3.2149 \(\int \frac {(a+b x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 51, 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} \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]

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]

-(((d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)*(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}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^m}{a+b x} \, dx\\ &=-\frac {(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.00 \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(b*x + a), x)

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

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)

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maple [F]  time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x +d \right )^{m}}{b^{2} x^{2}+2 a b x +a^{2}}\, dx \]

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

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

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)**m/(a + b*x), x)

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