Integrand size = 24, antiderivative size = 250 \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}} \]
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Time = 0.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\frac {(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \]
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {3 b d}{2}+\frac {5 a e}{2}\right )\right ) \int \sqrt {a+b x} (d+e x)^{3/2} \, dx}{4 b e} \\ & = -\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {((b d-a e) (3 b B d-8 A b e+5 a B e)) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{16 b^2 e} \\ & = -\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {\left ((b d-a e)^2 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b^3 e} \\ & = -\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^3 e^2} \\ & = -\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^4 e^2} \\ & = -\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^4 e^2} \\ & = -\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^3 B e^3-a^2 b e^2 (31 B d+24 A e+10 B e x)+a b^2 e \left (16 A e (4 d+e x)+B \left (9 d^2+20 d e x+8 e^2 x^2\right )\right )+b^3 \left (8 A e \left (3 d^2+14 d e x+8 e^2 x^2\right )+B \left (-9 d^3+6 d^2 e x+72 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^3 e^2}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(212)=424\).
Time = 1.07 (sec) , antiderivative size = 968, normalized size of antiderivative = 3.87
method | result | size |
default | \(\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (40 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} 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 ) a^{4} e^{4}+9 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^{4} d^{4}+48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e -48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3}-12 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^{3} d^{3} e -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^{2} b^{2} d^{2} e^{2}-72 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^{2} b^{2} d \,e^{3}+72 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^{3} d^{2} e^{2}+36 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} b d \,e^{3}-62 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}+18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e +30 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{3} e^{3}-18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{3}+24 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^{3} b \,e^{4}-24 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^{4} d^{3} e +32 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} e^{3} x +224 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d \,e^{2} x -20 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x +12 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e x +16 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+144 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+128 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} \sqrt {b e}}\) | \(968\) |
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Time = 0.27 (sec) , antiderivative size = 766, normalized size of antiderivative = 3.06 \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\left [\frac {3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\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 (48 \, B b^{4} e^{4} x^{3} - 9 \, B b^{4} d^{3} e + 3 \, {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2} e^{2} - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (9 \, B b^{4} d e^{3} + {\left (B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (3 \, B b^{4} d^{2} e^{2} + 2 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d e^{3} - {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{4} e^{3}}, -\frac {3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\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 (48 \, B b^{4} e^{4} x^{3} - 9 \, B b^{4} d^{3} e + 3 \, {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2} e^{2} - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (9 \, B b^{4} d e^{3} + {\left (B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (3 \, B b^{4} d^{2} e^{2} + 2 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d e^{3} - {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{4} e^{3}}\right ] \]
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\[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A + B x\right ) \sqrt {a + b x} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1423 vs. \(2 (212) = 424\).
Time = 0.47 (sec) , antiderivative size = 1423, normalized size of antiderivative = 5.69 \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A+B\,x\right )\,\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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