3.1920 \(\int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=534 \[ -\frac{143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac{143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

[Out]

(143*(c*d^2 - a*e^2)^8*(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])/(131072*c^7*d^7*e^3) - (143*(c*d^2 - a*e^2)^6*(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))/(49152*c^6*d^6*e^2) + (
143*(c*d^2 - a*e^2)^4*(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))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^2 - a*e^2)
*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d
+ e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 -
a*e^2)^10*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])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

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Rubi [A]  time = 1.58164, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{143 \left (c d^2-a e^2\right )^{10} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{262144 c^{15/2} d^{15/2} e^{7/2}}+\frac{143 \left (c d^2-a e^2\right )^8 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{131072 c^7 d^7 e^3}-\frac{143 \left (c d^2-a e^2\right )^6 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{49152 c^6 d^6 e^2}+\frac{143 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{15360 c^5 d^5 e}+\frac{143 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{4480 c^4 d^4}+\frac{143 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{2880 c^3 d^3}+\frac{13 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{180 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{10 c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(143*(c*d^2 - a*e^2)^8*(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])/(131072*c^7*d^7*e^3) - (143*(c*d^2 - a*e^2)^6*(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))/(49152*c^6*d^6*e^2) + (
143*(c*d^2 - a*e^2)^4*(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))/(15360*c^5*d^5*e) + (143*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(7/2))/(4480*c^4*d^4) + (143*(c*d^2 - a*e^2)^2*(d + e*x)*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(2880*c^3*d^3) + (13*(c*d^2 - a*e^2)
*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(180*c^2*d^2) + ((d
+ e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(10*c*d) - (143*(c*d^2 -
a*e^2)^10*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])])/(262144*c^(15/2)*d^(15/2)*e^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(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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Mathematica [A]  time = 2.41195, size = 748, normalized size = 1.4 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \left (45045 a^9 e^{18}-15015 a^8 c d e^{16} (29 d+2 e x)+12012 a^7 c^2 d^2 e^{14} \left (157 d^2+24 d e x+2 e^2 x^2\right )-1716 a^6 c^3 d^3 e^{12} \left (2805 d^3+722 d^2 e x+134 d e^2 x^2+12 e^3 x^3\right )+286 a^5 c^4 d^4 e^{10} \left (27985 d^4+10960 d^3 e x+3444 d^2 e^2 x^2+688 d e^3 x^3+64 e^4 x^4\right )-130 a^4 c^5 d^5 e^8 \left (69505 d^5+39690 d^4 e x+19100 d^3 e^2 x^2+6472 d^2 e^3 x^3+1344 d e^4 x^4+128 e^5 x^5\right )+20 a^3 c^6 d^6 e^6 \left (349155 d^6+287800 d^5 e x+203530 d^4 e^2 x^2+105840 d^3 e^3 x^3+37312 d^2 e^4 x^4+7936 d e^5 x^5+768 e^6 x^6\right )+4 a^2 c^7 d^7 e^4 \left (471471 d^7+8537622 d^6 e x+27771366 d^5 e^2 x^2+47303260 d^4 e^3 x^3+47700160 d^3 e^4 x^4+28732032 d^2 e^5 x^5+9597184 d e^6 x^6+1372672 e^7 x^7\right )+a c^8 d^8 e^2 \left (-435435 d^8+288288 d^7 e x+35159496 d^6 e^2 x^2+144375968 d^5 e^3 x^3+275076480 d^4 e^4 x^4+296381440 d^3 e^5 x^5+186500096 d^2 e^6 x^6+64253952 d e^7 x^7+9404416 e^8 x^8\right )+c^9 d^9 \left (45045 d^9-30030 d^8 e x+24024 d^7 e^2 x^2+11775888 d^6 e^3 x^3+53757824 d^5 e^4 x^4+108773120 d^4 e^5 x^5+121912320 d^3 e^6 x^6+78891008 d^2 e^7 x^7+27754496 d e^8 x^8+4128768 e^9 x^9\right )\right )}{315 c^7 d^7 e^3 (d+e x)^2 (a e+c d x)^2}-\frac{143 \left (c d^2-a e^2\right )^{10} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{15/2} d^{15/2} e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right )}{262144} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(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*(45045*a^9*e^18 - 15015*a^8*c*d*e^16*(29*d
+ 2*e*x) + 12012*a^7*c^2*d^2*e^14*(157*d^2 + 24*d*e*x + 2*e^2*x^2) - 1716*a^6*c^
3*d^3*e^12*(2805*d^3 + 722*d^2*e*x + 134*d*e^2*x^2 + 12*e^3*x^3) + 286*a^5*c^4*d
^4*e^10*(27985*d^4 + 10960*d^3*e*x + 3444*d^2*e^2*x^2 + 688*d*e^3*x^3 + 64*e^4*x
^4) - 130*a^4*c^5*d^5*e^8*(69505*d^5 + 39690*d^4*e*x + 19100*d^3*e^2*x^2 + 6472*
d^2*e^3*x^3 + 1344*d*e^4*x^4 + 128*e^5*x^5) + 20*a^3*c^6*d^6*e^6*(349155*d^6 + 2
87800*d^5*e*x + 203530*d^4*e^2*x^2 + 105840*d^3*e^3*x^3 + 37312*d^2*e^4*x^4 + 79
36*d*e^5*x^5 + 768*e^6*x^6) + 4*a^2*c^7*d^7*e^4*(471471*d^7 + 8537622*d^6*e*x +
27771366*d^5*e^2*x^2 + 47303260*d^4*e^3*x^3 + 47700160*d^3*e^4*x^4 + 28732032*d^
2*e^5*x^5 + 9597184*d*e^6*x^6 + 1372672*e^7*x^7) + a*c^8*d^8*e^2*(-435435*d^8 +
288288*d^7*e*x + 35159496*d^6*e^2*x^2 + 144375968*d^5*e^3*x^3 + 275076480*d^4*e^
4*x^4 + 296381440*d^3*e^5*x^5 + 186500096*d^2*e^6*x^6 + 64253952*d*e^7*x^7 + 940
4416*e^8*x^8) + c^9*d^9*(45045*d^9 - 30030*d^8*e*x + 24024*d^7*e^2*x^2 + 1177588
8*d^6*e^3*x^3 + 53757824*d^5*e^4*x^4 + 108773120*d^4*e^5*x^5 + 121912320*d^3*e^6
*x^6 + 78891008*d^2*e^7*x^7 + 27754496*d*e^8*x^8 + 4128768*e^9*x^9)))/(315*c^7*d
^7*e^3*(a*e + c*d*x)^2*(d + e*x)^2) - (143*(c*d^2 - a*e^2)^10*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^(15/2
)*d^(15/2)*e^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/262144

