3.8.72 \(\int \frac {(d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m}}{f+g x} \, dx\) [772]

Optimal. Leaf size=99 \[ \frac {(a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g) (1-m)} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {905, 70} \begin {gather*} \frac {(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m),x]

[Out]

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

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]

Rule 905

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*
(f + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2
- 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 82, normalized size = 0.83 \begin {gather*} -\frac {(d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \, _2F_1\left (1,1-m;2-m;\frac {g (a e+c d x)}{-c d f+a e g}\right )}{(c d f-a e g) (-1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m} \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{-m}}{g x +f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m/(g*x+f)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x)

[Out]

int((e*x+d)^m/(g*x+f)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),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/(g*x+f)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="maxima")

[Out]

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

[Out]

integral((x*e + d)^m/((g*x + f)*(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)^m), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m/(g*x+f)/((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**m),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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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/(g*x+f)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="giac")

[Out]

integrate((x*e + d)^m/((g*x + f)*(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^m), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{\left (f+g\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^m),x)

[Out]

int((d + e*x)^m/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^m), x)

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