3.21.100 \(\int (d+e x)^{-1-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2100]

Optimal. Leaf size=107 \[ -\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p} (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {693, 691, 72, 71} \begin {gather*} -\frac {(d+e x)^{-2 p} \left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-1 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

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

Rule 71

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

Rule 72

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

Rule 691

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^m*((a + b*x + c*x^2
)^FracPart[p]/((1 + e*(x/d))^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)
*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^IntPart[m]*((d + e*
x)^FracPart[m]/(1 + e*(x/d))^FracPart[m]), Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ
[d, 0])

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 95, normalized size = 0.89 \begin {gather*} -\frac {\left (\frac {e (a e+c d x)}{-c d^2+a e^2}\right )^{-p} (d+e x)^{-2 p} ((a e+c d x) (d+e x))^p \, _2F_1\left (-p,-p;1-p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(-1 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

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

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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{-1-2 p} \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-1-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)

[Out]

int((e*x+d)^(-1-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,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)^(-1-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^p*(x*e + d)^(-2*p - 1), 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)^(-1-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="fricas")

[Out]

integral((c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)^p*(x*e + d)^(-2*p - 1), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-1-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

Timed out

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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)^(-1-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="giac")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^p*(x*e + d)^(-2*p - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^(2*p + 1),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^(2*p + 1), x)

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