3.1886 \(\int \frac{(A+B x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=112 \[ \frac{(d+e x)^{m+1} (a B e (m+1)-b (A e m+B d)) \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)^2}-\frac{(A b-a B) (d+e x)^{m+1}}{b (a+b x) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.173426, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{(d+e x)^{m+1} (a B e (m+1)-b (A e m+B d)) \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)^2}-\frac{(A b-a B) (d+e x)^{m+1}}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.5738, size = 88, normalized size = 0.79 \[ - \frac{\left (d + e x\right )^{m + 1} \left (A b e m + B \left (- a e \left (m + 1\right ) + b d\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{b \left (m + 1\right ) \left (a e - b d\right )^{2}} + \frac{\left (d + e x\right )^{m + 1} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.153173, size = 0, normalized size = 0. \[ \int \frac{(A+B x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[((A + B*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

int((B*x+A)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{m}}{\left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Integral((A + B*x)*(d + e*x)**m/(a + b*x)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2), x)