3.1921 \(\int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=474 \[ -\frac{55 \left (c d^2-a e^2\right )^9 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac{55 \left (c d^2-a e^2\right )^7 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32768 c^6 d^6 e^3}-\frac{55 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac{11 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac{11 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac{11 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \]

[Out]

(55*(c*d^2 - a*e^2)^7*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(32768*c^6*d^6*e^3) - (55*(c*d^2 - a*e^2)^5*(c*d^2 + a*e^2 + 2*c*
d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12288*c^5*d^5*e^2) + (11*
(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(5/2))/(768*c^4*d^4*e) + (11*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(7/2))/(224*c^3*d^3) + (11*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(144*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) - (55*(c*d^2 - a*e^2)^9*ArcTanh[(c*d^2 + a*
e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])])/(65536*c^(13/2)*d^(13/2)*e^(7/2))

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Rubi [A]  time = 1.19235, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{55 \left (c d^2-a e^2\right )^9 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac{55 \left (c d^2-a e^2\right )^7 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32768 c^6 d^6 e^3}-\frac{55 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac{11 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac{11 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac{11 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(55*(c*d^2 - a*e^2)^7*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(32768*c^6*d^6*e^3) - (55*(c*d^2 - a*e^2)^5*(c*d^2 + a*e^2 + 2*c*
d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12288*c^5*d^5*e^2) + (11*
(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(5/2))/(768*c^4*d^4*e) + (11*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(7/2))/(224*c^3*d^3) + (11*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(144*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) - (55*(c*d^2 - a*e^2)^9*ArcTanh[(c*d^2 + a*
e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])])/(65536*c^(13/2)*d^(13/2)*e^(7/2))

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Rubi in Sympy [A]  time = 160.468, size = 459, normalized size = 0.97 \[ \frac{\left (d + e x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{9 c d} - \frac{11 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{144 c^{2} d^{2}} + \frac{11 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{224 c^{3} d^{3}} - \frac{11 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{768 c^{4} d^{4} e} + \frac{55 \left (a e^{2} - c d^{2}\right )^{5} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{12288 c^{5} d^{5} e^{2}} - \frac{55 \left (a e^{2} - c d^{2}\right )^{7} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{32768 c^{6} d^{6} e^{3}} + \frac{55 \left (a e^{2} - c d^{2}\right )^{9} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{65536 c^{\frac{13}{2}} d^{\frac{13}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

(d + e*x)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(9*c*d) - 11*(d +
e*x)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(144*c
**2*d**2) + 11*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**
(7/2)/(224*c**3*d**3) - 11*(a*e**2 - c*d**2)**3*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a
*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(768*c**4*d**4*e) + 55*(a*e**2 -
c*d**2)**5*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d*
*2))**(3/2)/(12288*c**5*d**5*e**2) - 55*(a*e**2 - c*d**2)**7*(a*e**2 + c*d**2 +
2*c*d*e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(32768*c**6*d**6*e**3)
+ 55*(a*e**2 - c*d**2)**9*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d
)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(65536*c**(13/2)*d**(
13/2)*e**(7/2))

