e) (5 b B d-4 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {\frac {\sqrt {e} \sqrt {a+b x} \left (4 A b e (3 d+e x)+a e (13 B d-8 A e+5 B e x)+b B \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{\sqrt {d+e x}}-\frac {6 (b d-a e) (5 b B d-4 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{\sqrt {b}}}{4 e^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(739\) vs. \(2(172)=344\).
Time = 1.09 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.66
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (12 A \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 \,e^{3} x -12 A \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^{2} d \,e^{2} x +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} e^{3} x -18 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 d \,e^{2} x +15 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^{2} d^{2} e x +4 B b \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+12 A \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 d \,e^{2}-12 A \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^{2} d^{2} e +8 A b \,e^{2} 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^{2} d \,e^{2}-18 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 \,d^{2} e +15 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^{2} d^{3}+10 B a \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-10 B b d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-16 A a \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+24 A b d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+26 B a d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-30 B b \,d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {e x +d}\, e^{3}}\) | \(740\) |
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Time = 0.57 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\left [\frac {3 \, {\left (5 \, B b^{2} d^{3} - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e + {\left (B a^{2} + 4 \, A a b\right )} d e^{2} + {\left (5 \, B b^{2} d^{2} e - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} + {\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} + {\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} - {\left (5 \, B b^{2} d e^{2} - {\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{16 \, {\left (b e^{5} x + b d e^{4}\right )}}, -\frac {3 \, {\left (5 \, B b^{2} d^{3} - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e + {\left (B a^{2} + 4 \, A a b\right )} d e^{2} + {\left (5 \, B b^{2} d^{2} e - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} + {\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, B b^{2} e^{3} x^{2} - 15 \, B b^{2} d^{2} e - 8 \, A a b e^{3} + {\left (13 \, B a b + 12 \, A b^{2}\right )} d e^{2} - {\left (5 \, B b^{2} d e^{2} - {\left (5 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{8 \, {\left (b e^{5} x + b d e^{4}\right )}}\right ] \]
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\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |}}{b e} - \frac {5 \, B b^{2} d e^{3} {\left | b \right |} - B a b e^{4} {\left | b \right |} - 4 \, A b^{2} e^{4} {\left | b \right |}}{b^{2} e^{5}}\right )} - \frac {3 \, {\left (5 \, B b^{3} d^{2} e^{2} {\left | b \right |} - 6 \, B a b^{2} d e^{3} {\left | b \right |} - 4 \, A b^{3} d e^{3} {\left | b \right |} + B a^{2} b e^{4} {\left | b \right |} + 4 \, A a b^{2} e^{4} {\left | b \right |}\right )}}{b^{2} e^{5}}\right )}}{4 \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} - \frac {3 \, {\left (5 \, B b^{2} d^{2} {\left | b \right |} - 6 \, B a b d e {\left | b \right |} - 4 \, A b^{2} d e {\left | b \right |} + B a^{2} e^{2} {\left | b \right |} + 4 \, A a b e^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt {b e} b e^{3}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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