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Maple [B]  time = 0.043, size = 2973, normalized size = 5.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

143/131072*d^11/e^3*c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-143/49152*d^8/e^
2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+715/32768*d^7*e*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)*a^2-10777/40320*e^2/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7
/2)*a+715/49152*e^6/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+67/180*e^2/c
*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)-143/5120*e^7/d^3/c^4*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)*a^4+143/7680*e^5/d/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)*a^3-1001/8192*e^8/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5+143
/1280*e^4/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^2+143/7680*e^3*d/c^2*(
a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2-143/49152*e^12/d^6/c^6*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*a^7-143/4480*e^6/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(7/2)*a^3+5863/40320*e^4/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a^
2+1423/2880*e*d/c*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+715/49152*e^10/d^4/c
^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^6-429/16384*e^8/d^2/c^4*(a*e*d+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5+143/131072*e^15/d^7/c^7*(a*e*d+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*a^9-1001/131072*e^13/d^5/c^6*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*a^8+715/32768*e^11/d^3/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-100
1/32768*e^9/d/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-1001/131072*d^9/e*
c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-1001/32768*d^5*e^3/c*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2)*a^3+2145/32768*d^7*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+143/40
96*d^5*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+1001/16384*d^6*e^2*(a*e*d+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+715/49152*d^2*e^4/c^2*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*a^3-143/24576*d^7/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2
)*x+143/65536*d^10/e^2*c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-143/262144*
d^13/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)-143/5120*d^3*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)*a+1001/65536*d*e^7/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+1
001/65536*d^3*e^5/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-143/8192*d^8*c
*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-429/16384*d^4*e^2/c*(a*e*d+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^2-715/8192*d^3*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(3/2)*x*a^2+715/131072*d^11/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-15015/131072*d^5*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+5005/32768*d^2*e^6/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*x*a^4-1001/8192*d^4*e^4/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*
a^3-15015/131072*d*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+9009/65536*d^3*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^5-6435/262144*d^9*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^2-143/1920*e^2
*d^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a+143/65536*e^14/d^6/c^6*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^8-143/8192*e^12/d^4/c^5*(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x*a^7+1001/16384*e^10/d^2/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*x*a^6+143/2880*e^5/d^3/c^3*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)
*a^2-143/262144*e^17/d^7/c^7*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^10+715/131072*e^15/d^5/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^9-6435/262144*e^13/d^3/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+143
/7680*e^8/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^4-13/180*e^4/d^2/c
^2*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*a-39/160*e^3/d/c^2*x*(a*e*d+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(7/2)*a+2145/32768*e^11/d/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^7-14
3/1920*e^6/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^3-143/24576*e^11/
d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^6+143/4096*e^9/d^3/c^4*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^5-715/8192*e^7/d/c^3*(a*e*d+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*x*a^4+715/6144*e^5*d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*x*a^3+1/10*e^3*x^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/d/c+143/15360*e^
9/d^5/c^5*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^5+1137/4480*d^2/c*(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+143/15360*d^5/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(5/2)+143/7680*d^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+715/49152*d^6*(a*e*
d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.512959, 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)^4,x, algorithm="fricas")

