Optimal. Leaf size=135 \[ \frac {B (a+b x) (d+e x)^{m+1}}{b e (m+1) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) (A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 80, 68} \[ \frac {B (a+b x) (d+e x)^{m+1}}{b e (m+1) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) (A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 80
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)^m}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {B (a+b x) (d+e x)^{1+m}}{b e (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (A b^2 e (1+m)-a b B e (1+m)\right ) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^m}{a b+b^2 x} \, dx}{b^2 e (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {B (a+b x) (d+e x)^{1+m}}{b e (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (a+b x) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e) (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 94, normalized size = 0.70 \[ \frac {(a+b x) (d+e x)^{m+1} \left ((a B e-A b e) \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )+B (b d-a e)\right )}{b e (m+1) \sqrt {(a+b x)^2} (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}} \,d x \]
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 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{m}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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