3.2093 \(\int (d+e x)^{-5-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=288 \[ \frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]

[Out]

((d + e*x)^(-5 - 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)*(4 + p)) + (6*c^2*d^2*(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)^3*(2 + p)*(3 + p)*(4 + p)) + (6*c^3*d^3*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)
*(3 + p)*(4 + p)*(d + e*x)^(2*(1 + p))) + (3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^2*(3 + p)*(4 + p)*(d + e*x)^(2*(2 + p)))

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Rubi [A]  time = 0.432876, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

((d + e*x)^(-5 - 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)*(4 + p)) + (6*c^2*d^2*(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)^3*(2 + p)*(3 + p)*(4 + p)) + (6*c^3*d^3*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)
*(3 + p)*(4 + p)*(d + e*x)^(2*(1 + p))) + (3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^2*(3 + p)*(4 + p)*(d + e*x)^(2*(2 + p)))

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Rubi in Sympy [A]  time = 126.583, size = 264, normalized size = 0.92 \[ \frac{6 c^{3} d^{3} \left (d + e x\right )^{- 2 p - 2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{4}} - \frac{6 c^{2} d^{2} \left (d + e x\right )^{- 2 p - 3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{3 c d \left (d + e x\right )^{- 2 p - 4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\left (d + e x\right )^{- 2 p - 5} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 4\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-5-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

6*c**3*d**3*(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)*(p + 4)*(a*e**2 - c*d**2)**4) - 6*c**2*d**2*(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)*(p + 4)*(a*e**2 - c*d**2)**3) + 3*c*d*(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)*(p + 4)*(a*e**2 - c*d**2)**2) - (
d + e*x)**(-2*p - 5)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(p + 1)/((p + 4
)*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.649324, size = 216, normalized size = 0.75 \[ \frac{(d+e x)^{-2 p} ((d+e x) (a e+c d x))^p \left (\frac{6 c^4 d^4}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^3 d^3 p}{(p+1) \left (p^3+9 p^2+26 p+24\right ) (d+e x) \left (c d^2-a e^2\right )^3}+\frac{3 c^2 d^2 p}{(p+2) \left (p^2+7 p+12\right ) (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{c d p}{(p+3) (p+4) (d+e x)^3 \left (c d^2-a e^2\right )}-\frac{1}{(p+4) (d+e x)^4}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(-5 - 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*((6*c^4*d^4)/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)*(3
+ p)*(4 + p)) - 1/((4 + p)*(d + e*x)^4) + (c*d*p)/((c*d^2 - a*e^2)*(3 + p)*(4 +
p)*(d + e*x)^3) + (3*c^2*d^2*p)/((c*d^2 - a*e^2)^2*(2 + p)*(12 + 7*p + p^2)*(d +
e*x)^2) + (6*c^3*d^3*p)/((c*d^2 - a*e^2)^3*(1 + p)*(24 + 26*p + 9*p^2 + p^3)*(d
+ e*x))))/(e*(d + e*x)^(2*p))