[Out]

[1/165150720*(4*(4128768*c^9*d^9*e^9*x^9 + 45045*c^9*d^18 - 435435*a*c^8*d^16*e^
2 + 1885884*a^2*c^7*d^14*e^4 + 6983100*a^3*c^6*d^12*e^6 - 9035650*a^4*c^5*d^10*e
^8 + 8003710*a^5*c^4*d^8*e^10 - 4813380*a^6*c^3*d^6*e^12 + 1885884*a^7*c^2*d^4*e
^14 - 435435*a^8*c*d^2*e^16 + 45045*a^9*e^18 + 229376*(121*c^9*d^10*e^8 + 41*a*c
^8*d^8*e^10)*x^8 + 14336*(5503*c^9*d^11*e^7 + 4482*a*c^8*d^9*e^9 + 383*a^2*c^7*d
^7*e^11)*x^7 + 1024*(119055*c^9*d^12*e^6 + 182129*a*c^8*d^10*e^8 + 37489*a^2*c^7
*d^8*e^10 + 15*a^3*c^6*d^6*e^12)*x^6 + 256*(424895*c^9*d^13*e^5 + 1157740*a*c^8*
d^11*e^7 + 448938*a^2*c^7*d^9*e^9 + 620*a^3*c^6*d^7*e^11 - 65*a^4*c^5*d^5*e^13)*
x^5 + 128*(419983*c^9*d^14*e^4 + 2149035*a*c^8*d^12*e^6 + 1490630*a^2*c^7*d^10*e
^8 + 5830*a^3*c^6*d^8*e^10 - 1365*a^4*c^5*d^6*e^12 + 143*a^5*c^4*d^4*e^14)*x^4 +
16*(735993*c^9*d^15*e^3 + 9023498*a*c^8*d^13*e^5 + 11825815*a^2*c^7*d^11*e^7 +
132300*a^3*c^6*d^9*e^9 - 52585*a^4*c^5*d^7*e^11 + 12298*a^5*c^4*d^5*e^13 - 1287*
a^6*c^3*d^3*e^15)*x^3 + 8*(3003*c^9*d^16*e^2 + 4394937*a*c^8*d^14*e^4 + 13885683
*a^2*c^7*d^12*e^6 + 508825*a^3*c^6*d^10*e^8 - 310375*a^4*c^5*d^8*e^10 + 123123*a
^5*c^4*d^6*e^12 - 28743*a^6*c^3*d^4*e^14 + 3003*a^7*c^2*d^2*e^16)*x^2 - 2*(15015
*c^9*d^17*e - 144144*a*c^8*d^15*e^3 - 17075244*a^2*c^7*d^13*e^5 - 2878000*a^3*c^
6*d^11*e^7 + 2579850*a^4*c^5*d^9*e^9 - 1567280*a^5*c^4*d^7*e^11 + 619476*a^6*c^3
*d^5*e^13 - 144144*a^7*c^2*d^3*e^15 + 15015*a^8*c*d*e^17)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 45045*(c^10*d^20 - 10*a*c^9*d^18*e^2 + 45
*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*a^5*c^5*d^
10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4*e^16 - 10
*a^9*c*d^2*e^18 + a^10*e^20)*log(-4*(2*c^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sq
rt(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^7
*d^7*e^3), 1/82575360*(2*(4128768*c^9*d^9*e^9*x^9 + 45045*c^9*d^18 - 435435*a*c^
8*d^16*e^2 + 1885884*a^2*c^7*d^14*e^4 + 6983100*a^3*c^6*d^12*e^6 - 9035650*a^4*c
