3.18.36 \(\int \frac {(d+e x)^m}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1736]

Optimal. Leaf size=53 \[ \frac {e^5 (d+e x)^{1+m} \, _2F_1\left (6,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^6 (1+m)} \]

[Out]

e^5*(e*x+d)^(1+m)*hypergeom([6, 1+m],[2+m],b*(e*x+d)/(-a*e+b*d))/(-a*e+b*d)^6/(1+m)

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 70} \begin {gather*} \frac {e^5 (d+e x)^{m+1} \, _2F_1\left (6,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^5*(d + e*x)^(1 + m)*Hypergeometric2F1[6, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a*e)])/((b*d - a*e)^6*(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 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^m}{(a+b x)^6} \, dx\\ &=\frac {e^5 (d+e x)^{1+m} \, _2F_1\left (6,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^6 (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 54, normalized size = 1.02 \begin {gather*} \frac {e^5 (d+e x)^{1+m} \, _2F_1\left (6,1+m;2+m;-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^6 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^5*(d + e*x)^(1 + m)*Hypergeometric2F1[6, 1 + m, 2 + m, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^6*
(1 + m))

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Maple [F]
time = 0.72, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

integral((x*e + d)^m/(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a
^6), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{m}}{\left (a + b x\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)**m/(a + b*x)**6, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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