3.1887 \(\int \frac{(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{e^2 (d+e x)^{m+1} (b (3 B d-A e (2-m))-a B e (m+1)) \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{3 b (m+1) (b d-a e)^4}-\frac{(A b-a B) (d+e x)^{m+1}}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

-((A*b - a*B)*(d + e*x)^(1 + m))/(3*b*(b*d - a*e)*(a + b*x)^3) - (e^2*(b*(3*B*d
- A*e*(2 - m)) - a*B*e*(1 + m))*(d + e*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2
+ m, (b*(d + e*x))/(b*d - a*e)])/(3*b*(b*d - a*e)^4*(1 + m))

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Rubi [A]  time = 0.183091, antiderivative size = 125, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{e^2 (d+e x)^{m+1} (-a B e (m+1)-A b e (2-m)+3 b B d) \, _2F_1\left (3,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{3 b (m+1) (b d-a e)^4}-\frac{(A b-a B) (d+e x)^{m+1}}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b - a*B)*(d + e*x)^(1 + m))/(3*b*(b*d - a*e)*(a + b*x)^3) - (e^2*(3*b*B*d -
A*b*e*(2 - m) - a*B*e*(1 + m))*(d + e*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2
+ m, (b*(d + e*x))/(b*d - a*e)])/(3*b*(b*d - a*e)^4*(1 + m))

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Rubi in Sympy [A]  time = 45.7669, size = 100, normalized size = 0.79 \[ - \frac{e^{2} \left (d + e x\right )^{m + 1} \left (- A b e \left (- m + 2\right ) + B \left (- a e \left (m + 1\right ) + 3 b d\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{3 b \left (m + 1\right ) \left (a e - b d\right )^{4}} + \frac{\left (d + e x\right )^{m + 1} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} \]

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)**2,x)

[Out]

-e**2*(d + e*x)**(m + 1)*(-A*b*e*(-m + 2) + B*(-a*e*(m + 1) + 3*b*d))*hyper((3,
m + 1), (m + 2,), b*(-d - e*x)/(a*e - b*d))/(3*b*(m + 1)*(a*e - b*d)**4) + (d +
e*x)**(m + 1)*(A*b - B*a)/(3*b*(a + b*x)**3*(a*e - b*d))

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Mathematica [A]  time = 0.231497, size = 0, normalized size = 0. \[ \int \frac{(A+B x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \]

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)^2,x]

[Out]

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

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x + d)^m/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*
x + a^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

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