3.2342 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=409 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{192 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{512 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{1024 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2
*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(512*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^2
) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)
*x)*(a + b*x + c*x^2)^(3/2))/(192*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^4) - (e*(a
+ b*x + c*x^2)^(5/2))/(6*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) - (7*e*(2*c*d - b
*e)*(a + b*x + c*x^2)^(5/2))/(60*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^5) + ((b^2
- 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b*d - 2*a*e +
(2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(1024*
(c*d^2 - b*d*e + a*e^2)^(9/2))

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Rubi [A]  time = 1.52301, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{192 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{512 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{1024 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2
*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(512*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^2
) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)
*x)*(a + b*x + c*x^2)^(3/2))/(192*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^4) - (e*(a
+ b*x + c*x^2)^(5/2))/(6*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) - (7*e*(2*c*d - b
*e)*(a + b*x + c*x^2)^(5/2))/(60*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^5) + ((b^2
- 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b*d - 2*a*e +
(2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(1024*
(c*d^2 - b*d*e + a*e^2)^(9/2))

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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((c*x**2+b*x+a)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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Mathematica [A]  time = 6.4953, size = 705, normalized size = 1.72 \[ \frac{(a+x (b+c x))^{3/2} \left (\frac{(b e-2 c d) \left (1296 a^2 c^2 e^4-760 a b^2 c e^4+448 a b c^2 d e^3-448 a c^3 d^2 e^2+105 b^4 e^4-80 b^3 c d e^3+16 b^2 c^2 d^2 e^2+128 b c^3 d^3 e-64 c^4 d^4\right )}{7680 e^3 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac{-240 a^2 c^2 e^4+216 a b^2 c e^4-384 a b c^2 d e^3+384 a c^3 d^2 e^2-35 b^4 e^4+64 b^3 c d e^3-128 b c^3 d^3 e+64 c^4 d^4}{3840 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{(b e-2 c d) \left (-36 a c e^2+7 b^2 e^2+8 b c d e-8 c^2 d^2\right )}{960 e^3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}+\frac{-140 a c e^2-3 b^2 e^2+152 b c d e-152 c^2 d^2}{480 e^3 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}+\frac{-a e^2+b d e-c d^2}{6 e^3 (d+e x)^6}+\frac{13 (2 c d-b e)}{60 e^3 (d+e x)^5}\right )}{a+b x+c x^2}+\frac{\left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2} \log (d+e x) \left (-4 a c e^2+7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{1024 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac{\left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2} \left (-4 a c e^2+7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \log \left (2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{1024 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

((a + x*(b + c*x))^(3/2)*((-(c*d^2) + b*d*e - a*e^2)/(6*e^3*(d + e*x)^6) + (13*(
2*c*d - b*e))/(60*e^3*(d + e*x)^5) + (-152*c^2*d^2 + 152*b*c*d*e - 3*b^2*e^2 - 1
40*a*c*e^2)/(480*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + ((-2*c*d + b*e)*(-8*
c^2*d^2 + 8*b*c*d*e + 7*b^2*e^2 - 36*a*c*e^2))/(960*e^3*(c*d^2 - b*d*e + a*e^2)^
