3.2195 \(\int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=562 \[ \frac{11 (2 c d-b e)^9 (-13 b e g+6 c d g+20 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{262144 c^{15/2} e^2}+\frac{11 (b+2 c x) (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-13 b e g+6 c d g+20 c e f)}{131072 c^7 e}+\frac{11 (b+2 c x) (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+6 c d g+20 c e f)}{49152 c^6 e}+\frac{11 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+6 c d g+20 c e f)}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{4480 c^4 e^2}-\frac{11 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{2880 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2} \]

[Out]

(11*(2*c*d - b*e)^7*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2])/(131072*c^7*e) + (11*(2*c*d - b*e)^5*(20*c*e*f + 6*c*
d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(49152*
c^6*e) + (11*(2*c*d - b*e)^3*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15360*c^5*e) - (11*(2*c*d - b*e)^2*(20*c*
e*f + 6*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(4480*c^4
*e^2) - (11*(2*c*d - b*e)*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2880*c^3*e^2) - ((20*c*e*f + 6*c*d*g - 13*b*e*
g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(180*c^2*e^2) - (g*(
d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(10*c*e^2) + (11*(2*c*d
- b*e)^9*(20*c*e*f + 6*c*d*g - 13*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[
d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(262144*c^(15/2)*e^2)

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Rubi [A]  time = 2.35424, antiderivative size = 562, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{11 (2 c d-b e)^9 (-13 b e g+6 c d g+20 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{262144 c^{15/2} e^2}+\frac{11 (b+2 c x) (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-13 b e g+6 c d g+20 c e f)}{131072 c^7 e}+\frac{11 (b+2 c x) (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+6 c d g+20 c e f)}{49152 c^6 e}+\frac{11 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+6 c d g+20 c e f)}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{4480 c^4 e^2}-\frac{11 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{2880 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(11*(2*c*d - b*e)^7*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2])/(131072*c^7*e) + (11*(2*c*d - b*e)^5*(20*c*e*f + 6*c*
d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(49152*
c^6*e) + (11*(2*c*d - b*e)^3*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15360*c^5*e) - (11*(2*c*d - b*e)^2*(20*c*
e*f + 6*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(4480*c^4
*e^2) - (11*(2*c*d - b*e)*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2880*c^3*e^2) - ((20*c*e*f + 6*c*d*g - 13*b*e*
g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(180*c^2*e^2) - (g*(
d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(10*c*e^2) + (11*(2*c*d
