3.3157 \(\int \frac{(A+B x) (d+e x)^m}{a+b x} \, dx\)

Optimal. Leaf size=85 \[ \frac{B (d+e x)^{m+1}}{b e (m+1)}-\frac{(A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \]

[Out]

(B*(d + e*x)^(1 + m))/(b*e*(1 + m)) - ((A*b - a*B)*(d + e*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(b*(b*d - a*e)*(1 + m))

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Rubi [A]  time = 0.122028, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{B (d+e x)^{m+1}}{b e (m+1)}-\frac{(A b-a B) (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^m)/(a + b*x),x]

[Out]

(B*(d + e*x)^(1 + m))/(b*e*(1 + m)) - ((A*b - a*B)*(d + e*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/(b*(b*d - a*e)*(1 + m))

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Rubi in Sympy [A]  time = 12.6737, size = 63, normalized size = 0.74 \[ \frac{B \left (d + e x\right )^{m + 1}}{b e \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A b - B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{b \left (m + 1\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m/(b*x+a),x)

[Out]

B*(d + e*x)**(m + 1)/(b*e*(m + 1)) + (d + e*x)**(m + 1)*(A*b - B*a)*hyper((1, m
+ 1), (m + 2,), b*(-d - e*x)/(a*e - b*d))/(b*(m + 1)*(a*e - b*d))

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Mathematica [A]  time = 0.110255, size = 78, normalized size = 0.92 \[ \frac{(d+e x)^{m+1} \left ((a B e-A b e) \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )+B (b d-a e)\right )}{b e (m+1) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^m)/(a + b*x),x]

[Out]

((d + e*x)^(1 + m)*(B*(b*d - a*e) + (-(A*b*e) + a*B*e)*Hypergeometric2F1[1, 1 +
m, 2 + m, (b*(d + e*x))/(b*d - a*e)]))/(b*e*(b*d - a*e)*(1 + m))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^m/(b*x+a),x)

[Out]

int((B*x+A)*(e*x+d)^m/(b*x+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b*x + a),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b*x + a),x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x + d)^m/(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{m}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**m/(b*x+a),x)

[Out]

Integral((A + B*x)*(d + e*x)**m/(a + b*x), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b*x + a),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b*x + a), x)