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Mathematica [A]  time = 2.60155, size = 638, normalized size = 1.35 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \left (-3465 a^8 e^{16}+2310 a^7 c d e^{14} (13 d+e x)-462 a^6 c^2 d^2 e^{12} \left (249 d^2+43 d e x+4 e^2 x^2\right )+198 a^5 c^3 d^3 e^{10} \left (1289 d^3+381 d^2 e x+80 d e^2 x^2+8 e^3 x^3\right )-22 a^4 c^4 d^4 e^8 \left (16384 d^4+7531 d^3 e x+2724 d^2 e^2 x^2+616 d e^3 x^3+64 e^4 x^4\right )+2 a^3 c^5 d^5 e^6 \left (167301 d^5+115609 d^4 e x+65536 d^3 e^2 x^2+25584 d^2 e^3 x^3+6016 d e^4 x^4+640 e^5 x^5\right )+6 a^2 c^6 d^6 e^4 \left (19173 d^6+282339 d^5 e x+763652 d^4 e^2 x^2+1040048 d^3 e^3 x^3+786432 d^2 e^4 x^4+315776 d e^5 x^5+52736 e^6 x^6\right )+2 a c^7 d^7 e^2 \left (-15015 d^7+9933 d^6 e x+876816 d^5 e^2 x^2+2988664 d^4 e^3 x^3+4548736 d^3 e^4 x^4+3672960 d^2 e^5 x^5+1540096 d e^6 x^6+265216 e^7 x^7\right )+c^8 d^8 \left (3465 d^8-2310 d^7 e x+1848 d^6 e^2 x^2+588240 d^5 e^3 x^3+2229632 d^4 e^4 x^4+3603200 d^3 e^5 x^5+3025920 d^2 e^6 x^6+1304576 d e^7 x^7+229376 e^8 x^8\right )\right )}{63 c^6 d^6 e^3 (d+e x)^2 (a e+c d x)^2}-\frac{55 \left (c d^2-a e^2\right )^9 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{13/2} d^{13/2} e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right )}{65536} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((2*(-3465*a^8*e^16 + 2310*a^7*c*d*e^14*(13*d +
e*x) - 462*a^6*c^2*d^2*e^12*(249*d^2 + 43*d*e*x + 4*e^2*x^2) + 198*a^5*c^3*d^3*
e^10*(1289*d^3 + 381*d^2*e*x + 80*d*e^2*x^2 + 8*e^3*x^3) - 22*a^4*c^4*d^4*e^8*(1
6384*d^4 + 7531*d^3*e*x + 2724*d^2*e^2*x^2 + 616*d*e^3*x^3 + 64*e^4*x^4) + 2*a^3
*c^5*d^5*e^6*(167301*d^5 + 115609*d^4*e*x + 65536*d^3*e^2*x^2 + 25584*d^2*e^3*x^
3 + 6016*d*e^4*x^4 + 640*e^5*x^5) + 6*a^2*c^6*d^6*e^4*(19173*d^6 + 282339*d^5*e*
x + 763652*d^4*e^2*x^2 + 1040048*d^3*e^3*x^3 + 786432*d^2*e^4*x^4 + 315776*d*e^5
*x^5 + 52736*e^6*x^6) + 2*a*c^7*d^7*e^2*(-15015*d^7 + 9933*d^6*e*x + 876816*d^5*
e^2*x^2 + 2988664*d^4*e^3*x^3 + 4548736*d^3*e^4*x^4 + 3672960*d^2*e^5*x^5 + 1540
096*d*e^6*x^6 + 265216*e^7*x^7) + c^8*d^8*(3465*d^8 - 2310*d^7*e*x + 1848*d^6*e^
2*x^2 + 588240*d^5*e^3*x^3 + 2229632*d^4*e^4*x^4 + 3603200*d^3*e^5*x^5 + 3025920
*d^2*e^6*x^6 + 1304576*d*e^7*x^7 + 229376*e^8*x^8)))/(63*c^6*d^6*e^3*(a*e + c*d*
x)^2*(d + e*x)^2) - (55*(c*d^2 - a*e^2)^9*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*
Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(13/2)*d^(13/2)*e^(7/2)*(
a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/65536

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Maple [B]  time = 0.023, size = 2368, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