^5*d^10*e^8 + 8003710*a^5*c^4*d^8*e^10 - 4813380*a^6*c^3*d^6*e^12 + 1885884*a^7*
c^2*d^4*e^14 - 435435*a^8*c*d^2*e^16 + 45045*a^9*e^18 + 229376*(121*c^9*d^10*e^8
+ 41*a*c^8*d^8*e^10)*x^8 + 14336*(5503*c^9*d^11*e^7 + 4482*a*c^8*d^9*e^9 + 383*
a^2*c^7*d^7*e^11)*x^7 + 1024*(119055*c^9*d^12*e^6 + 182129*a*c^8*d^10*e^8 + 3748
9*a^2*c^7*d^8*e^10 + 15*a^3*c^6*d^6*e^12)*x^6 + 256*(424895*c^9*d^13*e^5 + 11577
40*a*c^8*d^11*e^7 + 448938*a^2*c^7*d^9*e^9 + 620*a^3*c^6*d^7*e^11 - 65*a^4*c^5*d
^5*e^13)*x^5 + 128*(419983*c^9*d^14*e^4 + 2149035*a*c^8*d^12*e^6 + 1490630*a^2*c
^7*d^10*e^8 + 5830*a^3*c^6*d^8*e^10 - 1365*a^4*c^5*d^6*e^12 + 143*a^5*c^4*d^4*e^
14)*x^4 + 16*(735993*c^9*d^15*e^3 + 9023498*a*c^8*d^13*e^5 + 11825815*a^2*c^7*d^
11*e^7 + 132300*a^3*c^6*d^9*e^9 - 52585*a^4*c^5*d^7*e^11 + 12298*a^5*c^4*d^5*e^1
3 - 1287*a^6*c^3*d^3*e^15)*x^3 + 8*(3003*c^9*d^16*e^2 + 4394937*a*c^8*d^14*e^4 +
13885683*a^2*c^7*d^12*e^6 + 508825*a^3*c^6*d^10*e^8 - 310375*a^4*c^5*d^8*e^10 +
123123*a^5*c^4*d^6*e^12 - 28743*a^6*c^3*d^4*e^14 + 3003*a^7*c^2*d^2*e^16)*x^2 -
2*(15015*c^9*d^17*e - 144144*a*c^8*d^15*e^3 - 17075244*a^2*c^7*d^13*e^5 - 28780
00*a^3*c^6*d^11*e^7 + 2579850*a^4*c^5*d^9*e^9 - 1567280*a^5*c^4*d^7*e^11 + 61947
6*a^6*c^3*d^5*e^13 - 144144*a^7*c^2*d^3*e^15 + 15015*a^8*c*d*e^17)*x)*sqrt(c*d*e
*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 45045*(c^10*d^20 - 10*a*c^9*d^1
8*e^2 + 45*a^2*c^8*d^16*e^4 - 120*a^3*c^7*d^14*e^6 + 210*a^4*c^6*d^12*e^8 - 252*
a^5*c^5*d^10*e^10 + 210*a^6*c^4*d^8*e^12 - 120*a^7*c^3*d^6*e^14 + 45*a^8*c^2*d^4
*e^16 - 10*a^9*c*d^2*e^18 + a^10*e^20)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sq
rt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/(sqrt(-c*d*e)*c
^7*d^7*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)**4*(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 = 0.272116, 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)^4,x, algorithm="giac")

[Out]

Done