3.2092 \(\int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=206 \[ \frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]

[Out]

(2*c*d*(d + e*x)^(-3 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c
*d^2 - a*e^2)^2*(2 + p)*(3 + p)) + (2*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(1 + p))/((c*d^2 - a*e^2)^3*(1 + p)*(2 + p)*(3 + p)*(d + e*x)^(2*(1 + p)))
+ (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p)/((c*d^2 - a*e^2)*(3 + p)*(d +
e*x)^(2*(2 + p)))

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Rubi [A]  time = 0.257502, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

(2*c*d*(d + e*x)^(-3 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c
*d^2 - a*e^2)^2*(2 + p)*(3 + p)) + (2*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(1 + p))/((c*d^2 - a*e^2)^3*(1 + p)*(2 + p)*(3 + p)*(d + e*x)^(2*(1 + p)))
+ (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p)/((c*d^2 - a*e^2)*(3 + p)*(d +
e*x)^(2*(2 + p)))

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Rubi in Sympy [A]  time = 78.1269, size = 189, normalized size = 0.92 \[ - \frac{2 c^{2} d^{2} \left (d + e x\right )^{- 2 p - 2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{2 c d \left (d + e x\right )^{- 2 p - 3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right ) \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\left (d + e x\right )^{- 2 p - 4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 3\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-4-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

-2*c**2*d**2*(d + e*x)**(-2*p - 2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(
p + 1)/((p + 1)*(p + 2)*(p + 3)*(a*e**2 - c*d**2)**3) + 2*c*d*(d + e*x)**(-2*p -
3)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(p + 1)/((p + 2)*(p + 3)*(a*e**2
- c*d**2)**2) - (d + e*x)**(-2*p - 4)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)**(p + 1)/((p + 3)*(a*e**2 - c*d**2))

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Mathematica [C]  time = 0.149826, size = 101, normalized size = 0.49 \[ -\frac{(d+e x)^{-2 p-3} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p-3,-p;-p-2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

-(((d + e*x)^(-3 - 2*p)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-3 - p, -p
, -2 - p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e*(3 + p)*((e*(a*e + c*d*x))/(-(c*d
^2) + a*e^2))^p))

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Maple [A]  time = 0.009, size = 381, normalized size = 1.9 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-3-2\,p} \left ({a}^{2}{e}^{4}{p}^{2}-2\,ac{d}^{2}{e}^{2}{p}^{2}-2\,acd{e}^{3}px+{c}^{2}{d}^{4}{p}^{2}+2\,{c}^{2}{d}^{3}epx+2\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+3\,{a}^{2}{e}^{4}p-8\,ac{d}^{2}{e}^{2}p-2\,xacd{e}^{3}+5\,{c}^{2}{d}^{4}p+6\,x{c}^{2}{d}^{3}e+2\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}+6\,{c}^{2}{d}^{4} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{{a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}-{c}^{3}{d}^{6}{p}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-18\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}+18\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}-6\,{c}^{3}{d}^{6}{p}^{2}+11\,{a}^{3}{e}^{6}p-33\,{a}^{2}c{d}^{2}{e}^{4}p+33\,a{c}^{2}{d}^{4}{e}^{2}p-11\,{c}^{3}{d}^{6}p+6\,{a}^{3}{e}^{6}-18\,{a}^{2}c{d}^{2}{e}^{4}+18\,{c}^{2}{d}^{4}a{e}^{2}-6\,{c}^{3}{d}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-4-2*p)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)

[Out]

-(c*d*x+a*e)*(e*x+d)^(-3-2*p)*(a^2*e^4*p^2-2*a*c*d^2*e^2*p^2-2*a*c*d*e^3*p*x+c^2
*d^4*p^2+2*c^2*d^3*e*p*x+2*c^2*d^2*e^2*x^2+3*a^2*e^4*p-8*a*c*d^2*e^2*p-2*a*c*d*e
^3*x+5*c^2*d^4*p+6*c^2*d^3*e*x+2*a^2*e^4-6*a*c*d^2*e^2+6*c^2*d^4)*(c*d*e*x^2+a*e
^2*x+c*d^2*x+a*d*e)^p/(a^3*e^6*p^3-3*a^2*c*d^2*e^4*p^3+3*a*c^2*d^4*e^2*p^3-c^3*d
^6*p^3+6*a^3*e^6*p^2-18*a^2*c*d^2*e^4*p^2+18*a*c^2*d^4*e^2*p^2-6*c^3*d^6*p^2+11*
a^3*e^6*p-33*a^2*c*d^2*e^4*p+33*a*c^2*d^4*e^2*p-11*c^3*d^6*p+6*a^3*e^6-18*a^2*c*
d^2*e^4+18*a*c^2*d^4*e^2-6*c^3*d^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4), x)

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Fricas [A]  time = 0.239958, size = 783, normalized size = 3.8 \[ \frac{{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \,{\left (4 \, c^{3} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} +{\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} +{\left (12 \, c^{3} d^{5} e +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} +{\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} +{\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p +{\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} +{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} +{\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\right )} p\right )} x\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} +{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="fricas")

[Out]

(2*c^3*d^3*e^3*x^4 + 6*a*c^2*d^5*e - 6*a^2*c*d^3*e^3 + 2*a^3*d*e^5 + 2*(4*c^3*d^
4*e^2 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*p)*x^3 + (a*c^2*d^5*e - 2*a^2*c*d^3*e^3 +
a^3*d*e^5)*p^2 + (12*c^3*d^5*e + (c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*p^2
+ (7*c^3*d^5*e - 8*a*c^2*d^3*e^3 + a^2*c*d*e^5)*p)*x^2 + (5*a*c^2*d^5*e - 8*a^2
*c*d^3*e^3 + 3*a^3*d*e^5)*p + (6*c^3*d^6 + 6*a*c^2*d^4*e^2 - 6*a^2*c*d^2*e^4 + 2
*a^3*e^6 + (c^3*d^6 - a*c^2*d^4*e^2 - a^2*c*d^2*e^4 + a^3*e^6)*p^2 + (5*c^3*d^6
- a*c^2*d^4*e^2 - 7*a^2*c*d^2*e^4 + 3*a^3*e^6)*p)*x)*(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)^p*(e*x + d)^(-2*p - 4)/(6*c^3*d^6 - 18*a*c^2*d^4*e^2 + 18*a^2*c*d^2
*e^4 - 6*a^3*e^6 + (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p^3 +
6*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p^2 + 11*(c^3*d^6 - 3
*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(-4-2*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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4), x)