2*(d + e*x)^3) + (64*c^4*d^4 - 128*b*c^3*d^3*e + 384*a*c^3*d^2*e^2 + 64*b^3*c*d*
e^3 - 384*a*b*c^2*d*e^3 - 35*b^4*e^4 + 216*a*b^2*c*e^4 - 240*a^2*c^2*e^4)/(3840*
e^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) + ((-2*c*d + b*e)*(-64*c^4*d^4 + 128*
b*c^3*d^3*e + 16*b^2*c^2*d^2*e^2 - 448*a*c^3*d^2*e^2 - 80*b^3*c*d*e^3 + 448*a*b*
c^2*d*e^3 + 105*b^4*e^4 - 760*a*b^2*c*e^4 + 1296*a^2*c^2*e^4))/(7680*e^3*(c*d^2
- b*d*e + a*e^2)^4*(d + e*x))))/(a + b*x + c*x^2) + ((b^2 - 4*a*c)^2*(24*c^2*d^2
- 24*b*c*d*e + 7*b^2*e^2 - 4*a*c*e^2)*(a + x*(b + c*x))^(3/2)*Log[d + e*x])/(10
24*(c*d^2 - b*d*e + a*e^2)^(9/2)*(a + b*x + c*x^2)^(3/2)) - ((b^2 - 4*a*c)^2*(24
*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2 - 4*a*c*e^2)*(a + x*(b + c*x))^(3/2)*Log[-(b*d
) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2
]])/(1024*(c*d^2 - b*d*e + a*e^2)^(9/2)*(a + b*x + c*x^2)^(3/2))

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Maple [B]  time = 0.066, size = 28629, normalized size = 70. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(e*x+d)^7,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*x^2 + b*x + a)^(3/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 102.414, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

[1/30720*(4*(4736*a^4*b*d*e^6 - 1280*a^5*e^7 - 120*(3*b^3*c^2 - 20*a*b*c^3)*d^7
+ 24*(15*b^4*c - 110*a*b^2*c^2 - 328*a^2*c^3)*d^6*e - 5*(21*b^5 - 200*a*b^3*c -
3120*a^2*b*c^2)*d^5*e^2 - 2*(35*a*b^4 + 5832*a^2*b^2*c + 4272*a^3*c^2)*d^4*e^3 +
8*(377*a^2*b^3 + 1812*a^3*b*c)*d^3*e^4 - 16*(387*a^3*b^2 + 332*a^4*c)*d^2*e^5 +
(128*c^5*d^5*e^2 - 320*b*c^4*d^4*e^3 + 32*(3*b^2*c^3 + 28*a*c^4)*d^3*e^4 + 16*(
11*b^3*c^2 - 84*a*b*c^3)*d^2*e^5 - 2*(145*b^4*c - 984*a*b^2*c^2 + 1296*a^2*c^3)*
d*e^6 + (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^7)*x^5 + (768*c^5*d^6*e - 198
4*b*c^4*d^5*e^2 + 32*(23*b^2*c^3 + 168*a*c^4)*d^4*e^3 + 112*(9*b^3*c^2 - 76*a*b*
c^3)*d^3*e^4 - 16*(103*b^4*c - 690*a*b^2*c^2 + 792*a^2*c^3)*d^2*e^5 + (595*b^5 -
4104*a*b^3*c + 6192*a^2*b*c^2)*d*e^6 - 2*(35*a*b^4 - 216*a^2*b^2*c + 240*a^3*c^
2)*e^7)*x^4 + 2*(960*c^5*d^7 - 2592*b*c^4*d^6*e + 8*(153*b^2*c^3 + 836*a*c^4)*d^
5*e^2 + 4*(279*b^3*c^2 - 2836*a*b*c^3)*d^4*e^3 - 2*(963*b^4*c - 6480*a*b^2*c^2 +
5872*a^2*c^3)*d^3*e^4 + (693*b^5 - 4448*a*b^3*c + 4944*a^2*b*c^2)*d^2*e^5 - 4*(
49*a*b^4 - 282*a^2*b^2*c + 168*a^3*c^2)*d*e^6 + 4*(7*a^2*b^3 - 36*a^3*b*c)*e^7)*
x^3 + 2*(1440*b*c^4*d^7 - 192*(28*b^2*c^3 - 19*a*c^4)*d^6*e + 4*(1535*b^3*c^2 -
228*a*b*c^3)*d^5*e^2 - 4*(893*b^4*c - 306*a*b^2*c^2 + 4296*a^2*c^3)*d^4*e^3 + (8
43*b^5 + 80*a*b^3*c + 16752*a^2*b*c^2)*d^3*e^4 - 2*(225*a*b^4 + 2496*a^2*b^2*c +
2576*a^3*c^2)*d^2*e^5 + 4*(39*a^2*b^3 + 1036*a^3*b*c)*d*e^6 - 8*(3*a^3*b^2 + 14
0*a^4*c)*e^7)*x^2 - (1664*a^4*b*e^7 - 240*(b^2*c^3 + 20*a*c^4)*d^7 + 24*(95*b^3*
c^2 + 396*a*b*c^3)*d^6*e - 2*(1055*b^4*c + 10680*a*b^2*c^2 - 8016*a^2*c^3)*d^5*e
^2 + (595*b^5 + 18888*a*b^3*c - 5712*a^2*b*c^2)*d^4*e^3 - 8*(719*a*b^4 + 1866*a^
2*b^2*c - 648*a^3*c^2)*d^3*e^4 + 8*(1191*a^2*b^3 + 316*a^3*b*c)*d^2*e^5 - 32*(20
5*a^3*b^2 - 36*a^4*c)*d*e^6)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a
) - 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 - 24*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d^7*e + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^6*e^
2 + (24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e^6 - 24*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*d*e^7 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*
x^6 + 6*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c
^2 + 16*a^2*b*c^3)*d^2*e^6 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)
*d*e^7)*x^5 + 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4*e^4 - 24*(b^5*c -
