Integrand size = 28, antiderivative size = 219 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {(b d-a e)^3 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (1+m) (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (2+m) (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (3+m) (a+b x)}+\frac {b^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (4+m) (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^m \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (d+e x)^m}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{1+m}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{2+m}}{e^3}+\frac {b^6 (d+e x)^{3+m}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {(b d-a e)^3 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (1+m) (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (2+m) (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (3+m) (a+b x)}+\frac {b^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (4+m) (a+b x)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.52 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {\left ((a+b x)^2\right )^{3/2} (d+e x)^{1+m} \left (-\frac {(b d-a e)^3}{1+m}+\frac {3 b (b d-a e)^2 (d+e x)}{2+m}-\frac {3 b^2 (b d-a e) (d+e x)^2}{3+m}+\frac {b^3 (d+e x)^3}{4+m}\right )}{e^4 (a+b x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs. \(2(175)=350\).
Time = 2.25 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.84
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 e^{3} x^{3} b^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 x^{2} a \,b^{2} e^{3}+6 b^{3} d^{2} e m x -6 x^{2} b^{3} d \,e^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +24 a^{3} e^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right )}{e^{4} \left (b x +a \right )^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(402\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (b^{3} e^{4} m^{3} x^{4}+3 a \,b^{2} e^{4} m^{3} x^{3}+b^{3} d \,e^{3} m^{3} x^{3}+6 b^{3} e^{4} m^{2} x^{4}+3 a^{2} b \,e^{4} m^{3} x^{2}+3 a \,b^{2} d \,e^{3} m^{3} x^{2}+21 a \,b^{2} e^{4} m^{2} x^{3}+3 b^{3} d \,e^{3} m^{2} x^{3}+11 b^{3} e^{4} m \,x^{4}+a^{3} e^{4} m^{3} x +3 a^{2} b d \,e^{3} m^{3} x +24 a^{2} b \,e^{4} m^{2} x^{2}+15 a \,b^{2} d \,e^{3} m^{2} x^{2}+42 a \,b^{2} e^{4} m \,x^{3}-3 b^{3} d^{2} e^{2} m^{2} x^{2}+2 b^{3} d \,e^{3} m \,x^{3}+6 b^{3} x^{4} e^{4}+a^{3} d \,e^{3} m^{3}+9 a^{3} e^{4} m^{2} x +21 a^{2} b d \,e^{3} m^{2} x +57 a^{2} b \,e^{4} m \,x^{2}-6 a \,b^{2} d^{2} e^{2} m^{2} x +12 a \,b^{2} d \,e^{3} m \,x^{2}+24 a \,b^{2} e^{4} x^{3}-3 b^{3} d^{2} e^{2} m \,x^{2}+9 a^{3} d \,e^{3} m^{2}+26 a^{3} e^{4} m x -3 a^{2} b \,d^{2} e^{2} m^{2}+36 a^{2} b d \,e^{3} m x +36 a^{2} b \,e^{4} x^{2}-24 a \,b^{2} d^{2} e^{2} m x +6 b^{3} d^{3} e m x +26 a^{3} d \,e^{3} m +24 e^{4} a^{3} x -21 a^{2} b \,d^{2} e^{2} m +6 a \,b^{2} d^{3} e m +24 a^{3} d \,e^{3}-36 a^{2} b \,d^{2} e^{2}+24 a \,b^{2} d^{3} e -6 b^{3} d^{4}\right ) \left (e x +d \right )^{m}}{\left (b x +a \right ) \left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(563\) |
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Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (175) = 350\).
Time = 0.58 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.26 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} + {\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} + {\left (24 \, a b^{2} e^{4} + {\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \, {\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \, {\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \, {\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \, {\left (12 \, a^{2} b e^{4} + {\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} - {\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} - {\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} + {\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m + {\left (24 \, a^{3} e^{4} + {\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \, {\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \, {\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
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\[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (d + e x\right )^{m} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.39 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3} e^{4} x^{4} - 3 \, {\left (m^{2} + 7 \, m + 12\right )} a^{2} b d^{2} e^{2} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} d e^{3} + 6 \, a b^{2} d^{3} e {\left (m + 4\right )} - 6 \, b^{3} d^{4} + {\left ({\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d e^{3} + 3 \, {\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a b^{2} e^{4}\right )} x^{3} - 3 \, {\left ({\left (m^{2} + m\right )} b^{3} d^{2} e^{2} - {\left (m^{3} + 5 \, m^{2} + 4 \, m\right )} a b^{2} d e^{3} - {\left (m^{3} + 8 \, m^{2} + 19 \, m + 12\right )} a^{2} b e^{4}\right )} x^{2} - {\left (6 \, {\left (m^{2} + 4 \, m\right )} a b^{2} d^{2} e^{2} - 3 \, {\left (m^{3} + 7 \, m^{2} + 12 \, m\right )} a^{2} b d e^{3} - {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} e^{4} - 6 \, b^{3} d^{3} e m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (175) = 350\).
Time = 0.31 (sec) , antiderivative size = 1073, normalized size of antiderivative = 4.90 \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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