- b*e)^9*(20*c*e*f + 6*c*d*g - 13*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[
d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(262144*c^(15/2)*e^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)**3*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.52437, size = 1491, normalized size = 2.65 \[ \frac{\left (\frac{1}{10} c^2 e^7 g x^9+\frac{1}{180} c e^6 (20 c e f+60 c d g+41 b e g) x^8+\frac{e^5 \left (1080 d e f c^2+324 d^2 g c^2+740 b e^2 f c+2976 b d e g c+383 b^2 e^2 g\right ) x^7}{2880}+\frac{e^4 \left (5120 d^2 e f c^3-30720 d^3 g c^3+47800 b d e^2 f c^2+59396 b d^2 e g c^2+6180 b^2 e^3 f c+31264 b^2 d e^2 g c+15 b^3 e^3 g\right ) x^6}{40320 c}+\frac{e^3 \left (-144480 d^3 e f c^4-140112 d^4 g c^4+278160 b d^2 e^2 f c^3-16176 b d^3 e g c^3+147720 b^2 d e^3 f c^2+298968 b^2 d^2 e^2 g c^2+100 b^3 e^4 f c+780 b^3 d e^3 g c-65 b^4 e^4 g\right ) x^5}{161280 c^2}-\frac{e^2 \left (337920 d^4 e f c^5-92160 d^5 g c^5+46560 b d^3 e^2 f c^4+762000 b d^4 e g c^4-730320 b^2 d^2 e^3 f c^3-733200 b^2 d^3 e^2 g c^3-2760 b^3 d e^4 f c^2-9960 b^3 d^2 e^3 g c^2+220 b^4 e^5 f c+1860 b^4 d e^4 g c-143 b^5 e^5 g\right ) x^4}{322560 c^3}+\frac{e \left (981120 d^5 e f c^6+2358720 d^6 g c^6-7859520 b d^4 e^2 f c^5-6092160 b d^5 e g c^5+7487040 b^2 d^3 e^3 f c^4+3484080 b^2 d^4 e^2 g c^4+151520 b^3 d^2 e^4 f c^3+339840 b^3 d^3 e^3 g c^3-26840 b^4 d e^5 f c^2-106540 b^4 d^2 e^4 g c^2+1980 b^5 e^6 f c+18040 b^5 d e^5 g c-1287 b^6 e^6 g\right ) x^3}{2580480 c^4}+\frac{\left (6553600 d^6 e f c^7+1966080 d^7 g c^7-16717440 b d^5 e^2 f c^6-3081920 b d^6 e g c^6+9107520 b^2 d^4 e^3 f c^5-336000 b^2 d^5 e^2 g c^5+1415360 b^3 d^3 e^4 f c^4+2246160 b^3 d^4 e^3 g c^4-417120 b^4 d^2 e^5 f c^3-1045120 b^4 d^3 e^4 g c^3+67320 b^5 d e^6 f c^2+291324 b^5 d^2 e^5 g c^2-4620 b^6 e^7 f c-45144 b^6 d e^6 g c+3003 b^7 e^7 g\right ) x^2}{5160960 c^5}+\frac{\left (11773440 d^7 e f c^8-2661120 d^8 g c^8-14992640 b d^6 e^2 f c^7+12622080 b d^7 e g c^7-10945920 b^2 d^5 e^3 f c^6-24504320 b^2 d^6 e^2 g c^6+21264960 b^3 d^4 e^4 f c^5+25880640 b^3 d^5 e^3 g c^5-9217120 b^4 d^3 e^5 f c^4-16587360 b^4 d^4 e^4 g c^4+2431440 b^5 d^2 e^6 f c^3+6720560 b^5 d^3 e^5 g c^3-360360 b^6 d e^7 f c^2-1688544 b^6 d^2 e^6 g c^2+23100 b^7 e^8 f c+241164 b^7 d e^7 g c-15015 b^8 e^8 g\right ) x}{20643840 c^6 e}+\frac{-19005440 d^8 e f c^9-9830400 d^9 g c^9+87795200 b d^7 e^2 f c^8+51078400 b d^8 e g c^8-161137920 b^2 d^6 e^3 f c^7-117794560 b^2 d^7 e^2 g c^7+157489280 b^3 d^5 e^4 f c^6+156115200 b^3 d^6 e^3 g c^6-93114560 b^4 d^4 e^5 f c^5-130302400 b^4 d^5 e^4 g c^5+35402400 b^5 d^3 e^6 f c^4+71145184 b^5 d^4 e^5 g c^4-8445360 b^6 d^2 e^7 f c^3-25545168 b^6 d^3 e^6 g c^3+1155000 b^7 d e^8 f c^2+5835984 b^7 d^2 e^7 g c^2-69300 b^8 e^9 f c-771540 b^8 d e^8 g c+45045 b^9 e^9 g}{41287680 c^7 e^2}\right ) ((d+e x) (c (d-e x)-b e))^{5/2}}{(d+e x)^2 (c d-b e-c e x)^2}-\frac{11 i (b e-2 c d)^9 (20 c e f+6 c d g-13 b e g) ((d+e x) (c (d-e x)-b e))^{5/2} \log \left (2 \sqrt{d+e x} \sqrt{c d-b e-c e x}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{262144 c^{15/2} e^2 (d+e x)^{5/2} (c d-b e-c e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(((-19005440*c^9*d^8*e*f + 87795200*b*c^8*d^7*e^2*f - 161137920*b^2*c^7*d^6*e^3*
