Integrand size = 33, antiderivative size = 171 \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (1+m) (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (2+m) (a+b x)}+\frac {b B (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (3+m) (a+b x)} \]
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Time = 0.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {784, 78} \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1) (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2) (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+3}}{e^3 (m+3) (a+b x)} \]
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Rule 78
Rule 784
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 ) (A+B x) (d+e x)^m \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e) (d+e x)^m}{e^2}+\frac {b (-2 b B d+A b e+a B e) (d+e x)^{1+m}}{e^2}+\frac {b^2 B (d+e x)^{2+m}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e) (B d-A e) (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (1+m) (a+b x)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (2+m) (a+b x)}+\frac {b B (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (3+m) (a+b x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.71 \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {(a+b x)^2} (d+e x)^{1+m} \left (a e (3+m) (-B d+A e (2+m)+B e (1+m) x)+b \left (A e (3+m) (-d+e (1+m) x)+B \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )\right )}{e^3 (1+m) (2+m) (3+m) (a+b x)} \]
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Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.20
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \sqrt {\left (b x +a \right )^{2}}\, \left (B b \,e^{2} m^{2} x^{2}+A b \,e^{2} m^{2} x +B a \,e^{2} m^{2} x +3 B b \,e^{2} m \,x^{2}+A a \,e^{2} m^{2}+4 A b \,e^{2} m x +4 B a \,e^{2} m x -2 B b d e m x +2 B b \,e^{2} x^{2}+5 A a \,e^{2} m -A b d e m +3 A b \,e^{2} x -a B d e m +3 B a \,e^{2} x -2 B b d e x +6 A a \,e^{2}-3 A b d e -3 B a d e +2 B b \,d^{2}\right )}{e^{3} \left (b x +a \right ) \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(205\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (B b \,e^{3} m^{2} x^{3}+A b \,e^{3} m^{2} x^{2}+B a \,e^{3} m^{2} x^{2}+B b d \,e^{2} m^{2} x^{2}+3 B b \,e^{3} m \,x^{3}+A a \,e^{3} m^{2} x +A b d \,e^{2} m^{2} x +4 A b \,e^{3} m \,x^{2}+B a d \,e^{2} m^{2} x +4 B a \,e^{3} m \,x^{2}+B b d \,e^{2} m \,x^{2}+2 x^{3} B b \,e^{3}+A a d \,e^{2} m^{2}+5 A a \,e^{3} m x +3 A b d \,e^{2} m x +3 A b \,e^{3} x^{2}+3 B a d \,e^{2} m x +3 B \,x^{2} a \,e^{3}-2 B b \,d^{2} e m x +5 A a d \,e^{2} m +6 A a \,e^{3} x -A b \,d^{2} e m -B a \,d^{2} e m +6 A a d \,e^{2}-3 A b \,d^{2} e -3 B a \,d^{2} e +2 B b \,d^{3}\right ) \left (e x +d \right )^{m}}{\left (b x +a \right ) \left (2+m \right ) \left (3+m \right ) \left (1+m \right ) e^{3}}\) | \(314\) |
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Time = 0.31 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.50 \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \, {\left (B a + A b\right )} d^{2} e + {\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} + {\left (3 \, {\left (B a + A b\right )} e^{3} + {\left (B b d e^{2} + {\left (B a + A b\right )} e^{3}\right )} m^{2} + {\left (B b d e^{2} + 4 \, {\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} + {\left (5 \, A a d e^{2} - {\left (B a + A b\right )} d^{2} e\right )} m + {\left (6 \, A a e^{3} + {\left (A a e^{3} + {\left (B a + A b\right )} d e^{2}\right )} m^{2} - {\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \, {\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
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\[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \left (A + B x\right ) \left (d + e x\right )^{m} \sqrt {\left (a + b x\right )^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (b e^{2} {\left (m + 1\right )} x^{2} + a d e {\left (m + 2\right )} - b d^{2} + {\left (a e^{2} {\left (m + 2\right )} + b d e m\right )} x\right )} {\left (e x + d\right )}^{m} A}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b e^{3} x^{3} - a d^{2} e {\left (m + 3\right )} + 2 \, b d^{3} + {\left ({\left (m^{2} + m\right )} b d e^{2} + {\left (m^{2} + 4 \, m + 3\right )} a e^{3}\right )} x^{2} + {\left ({\left (m^{2} + 3 \, m\right )} a d e^{2} - 2 \, b d^{2} e m\right )} x\right )} {\left (e x + d\right )}^{m} B}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (138) = 276\).
Time = 0.30 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.81 \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (e x + d\right )}^{m} B b e^{3} m^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} B b d e^{2} m^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} B a e^{3} m^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} A b e^{3} m^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )}^{m} B b e^{3} m x^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} B a d e^{2} m^{2} x \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} A b d e^{2} m^{2} x \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} A a e^{3} m^{2} x \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} B b d e^{2} m x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (e x + d\right )}^{m} B a e^{3} m x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (e x + d\right )}^{m} A b e^{3} m x^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (e x + d\right )}^{m} B b e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (e x + d\right )}^{m} A a d e^{2} m^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, {\left (e x + d\right )}^{m} B b d^{2} e m x \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )}^{m} B a d e^{2} m x \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )}^{m} A b d e^{2} m x \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (e x + d\right )}^{m} A a e^{3} m x \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )}^{m} B a e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (e x + d\right )}^{m} A b e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) - {\left (e x + d\right )}^{m} B a d^{2} e m \mathrm {sgn}\left (b x + a\right ) - {\left (e x + d\right )}^{m} A b d^{2} e m \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (e x + d\right )}^{m} A a d e^{2} m \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (e x + d\right )}^{m} A a e^{3} x \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (e x + d\right )}^{m} B b d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (e x + d\right )}^{m} B a d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (e x + d\right )}^{m} A b d^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (e x + d\right )}^{m} A a d e^{2} \mathrm {sgn}\left (b x + a\right )}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
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Timed out. \[ \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^m\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2} \,d x \]
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