385/16384*d^6*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+55/32768*d^10/e^3*c^
2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-55/12288*d^7/e^2*c*(a*e*d+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)+43/144*e/c*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+55/655
36*e^15/d^6/c^6*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^9-495/65536*e^13/d^4/c^5*ln((1/2*a*e^2+1
/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)
^(1/2)*a^8-1925/16384*d^3*e^4/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+34
65/32768*d^2*e^7/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^5-3465/32768*d^4*e^5/c*ln((1/2*a*e^
2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e
*c)^(1/2)*a^4+1925/16384*d*e^6/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4
-495/16384*d^8*e*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^2-275/3072*d^2*e^3/c*(a*e*d+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+495/65536*d^10/e*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e
*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a+11/38
4*e^5/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^3-55/32768*e^13/d^6/c^6*
(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^8+165/16384*e^11/d^4/c^5*(a*e*d+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-385/16384*e^9/d^2/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*a^6+275/3072*e^5/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^3-
11/384*d^2*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a-385/16384*d^7*c*(a*e*d+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+55/16384*d^9/e^2*c^2*(a*e*d+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x-1155/16384*e^9/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^
(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^6-11/768*e^7/d^4/
c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^4+11/224*e^4/d^3/c^3*(a*e*d+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(7/2)*a^2+55/12288*e^10/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)*a^6-55/3072*e^8/d^3/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5
-11/63*e^2/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a+275/12288*e^6/d/c^3*(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+1/9*e^2*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(7/2)/d/c-385/16384*d^4*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
^3-165/16384*d^8/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1155/16384*d^6*e^
3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2))/(d*e*c)^(1/2)*a^3-55/6144*d^6/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)*x-55/65536*d^12/e^3*c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+1155/16384*d^5*e^2*(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+275/6144*d^4*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)*x*a+53/224*d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+11/384*d^3*(a*e
*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+11/768*d^4/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)+55/3072*d^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+495/16384*e^11/
d^2/c^4*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^7-11/144*e^3/d^2/c^2*x*(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(7/2)*a-1155/16384*e^8/d/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*
a^5-11/128*e^2*d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a-55/16384*e^12/d^5
/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^7+385/16384*e^10/d^3/c^4*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6-11/384*e^6/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(5/2)*x*a^3+11/128*e^4/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)*x*a^2+55/6144*e^9/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^5-275/61
44*e^7/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^4+385/16384*e^7/c^3*(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5-275/12288*d^3*e^2/c*(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^(5/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.463948, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(5/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/8257536*(4*(229376*c^8*d^8*e^8*x^8 + 3465*c^8*d^16 - 30030*a*c^7*d^14*e^2 + 1
15038*a^2*c^6*d^12*e^4 + 334602*a^3*c^5*d^10*e^6 - 360448*a^4*c^4*d^8*e^8 + 2552
22*a^5*c^3*d^6*e^10 - 115038*a^6*c^2*d^4*e^12 + 30030*a^7*c*d^2*e^14 - 3465*a^8*
e^16 + 14336*(91*c^8*d^9*e^7 + 37*a*c^7*d^7*e^9)*x^7 + 1024*(2955*c^8*d^10*e^6 +
3008*a*c^7*d^8*e^8 + 309*a^2*c^6*d^6*e^10)*x^6 + 256*(14075*c^8*d^11*e^5 + 2869
5*a*c^7*d^9*e^7 + 7401*a^2*c^6*d^7*e^9 + 5*a^3*c^5*d^5*e^11)*x^5 + 128*(17419*c^
8*d^12*e^4 + 71074*a*c^7*d^10*e^6 + 36864*a^2*c^6*d^8*e^8 + 94*a^3*c^5*d^6*e^10
- 11*a^4*c^4*d^4*e^12)*x^4 + 16*(36765*c^8*d^13*e^3 + 373583*a*c^7*d^11*e^5 + 39
0018*a^2*c^6*d^9*e^7 + 3198*a^3*c^5*d^7*e^9 - 847*a^4*c^4*d^5*e^11 + 99*a^5*c^3*
d^3*e^13)*x^3 + 8*(231*c^8*d^14*e^2 + 219204*a*c^7*d^12*e^4 + 572739*a^2*c^6*d^1
0*e^6 + 16384*a^3*c^5*d^8*e^8 - 7491*a^4*c^4*d^6*e^10 + 1980*a^5*c^3*d^4*e^12 -
231*a^6*c^2*d^2*e^14)*x^2 - 2*(1155*c^8*d^15*e - 9933*a*c^7*d^13*e^3 - 847017*a^
2*c^6*d^11*e^5 - 115609*a^3*c^5*d^9*e^7 + 82841*a^4*c^4*d^7*e^9 - 37719*a^5*c^3*
d^5*e^11 + 9933*a^6*c^2*d^3*e^13 - 1155*a^7*c*d*e^15)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^
7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 +
84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*log(-4
*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e +
a*c*d*e^3)*x)*sqrt(c*d*e)))/(sqrt(c*d*e)*c^6*d^6*e^3), 1/4128768*(2*(229376*c^8
*d^8*e^8*x^8 + 3465*c^8*d^16 - 30030*a*c^7*d^14*e^2 + 115038*a^2*c^6*d^12*e^4 +
334602*a^3*c^5*d^10*e^6 - 360448*a^4*c^4*d^8*e^8 + 255222*a^5*c^3*d^6*e^10 - 115
038*a^6*c^2*d^4*e^12 + 30030*a^7*c*d^2*e^14 - 3465*a^8*e^16 + 14336*(91*c^8*d^9*
e^7 + 37*a*c^7*d^7*e^9)*x^7 + 1024*(2955*c^8*d^10*e^6 + 3008*a*c^7*d^8*e^8 + 309
*a^2*c^6*d^6*e^10)*x^6 + 256*(14075*c^8*d^11*e^5 + 28695*a*c^7*d^9*e^7 + 7401*a^
2*c^6*d^7*e^9 + 5*a^3*c^5*d^5*e^11)*x^5 + 128*(17419*c^8*d^12*e^4 + 71074*a*c^7*
d^10*e^6 + 36864*a^2*c^6*d^8*e^8 + 94*a^3*c^5*d^6*e^10 - 11*a^4*c^4*d^4*e^12)*x^
4 + 16*(36765*c^8*d^13*e^3 + 373583*a*c^7*d^11*e^5 + 390018*a^2*c^6*d^9*e^7 + 31
98*a^3*c^5*d^7*e^9 - 847*a^4*c^4*d^5*e^11 + 99*a^5*c^3*d^3*e^13)*x^3 + 8*(231*c^
8*d^14*e^2 + 219204*a*c^7*d^12*e^4 + 572739*a^2*c^6*d^10*e^6 + 16384*a^3*c^5*d^8
*e^8 - 7491*a^4*c^4*d^6*e^10 + 1980*a^5*c^3*d^4*e^12 - 231*a^6*c^2*d^2*e^14)*x^2
- 2*(1155*c^8*d^15*e - 9933*a*c^7*d^13*e^3 - 847017*a^2*c^6*d^11*e^5 - 115609*a
^3*c^5*d^9*e^7 + 82841*a^4*c^4*d^7*e^9 - 37719*a^5*c^3*d^5*e^11 + 9933*a^6*c^2*d
^3*e^13 - 1155*a^7*c*d*e^15)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt
(-c*d*e) - 3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*
d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 3
6*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*arctan(1/2*(2*c*d*e*x + c*d^2
+ a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/(sqr
t(-c*d*e)*c^6*d^6*e^3)]

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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)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254943, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^(5/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

Done