f + 157489280*b^3*c^6*d^5*e^4*f - 93114560*b^4*c^5*d^4*e^5*f + 35402400*b^5*c^4*
d^3*e^6*f - 8445360*b^6*c^3*d^2*e^7*f + 1155000*b^7*c^2*d*e^8*f - 69300*b^8*c*e^
9*f - 9830400*c^9*d^9*g + 51078400*b*c^8*d^8*e*g - 117794560*b^2*c^7*d^7*e^2*g +
156115200*b^3*c^6*d^6*e^3*g - 130302400*b^4*c^5*d^5*e^4*g + 71145184*b^5*c^4*d^
4*e^5*g - 25545168*b^6*c^3*d^3*e^6*g + 5835984*b^7*c^2*d^2*e^7*g - 771540*b^8*c*
d*e^8*g + 45045*b^9*e^9*g)/(41287680*c^7*e^2) + ((11773440*c^8*d^7*e*f - 1499264
0*b*c^7*d^6*e^2*f - 10945920*b^2*c^6*d^5*e^3*f + 21264960*b^3*c^5*d^4*e^4*f - 92
17120*b^4*c^4*d^3*e^5*f + 2431440*b^5*c^3*d^2*e^6*f - 360360*b^6*c^2*d*e^7*f + 2
3100*b^7*c*e^8*f - 2661120*c^8*d^8*g + 12622080*b*c^7*d^7*e*g - 24504320*b^2*c^6
*d^6*e^2*g + 25880640*b^3*c^5*d^5*e^3*g - 16587360*b^4*c^4*d^4*e^4*g + 6720560*b
^5*c^3*d^3*e^5*g - 1688544*b^6*c^2*d^2*e^6*g + 241164*b^7*c*d*e^7*g - 15015*b^8*
e^8*g)*x)/(20643840*c^6*e) + ((6553600*c^7*d^6*e*f - 16717440*b*c^6*d^5*e^2*f +
9107520*b^2*c^5*d^4*e^3*f + 1415360*b^3*c^4*d^3*e^4*f - 417120*b^4*c^3*d^2*e^5*f
+ 67320*b^5*c^2*d*e^6*f - 4620*b^6*c*e^7*f + 1966080*c^7*d^7*g - 3081920*b*c^6*
d^6*e*g - 336000*b^2*c^5*d^5*e^2*g + 2246160*b^3*c^4*d^4*e^3*g - 1045120*b^4*c^3
*d^3*e^4*g + 291324*b^5*c^2*d^2*e^5*g - 45144*b^6*c*d*e^6*g + 3003*b^7*e^7*g)*x^
2)/(5160960*c^5) + (e*(981120*c^6*d^5*e*f - 7859520*b*c^5*d^4*e^2*f + 7487040*b^
2*c^4*d^3*e^3*f + 151520*b^3*c^3*d^2*e^4*f - 26840*b^4*c^2*d*e^5*f + 1980*b^5*c*
e^6*f + 2358720*c^6*d^6*g - 6092160*b*c^5*d^5*e*g + 3484080*b^2*c^4*d^4*e^2*g +
339840*b^3*c^3*d^3*e^3*g - 106540*b^4*c^2*d^2*e^4*g + 18040*b^5*c*d*e^5*g - 1287
*b^6*e^6*g)*x^3)/(2580480*c^4) - (e^2*(337920*c^5*d^4*e*f + 46560*b*c^4*d^3*e^2*
f - 730320*b^2*c^3*d^2*e^3*f - 2760*b^3*c^2*d*e^4*f + 220*b^4*c*e^5*f - 92160*c^
5*d^5*g + 762000*b*c^4*d^4*e*g - 733200*b^2*c^3*d^3*e^2*g - 9960*b^3*c^2*d^2*e^3
*g + 1860*b^4*c*d*e^4*g - 143*b^5*e^5*g)*x^4)/(322560*c^3) + (e^3*(-144480*c^4*d
^3*e*f + 278160*b*c^3*d^2*e^2*f + 147720*b^2*c^2*d*e^3*f + 100*b^3*c*e^4*f - 140
112*c^4*d^4*g - 16176*b*c^3*d^3*e*g + 298968*b^2*c^2*d^2*e^2*g + 780*b^3*c*d*e^3
*g - 65*b^4*e^4*g)*x^5)/(161280*c^2) + (e^4*(5120*c^3*d^2*e*f + 47800*b*c^2*d*e^
2*f + 6180*b^2*c*e^3*f - 30720*c^3*d^3*g + 59396*b*c^2*d^2*e*g + 31264*b^2*c*d*e
^2*g + 15*b^3*e^3*g)*x^6)/(40320*c) + (e^5*(1080*c^2*d*e*f + 740*b*c*e^2*f + 324
*c^2*d^2*g + 2976*b*c*d*e*g + 383*b^2*e^2*g)*x^7)/2880 + (c*e^6*(20*c*e*f + 60*c
*d*g + 41*b*e*g)*x^8)/180 + (c^2*e^7*g*x^9)/10)*((d + e*x)*(-(b*e) + c*(d - e*x)
))^(5/2))/((d + e*x)^2*(c*d - b*e - c*e*x)^2) - (((11*I)/262144)*(-2*c*d + b*e)^
9*(20*c*e*f + 6*c*d*g - 13*b*e*g)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*Log[(
(-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]])/(c^(15/2
)*e^2*(d + e*x)^(5/2)*(c*d - b*e - c*e*x)^(5/2))