8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*e^5 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64
*a^3*c^3)*d^2*e^6)*x^4 + 20*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 2
4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4*e^4 + (7*b^6 - 60*a*b^4*c + 144*a^2*b
^2*c^2 - 64*a^3*c^3)*d^3*e^5)*x^3 + 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*
d^6*e^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c
+ 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^4*e^4)*x^2 + 6*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16
*a^2*c^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2 + (7*b^6 - 60*
a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3)*x)*log(((8*a*b*d*e - 8*a^2*e^2
- (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c
*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*
d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e -
a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d
^2)))/((c^4*d^14 - 4*b*c^3*d^13*e - 4*a^3*b*d^7*e^7 + a^4*d^6*e^8 + 2*(3*b^2*c^2
+ 2*a*c^3)*d^12*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^11*e^3 + (b^4 + 12*a*b^2*c + 6*a^
2*c^2)*d^10*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^9*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^8*e^
6 + (c^4*d^8*e^6 - 4*b*c^3*d^7*e^7 - 4*a^3*b*d*e^13 + a^4*e^14 + 2*(3*b^2*c^2 +
2*a*c^3)*d^6*e^8 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^9 + (b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d^4*e^10 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^11 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^12)*
x^6 + 6*(c^4*d^9*e^5 - 4*b*c^3*d^8*e^6 - 4*a^3*b*d^2*e^12 + a^4*d*e^13 + 2*(3*b^
2*c^2 + 2*a*c^3)*d^7*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^8 + (b^4 + 12*a*b^2*c + 6
*a^2*c^2)*d^5*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3
*e^11)*x^5 + 15*(c^4*d^10*e^4 - 4*b*c^3*d^9*e^5 - 4*a^3*b*d^3*e^11 + a^4*d^2*e^1
2 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^7 + (b^4 + 12*
a*b^2*c + 6*a^2*c^2)*d^6*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^9 + 2*(3*a^2*b^2 + 2*
a^3*c)*d^4*e^10)*x^4 + 20*(c^4*d^11*e^3 - 4*b*c^3*d^10*e^4 - 4*a^3*b*d^4*e^10 +
a^4*d^3*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^6 +
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^8 + 2*(3*a
^2*b^2 + 2*a^3*c)*d^5*e^9)*x^3 + 15*(c^4*d^12*e^2 - 4*b*c^3*d^11*e^3 - 4*a^3*b*d
^5*e^9 + a^4*d^4*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^4 - 4*(b^3*c + 3*a*b*c^2)
*d^9*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^
7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^8)*x^2 + 6*(c^4*d^13*e - 4*b*c^3*d^12*e^2 - 4*
a^3*b*d^6*e^8 + a^4*d^5*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^3 - 4*(b^3*c + 3*a*
b*c^2)*d^10*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^5 - 4*(a*b^3 + 3*a^2*b*c)
*d^8*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/1
5360*(2*(4736*a^4*b*d*e^6 - 1280*a^5*e^7 - 120*(3*b^3*c^2 - 20*a*b*c^3)*d^7 + 24
*(15*b^4*c - 110*a*b^2*c^2 - 328*a^2*c^3)*d^6*e - 5*(21*b^5 - 200*a*b^3*c - 3120
*a^2*b*c^2)*d^5*e^2 - 2*(35*a*b^4 + 5832*a^2*b^2*c + 4272*a^3*c^2)*d^4*e^3 + 8*(
377*a^2*b^3 + 1812*a^3*b*c)*d^3*e^4 - 16*(387*a^3*b^2 + 332*a^4*c)*d^2*e^5 + (12
8*c^5*d^5*e^2 - 320*b*c^4*d^4*e^3 + 32*(3*b^2*c^3 + 28*a*c^4)*d^3*e^4 + 16*(11*b
^3*c^2 - 84*a*b*c^3)*d^2*e^5 - 2*(145*b^4*c - 984*a*b^2*c^2 + 1296*a^2*c^3)*d*e^
6 + (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^7)*x^5 + (768*c^5*d^6*e - 1984*b*
c^4*d^5*e^2 + 32*(23*b^2*c^3 + 168*a*c^4)*d^4*e^3 + 112*(9*b^3*c^2 - 76*a*b*c^3)
*d^3*e^4 - 16*(103*b^4*c - 690*a*b^2*c^2 + 792*a^2*c^3)*d^2*e^5 + (595*b^5 - 410
4*a*b^3*c + 6192*a^2*b*c^2)*d*e^6 - 2*(35*a*b^4 - 216*a^2*b^2*c + 240*a^3*c^2)*e