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Maple [B]  time = 0.016, size = 745, normalized size = 2.6 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-4-2\,p} \left ({a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}-3\,{a}^{2}cd{e}^{5}{p}^{2}x+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}+6\,a{c}^{2}{d}^{3}{e}^{3}{p}^{2}x+6\,a{c}^{2}{d}^{2}{e}^{4}p{x}^{2}-{c}^{3}{d}^{6}{p}^{3}-3\,{c}^{3}{d}^{5}e{p}^{2}x-6\,{c}^{3}{d}^{4}{e}^{2}p{x}^{2}-6\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-21\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}-9\,{a}^{2}cd{e}^{5}px+24\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}+30\,a{c}^{2}{d}^{3}{e}^{3}px+6\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-9\,{c}^{3}{d}^{6}{p}^{2}-21\,{c}^{3}{d}^{5}epx-24\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+11\,{a}^{3}{e}^{6}p-42\,{a}^{2}c{d}^{2}{e}^{4}p-6\,x{a}^{2}cd{e}^{5}+57\,a{c}^{2}{d}^{4}{e}^{2}p+24\,xa{c}^{2}{d}^{3}{e}^{3}-26\,{c}^{3}{d}^{6}p-36\,{c}^{3}{d}^{5}ex+6\,{a}^{3}{e}^{6}-24\,{a}^{2}c{d}^{2}{e}^{4}+36\,{c}^{2}{d}^{4}a{e}^{2}-24\,{c}^{3}{d}^{6} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{{a}^{4}{e}^{8}{p}^{4}-4\,{a}^{3}c{d}^{2}{e}^{6}{p}^{4}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}{p}^{4}+{c}^{4}{d}^{8}{p}^{4}+10\,{a}^{4}{e}^{8}{p}^{3}-40\,{a}^{3}c{d}^{2}{e}^{6}{p}^{3}+60\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{3}-40\,a{c}^{3}{d}^{6}{e}^{2}{p}^{3}+10\,{c}^{4}{d}^{8}{p}^{3}+35\,{a}^{4}{e}^{8}{p}^{2}-140\,{a}^{3}c{d}^{2}{e}^{6}{p}^{2}+210\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{2}-140\,a{c}^{3}{d}^{6}{e}^{2}{p}^{2}+35\,{c}^{4}{d}^{8}{p}^{2}+50\,{a}^{4}{e}^{8}p-200\,{a}^{3}c{d}^{2}{e}^{6}p+300\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}p-200\,a{c}^{3}{d}^{6}{e}^{2}p+50\,{c}^{4}{d}^{8}p+24\,{a}^{4}{e}^{8}-96\,{a}^{3}c{d}^{2}{e}^{6}+144\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-96\,a{c}^{3}{d}^{6}{e}^{2}+24\,{c}^{4}{d}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-5-2*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)^(-4-2*p)*(a^3*e^6*p^3-3*a^2*c*d^2*e^4*p^3-3*a^2*c*d*e^5*p^2
*x+3*a*c^2*d^4*e^2*p^3+6*a*c^2*d^3*e^3*p^2*x+6*a*c^2*d^2*e^4*p*x^2-c^3*d^6*p^3-3
*c^3*d^5*e*p^2*x-6*c^3*d^4*e^2*p*x^2-6*c^3*d^3*e^3*x^3+6*a^3*e^6*p^2-21*a^2*c*d^
2*e^4*p^2-9*a^2*c*d*e^5*p*x+24*a*c^2*d^4*e^2*p^2+30*a*c^2*d^3*e^3*p*x+6*a*c^2*d^
2*e^4*x^2-9*c^3*d^6*p^2-21*c^3*d^5*e*p*x-24*c^3*d^4*e^2*x^2+11*a^3*e^6*p-42*a^2*
c*d^2*e^4*p-6*a^2*c*d*e^5*x+57*a*c^2*d^4*e^2*p+24*a*c^2*d^3*e^3*x-26*c^3*d^6*p-3
6*c^3*d^5*e*x+6*a^3*e^6-24*a^2*c*d^2*e^4+36*a*c^2*d^4*e^2-24*c^3*d^6)*(c*d*e*x^2
+a*e^2*x+c*d^2*x+a*d*e)^p/(a^4*e^8*p^4-4*a^3*c*d^2*e^6*p^4+6*a^2*c^2*d^4*e^4*p^4
-4*a*c^3*d^6*e^2*p^4+c^4*d^8*p^4+10*a^4*e^8*p^3-40*a^3*c*d^2*e^6*p^3+60*a^2*c^2*
d^4*e^4*p^3-40*a*c^3*d^6*e^2*p^3+10*c^4*d^8*p^3+35*a^4*e^8*p^2-140*a^3*c*d^2*e^6
*p^2+210*a^2*c^2*d^4*e^4*p^2-140*a*c^3*d^6*e^2*p^2+35*c^4*d^8*p^2+50*a^4*e^8*p-2
00*a^3*c*d^2*e^6*p+300*a^2*c^2*d^4*e^4*p-200*a*c^3*d^6*e^2*p+50*c^4*d^8*p+24*a^4
*e^8-96*a^3*c*d^2*e^6+144*a^2*c^2*d^4*e^4-96*a*c^3*d^6*e^2+24*c^4*d^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

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 - 5),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 - 5), x)

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Fricas [A]  time = 0.244581, size = 1419, normalized size = 4.93 \[ \text{result too large to display} \]

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 - 5),x, algorithm="fricas")

[Out]

(6*c^4*d^4*e^4*x^5 + 24*a*c^3*d^7*e - 36*a^2*c^2*d^5*e^3 + 24*a^3*c*d^3*e^5 - 6*
a^4*d*e^7 + 6*(5*c^4*d^5*e^3 + (c^4*d^5*e^3 - a*c^3*d^3*e^5)*p)*x^4 + (a*c^3*d^7
*e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7)*p^3 + 3*(20*c^4*d^6*e^2 +
(c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*p^2 + (9*c^4*d^6*e^2 - 10*a*c^
3*d^4*e^4 + a^2*c^2*d^2*e^6)*p)*x^3 + 3*(3*a*c^3*d^7*e - 8*a^2*c^2*d^5*e^3 + 7*a
^3*c*d^3*e^5 - 2*a^4*d*e^7)*p^2 + (60*c^4*d^7*e + (c^4*d^7*e - 3*a*c^3*d^5*e^3 +
3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*p^3 + 3*(4*c^4*d^7*e - 9*a*c^3*d^5*e^3 + 6*a^2
*c^2*d^3*e^5 - a^3*c*d*e^7)*p^2 + (47*c^4*d^7*e - 60*a*c^3*d^5*e^3 + 15*a^2*c^2*
d^3*e^5 - 2*a^3*c*d*e^7)*p)*x^2 + (26*a*c^3*d^7*e - 57*a^2*c^2*d^5*e^3 + 42*a^3*
c*d^3*e^5 - 11*a^4*d*e^7)*p + (24*c^4*d^8 + 24*a*c^3*d^6*e^2 - 36*a^2*c^2*d^4*e^
4 + 24*a^3*c*d^2*e^6 - 6*a^4*e^8 + (c^4*d^8 - 2*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6
- a^4*e^8)*p^3 + 3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 6*a^3*c*d^
2*e^6 - 2*a^4*e^8)*p^2 + (26*c^4*d^8 - 10*a*c^3*d^6*e^2 - 45*a^2*c^2*d^4*e^4 + 4
0*a^3*c*d^2*e^6 - 11*a^4*e^8)*p)*x)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e
*x + d)^(-2*p - 5)/(24*c^4*d^8 - 96*a*c^3*d^6*e^2 + 144*a^2*c^2*d^4*e^4 - 96*a^3
*c*d^2*e^6 + 24*a^4*e^8 + (c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3
*c*d^2*e^6 + a^4*e^8)*p^4 + 10*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 -
4*a^3*c*d^2*e^6 + a^4*e^8)*p^3 + 35*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e
^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^2 + 50*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*
d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*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)**(-5-2*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. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

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 - 5),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 - 5), x)