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Maple [B]  time = 0.053, size = 5287, normalized size = 9.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

result too large to display

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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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^3*(g*x + f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 43.1632, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^3*(g*x + f),x, algorithm="fricas")

[Out]

[1/165150720*(4*(4128768*c^9*e^9*g*x^9 + 229376*(20*c^9*e^9*f + (60*c^9*d*e^8 +
41*b*c^8*e^9)*g)*x^8 + 14336*(20*(54*c^9*d*e^8 + 37*b*c^8*e^9)*f + (324*c^9*d^2*
e^7 + 2976*b*c^8*d*e^8 + 383*b^2*c^7*e^9)*g)*x^7 + 1024*(20*(256*c^9*d^2*e^7 + 2
390*b*c^8*d*e^8 + 309*b^2*c^7*e^9)*f - (30720*c^9*d^3*e^6 - 59396*b*c^8*d^2*e^7
- 31264*b^2*c^7*d*e^8 - 15*b^3*c^6*e^9)*g)*x^6 - 256*(20*(7224*c^9*d^3*e^6 - 139
08*b*c^8*d^2*e^7 - 7386*b^2*c^7*d*e^8 - 5*b^3*c^6*e^9)*f + (140112*c^9*d^4*e^5 +
16176*b*c^8*d^3*e^6 - 298968*b^2*c^7*d^2*e^7 - 780*b^3*c^6*d*e^8 + 65*b^4*c^5*e
^9)*g)*x^5 - 128*(20*(16896*c^9*d^4*e^5 + 2328*b*c^8*d^3*e^6 - 36516*b^2*c^7*d^2
*e^7 - 138*b^3*c^6*d*e^8 + 11*b^4*c^5*e^9)*f - (92160*c^9*d^5*e^4 - 762000*b*c^8
*d^4*e^5 + 733200*b^2*c^7*d^3*e^6 + 9960*b^3*c^6*d^2*e^7 - 1860*b^4*c^5*d*e^8 +
143*b^5*c^4*e^9)*g)*x^4 + 16*(20*(49056*c^9*d^5*e^4 - 392976*b*c^8*d^4*e^5 + 374
352*b^2*c^7*d^3*e^6 + 7576*b^3*c^6*d^2*e^7 - 1342*b^4*c^5*d*e^8 + 99*b^5*c^4*e^9
)*f + (2358720*c^9*d^6*e^3 - 6092160*b*c^8*d^5*e^4 + 3484080*b^2*c^7*d^4*e^5 + 3
39840*b^3*c^6*d^3*e^6 - 106540*b^4*c^5*d^2*e^7 + 18040*b^5*c^4*d*e^8 - 1287*b^6*
c^3*e^9)*g)*x^3 + 8*(20*(327680*c^9*d^6*e^3 - 835872*b*c^8*d^5*e^4 + 455376*b^2*
c^7*d^4*e^5 + 70768*b^3*c^6*d^3*e^6 - 20856*b^4*c^5*d^2*e^7 + 3366*b^5*c^4*d*e^8
- 231*b^6*c^3*e^9)*f + (1966080*c^9*d^7*e^2 - 3081920*b*c^8*d^6*e^3 - 336000*b^
2*c^7*d^5*e^4 + 2246160*b^3*c^6*d^4*e^5 - 1045120*b^4*c^5*d^3*e^6 + 291324*b^5*c
^4*d^2*e^7 - 45144*b^6*c^3*d*e^8 + 3003*b^7*c^2*e^9)*g)*x^2 - 20*(950272*c^9*d^8
*e - 4389760*b*c^8*d^7*e^2 + 8056896*b^2*c^7*d^6*e^3 - 7874464*b^3*c^6*d^5*e^4 +
4655728*b^4*c^5*d^4*e^5 - 1770120*b^5*c^4*d^3*e^6 + 422268*b^6*c^3*d^2*e^7 - 57
750*b^7*c^2*d*e^8 + 3465*b^8*c*e^9)*f - (9830400*c^9*d^9 - 51078400*b*c^8*d^8*e
