3.2150 \(\int \frac {(a+b x) (d+e x)^m}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3} \]

[Out]

-e^2*(e*x+d)^(1+m)*hypergeom([3, 1+m],[2+m],b*(e*x+d)/(-a*e+b*d))/(-a*e+b*d)^3/(1+m)

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Rubi [A]  time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 68} \[ -\frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3} \]

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]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^m}{(a+b x)^3} \, dx\\ &=-\frac {e^2 (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.00 \[ \frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac {b (d+e x)}{a e-b d}\right )}{(m+1) (a e-b d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]

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((e*x + d)^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]

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)

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maple [F]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{m}}{\left (a + b x\right )^{3}}\, dx \]

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)**2,x)

[Out]

Integral((d + e*x)**m/(a + b*x)**3, x)

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