3.2577 \(\int (d+e x)^{-3-2 p} (a+b x+c x^2)^p \, dx\)

Optimal. Leaf size=332 \[ \frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-1/2*e*(c*x^2+b*x+a)^(1+p)/(a*e^2-b*d*e+c*d^2)/(1+p)/((e*x+d)^(2+2*p))+1/2*(-b*e+2*c*d)*(e*x+d)^(-1-2*p)*(c*x^
2+b*x+a)^p*hypergeom([-p, -1-2*p],[-2*p],-4*c*(e*x+d)*(-4*a*c+b^2)^(1/2)/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*d-e
*(b+(-4*a*c+b^2)^(1/2))))*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/(1+2*p)/(2*c*d-e*(b-(-4*a*c+b^2)^(1
/2)))/(((2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*d-e*(b
+(-4*a*c+b^2)^(1/2))))^p)

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Rubi [A]  time = 0.14, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {730, 726} \[ \frac {\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}{\left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

-(e*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))) + ((2*c*d - b*e)*(b -
Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4
*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*
c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + 2*p)*(((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(b +
Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p)

Rule 726

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b - Rt[b^2 - 4*a*
c, 2] + 2*c*x)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Hypergeometric2F1[m + 1, -p, m + 2, (-4*c*Rt[b^2 - 4*a*c,
2]*(d + e*x))/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x))])/((m + 1)*(2*c*d - b*e
+ e*Rt[b^2 - 4*a*c, 2])*(((2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*(b + Rt[b^2 - 4*a*c, 2] + 2*c*x))/((2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)))^p), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0
]

Rule 730

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

Rubi steps

\begin {align*} \int (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^p \, dx &=-\frac {e (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (1+p)}+\frac {(2 c d-b e) \int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (1+p)}+\frac {(2 c d-b e) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+2 p)}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 294, normalized size = 0.89 \[ \frac {(d+e x)^{-2 (p+1)} (a+x (b+c x))^p \left (\frac {(d+e x) \left (\sqrt {b^2-4 a c}+b+2 c x\right ) (b e-2 c d) \left (\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (e \left (\sqrt {b^2-4 a c}+b\right )-2 c d\right )}\right )^{-p-1} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}\right )}{(2 p+1) \left (e \left (\sqrt {b^2-4 a c}+b\right )-2 c d\right )}-\frac {e (a+x (b+c x))}{p+1}\right )}{2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

((a + x*(b + c*x))^p*(-((e*(a + x*(b + c*x)))/(1 + p)) + ((-2*c*d + b*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(((2*
c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b +
Sqrt[b^2 - 4*a*c] - 2*c*x)))^(-1 - p)*(d + e*x)*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]
*(d + e*x))/((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))])/((-2*c*d + (b + Sqrt[b^2
- 4*a*c])*e)*(1 + 2*p))))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^(2*(1 + p)))

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fricas [F]  time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 3), x)

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maple [F]  time = 1.87, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{-2 p -3} \left (c \,x^{2}+b x +a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-2*p-3)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(-2*p-3)*(c*x^2+b*x+a)^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-3-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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