+ 117794560*b^2*c^7*d^7*e^2 - 156115200*b^3*c^6*d^6*e^3 + 130302400*b^4*c^5*d^5*
e^4 - 71145184*b^5*c^4*d^4*e^5 + 25545168*b^6*c^3*d^3*e^6 - 5835984*b^7*c^2*d^2*
e^7 + 771540*b^8*c*d*e^8 - 45045*b^9*e^9)*g + 2*(20*(588672*c^9*d^7*e^2 - 749632
*b*c^8*d^6*e^3 - 547296*b^2*c^7*d^5*e^4 + 1063248*b^3*c^6*d^4*e^5 - 460856*b^4*c
^5*d^3*e^6 + 121572*b^5*c^4*d^2*e^7 - 18018*b^6*c^3*d*e^8 + 1155*b^7*c^2*e^9)*f
- (2661120*c^9*d^8*e - 12622080*b*c^8*d^7*e^2 + 24504320*b^2*c^7*d^6*e^3 - 25880
640*b^3*c^6*d^5*e^4 + 16587360*b^4*c^5*d^4*e^5 - 6720560*b^5*c^4*d^3*e^6 + 16885
44*b^6*c^3*d^2*e^7 - 241164*b^7*c^2*d*e^8 + 15015*b^8*c*e^9)*g)*x)*sqrt(-c*e^2*x
^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 3465*(20*(512*c^10*d^9*e - 2304*b*c^9*d
^8*e^2 + 4608*b^2*c^8*d^7*e^3 - 5376*b^3*c^7*d^6*e^4 + 4032*b^4*c^6*d^5*e^5 - 20
16*b^5*c^5*d^4*e^6 + 672*b^6*c^4*d^3*e^7 - 144*b^7*c^3*d^2*e^8 + 18*b^8*c^2*d*e^
9 - b^9*c*e^10)*f + (3072*c^10*d^10 - 20480*b*c^9*d^9*e + 57600*b^2*c^8*d^8*e^2
- 92160*b^3*c^7*d^7*e^3 + 94080*b^4*c^6*d^6*e^4 - 64512*b^5*c^5*d^5*e^5 + 30240*
b^6*c^4*d^4*e^6 - 9600*b^7*c^3*d^3*e^7 + 1980*b^8*c^2*d^2*e^8 - 240*b^9*c*d*e^9
+ 13*b^10*e^10)*g)*log(4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c^2*e*x +
b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*sqrt(-
c)))/(sqrt(-c)*c^7*e^2), 1/82575360*(2*(4128768*c^9*e^9*g*x^9 + 229376*(20*c^9*e
^9*f + (60*c^9*d*e^8 + 41*b*c^8*e^9)*g)*x^8 + 14336*(20*(54*c^9*d*e^8 + 37*b*c^8
*e^9)*f + (324*c^9*d^2*e^7 + 2976*b*c^8*d*e^8 + 383*b^2*c^7*e^9)*g)*x^7 + 1024*(
20*(256*c^9*d^2*e^7 + 2390*b*c^8*d*e^8 + 309*b^2*c^7*e^9)*f - (30720*c^9*d^3*e^6
- 59396*b*c^8*d^2*e^7 - 31264*b^2*c^7*d*e^8 - 15*b^3*c^6*e^9)*g)*x^6 - 256*(20*
(7224*c^9*d^3*e^6 - 13908*b*c^8*d^2*e^7 - 7386*b^2*c^7*d*e^8 - 5*b^3*c^6*e^9)*f
+ (140112*c^9*d^4*e^5 + 16176*b*c^8*d^3*e^6 - 298968*b^2*c^7*d^2*e^7 - 780*b^3*c
^6*d*e^8 + 65*b^4*c^5*e^9)*g)*x^5 - 128*(20*(16896*c^9*d^4*e^5 + 2328*b*c^8*d^3*
e^6 - 36516*b^2*c^7*d^2*e^7 - 138*b^3*c^6*d*e^8 + 11*b^4*c^5*e^9)*f - (92160*c^9
*d^5*e^4 - 762000*b*c^8*d^4*e^5 + 733200*b^2*c^7*d^3*e^6 + 9960*b^3*c^6*d^2*e^7
- 1860*b^4*c^5*d*e^8 + 143*b^5*c^4*e^9)*g)*x^4 + 16*(20*(49056*c^9*d^5*e^4 - 392
976*b*c^8*d^4*e^5 + 374352*b^2*c^7*d^3*e^6 + 7576*b^3*c^6*d^2*e^7 - 1342*b^4*c^5
*d*e^8 + 99*b^5*c^4*e^9)*f + (2358720*c^9*d^6*e^3 - 6092160*b*c^8*d^5*e^4 + 3484
