Optimal. Leaf size=171 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1) (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2) (a+b x)}+\frac {b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+3}}{e^3 (m+3) (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \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} \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*}
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Mathematica [A] time = 0.12, size = 121, normalized size = 0.71 \begin {gather*} \frac {\sqrt {(a+b x)^2} (d+e x)^{m+1} \left (a e (m+3) (A e (m+2)-B d+B e (m+1) x)+b \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )\right )}{e^3 (m+1) (m+2) (m+3) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.08, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^m \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 256, normalized size = 1.50 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 659, normalized size = 3.85 \begin {gather*} \frac {{\left (x e + d\right )}^{m} B b m^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} B b d m^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} B a m^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} A b m^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} B b m x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} B a d m^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} A b d m^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} B b d m x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, {\left (x e + d\right )}^{m} B b d^{2} m x e \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} A a m^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (x e + d\right )}^{m} B a m x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (x e + d\right )}^{m} A b m x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (x e + d\right )}^{m} B b x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} A a d m^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} B a d m x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} A b d m x e^{2} \mathrm {sgn}\left (b x + a\right ) - {\left (x e + d\right )}^{m} B a d^{2} m e \mathrm {sgn}\left (b x + a\right ) - {\left (x e + d\right )}^{m} A b d^{2} m e \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (x e + d\right )}^{m} B b d^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (x e + d\right )}^{m} A a m x e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} B a x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} A b x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (x e + d\right )}^{m} A a d m e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )}^{m} B a d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )}^{m} A b d^{2} e \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} A a x e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} A a d e^{2} \mathrm {sgn}\left (b x + a\right )}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 205, normalized size = 1.20 \begin {gather*} \frac {\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 \,x^{2} e^{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 ) \sqrt {\left (b x +a \right )^{2}}\, \left (e x +d \right )^{m +1}}{\left (b x +a \right ) \left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 177, normalized size = 1.04 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \left (d + e x\right )^{m} \sqrt {\left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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