080*b^2*c^7*d^4*e^5 + 339840*b^3*c^6*d^3*e^6 - 106540*b^4*c^5*d^2*e^7 + 18040*b^
5*c^4*d*e^8 - 1287*b^6*c^3*e^9)*g)*x^3 + 8*(20*(327680*c^9*d^6*e^3 - 835872*b*c^
8*d^5*e^4 + 455376*b^2*c^7*d^4*e^5 + 70768*b^3*c^6*d^3*e^6 - 20856*b^4*c^5*d^2*e
^7 + 3366*b^5*c^4*d*e^8 - 231*b^6*c^3*e^9)*f + (1966080*c^9*d^7*e^2 - 3081920*b*
c^8*d^6*e^3 - 336000*b^2*c^7*d^5*e^4 + 2246160*b^3*c^6*d^4*e^5 - 1045120*b^4*c^5
*d^3*e^6 + 291324*b^5*c^4*d^2*e^7 - 45144*b^6*c^3*d*e^8 + 3003*b^7*c^2*e^9)*g)*x
^2 - 20*(950272*c^9*d^8*e - 4389760*b*c^8*d^7*e^2 + 8056896*b^2*c^7*d^6*e^3 - 78
74464*b^3*c^6*d^5*e^4 + 4655728*b^4*c^5*d^4*e^5 - 1770120*b^5*c^4*d^3*e^6 + 4222
68*b^6*c^3*d^2*e^7 - 57750*b^7*c^2*d*e^8 + 3465*b^8*c*e^9)*f - (9830400*c^9*d^9
- 51078400*b*c^8*d^8*e + 117794560*b^2*c^7*d^7*e^2 - 156115200*b^3*c^6*d^6*e^3 +
130302400*b^4*c^5*d^5*e^4 - 71145184*b^5*c^4*d^4*e^5 + 25545168*b^6*c^3*d^3*e^6
- 5835984*b^7*c^2*d^2*e^7 + 771540*b^8*c*d*e^8 - 45045*b^9*e^9)*g + 2*(20*(5886
72*c^9*d^7*e^2 - 749632*b*c^8*d^6*e^3 - 547296*b^2*c^7*d^5*e^4 + 1063248*b^3*c^6
*d^4*e^5 - 460856*b^4*c^5*d^3*e^6 + 121572*b^5*c^4*d^2*e^7 - 18018*b^6*c^3*d*e^8
+ 1155*b^7*c^2*e^9)*f - (2661120*c^9*d^8*e - 12622080*b*c^8*d^7*e^2 + 24504320*
b^2*c^7*d^6*e^3 - 25880640*b^3*c^6*d^5*e^4 + 16587360*b^4*c^5*d^4*e^5 - 6720560*
b^5*c^4*d^3*e^6 + 1688544*b^6*c^3*d^2*e^7 - 241164*b^7*c^2*d*e^8 + 15015*b^8*c*e
^9)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) + 3465*(20*(512*c^1
0*d^9*e - 2304*b*c^9*d^8*e^2 + 4608*b^2*c^8*d^7*e^3 - 5376*b^3*c^7*d^6*e^4 + 403
2*b^4*c^6*d^5*e^5 - 2016*b^5*c^5*d^4*e^6 + 672*b^6*c^4*d^3*e^7 - 144*b^7*c^3*d^2
*e^8 + 18*b^8*c^2*d*e^9 - b^9*c*e^10)*f + (3072*c^10*d^10 - 20480*b*c^9*d^9*e +
57600*b^2*c^8*d^8*e^2 - 92160*b^3*c^7*d^7*e^3 + 94080*b^4*c^6*d^6*e^4 - 64512*b^
5*c^5*d^5*e^5 + 30240*b^6*c^4*d^4*e^6 - 9600*b^7*c^3*d^3*e^7 + 1980*b^8*c^2*d^2*
e^8 - 240*b^9*c*d*e^9 + 13*b^10*e^10)*g)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2
*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c))))/(c^(15/2)*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{3} \left (f + g x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)**3*(f + g*x), x)

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GIAC/XCAS [A]  time = 0.345247, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^3*(g*x + f),x, algorithm="giac")

[Out]

Done