^7)*x^4 + 2*(960*c^5*d^7 - 2592*b*c^4*d^6*e + 8*(153*b^2*c^3 + 836*a*c^4)*d^5*e^
2 + 4*(279*b^3*c^2 - 2836*a*b*c^3)*d^4*e^3 - 2*(963*b^4*c - 6480*a*b^2*c^2 + 587
2*a^2*c^3)*d^3*e^4 + (693*b^5 - 4448*a*b^3*c + 4944*a^2*b*c^2)*d^2*e^5 - 4*(49*a
*b^4 - 282*a^2*b^2*c + 168*a^3*c^2)*d*e^6 + 4*(7*a^2*b^3 - 36*a^3*b*c)*e^7)*x^3
+ 2*(1440*b*c^4*d^7 - 192*(28*b^2*c^3 - 19*a*c^4)*d^6*e + 4*(1535*b^3*c^2 - 228*
a*b*c^3)*d^5*e^2 - 4*(893*b^4*c - 306*a*b^2*c^2 + 4296*a^2*c^3)*d^4*e^3 + (843*b
^5 + 80*a*b^3*c + 16752*a^2*b*c^2)*d^3*e^4 - 2*(225*a*b^4 + 2496*a^2*b^2*c + 257
6*a^3*c^2)*d^2*e^5 + 4*(39*a^2*b^3 + 1036*a^3*b*c)*d*e^6 - 8*(3*a^3*b^2 + 140*a^
4*c)*e^7)*x^2 - (1664*a^4*b*e^7 - 240*(b^2*c^3 + 20*a*c^4)*d^7 + 24*(95*b^3*c^2
+ 396*a*b*c^3)*d^6*e - 2*(1055*b^4*c + 10680*a*b^2*c^2 - 8016*a^2*c^3)*d^5*e^2 +
(595*b^5 + 18888*a*b^3*c - 5712*a^2*b*c^2)*d^4*e^3 - 8*(719*a*b^4 + 1866*a^2*b^
2*c - 648*a^3*c^2)*d^3*e^4 + 8*(1191*a^2*b^3 + 316*a^3*b*c)*d^2*e^5 - 32*(205*a^
3*b^2 - 36*a^4*c)*d*e^6)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) -
15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 - 24*(b^5*c - 8*a*b^3*c^2 + 16*
a^2*b*c^3)*d^7*e + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^6*e^2 +
(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e^6 - 24*(b^5*c - 8*a*b^3*c^2 + 16
*a^2*b*c^3)*d*e^7 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^8)*x^6
+ 6*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*e^5 - 24*(b^5*c - 8*a*b^3*c^2
+ 16*a^2*b*c^3)*d^2*e^6 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d*
e^7)*x^5 + 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4*e^4 - 24*(b^5*c - 8*a
*b^3*c^2 + 16*a^2*b*c^3)*d^3*e^5 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^
3*c^3)*d^2*e^6)*x^4 + 20*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5*e^3 - 24*(
b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4*e^4 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*
c^2 - 64*a^3*c^3)*d^3*e^5)*x^3 + 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6
*e^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5*e^3 + (7*b^6 - 60*a*b^4*c + 1
44*a^2*b^2*c^2 - 64*a^3*c^3)*d^4*e^4)*x^2 + 6*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^
2*c^4)*d^7*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^6*e^2 + (7*b^6 - 60*a*b
^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*e^3)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e
- a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 +
b*x + a))))/((c^4*d^14 - 4*b*c^3*d^13*e - 4*a^3*b*d^7*e^7 + a^4*d^6*e^8 + 2*(3*b
^2*c^2 + 2*a*c^3)*d^12*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^11*e^3 + (b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^10*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^9*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*
d^8*e^6 + (c^4*d^8*e^6 - 4*b*c^3*d^7*e^7 - 4*a^3*b*d*e^13 + a^4*e^14 + 2*(3*b^2*
c^2 + 2*a*c^3)*d^6*e^8 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^9 + (b^4 + 12*a*b^2*c + 6*a
^2*c^2)*d^4*e^10 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^11 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*
e^12)*x^6 + 6*(c^4*d^9*e^5 - 4*b*c^3*d^8*e^6 - 4*a^3*b*d^2*e^12 + a^4*d*e^13 + 2
*(3*b^2*c^2 + 2*a*c^3)*d^7*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^8 + (b^4 + 12*a*b^2
*c + 6*a^2*c^2)*d^5*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^10 + 2*(3*a^2*b^2 + 2*a^3*
c)*d^3*e^11)*x^5 + 15*(c^4*d^10*e^4 - 4*b*c^3*d^9*e^5 - 4*a^3*b*d^3*e^11 + a^4*d
^2*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^7 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^6*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^9 + 2*(3*a^2*b^
2 + 2*a^3*c)*d^4*e^10)*x^4 + 20*(c^4*d^11*e^3 - 4*b*c^3*d^10*e^4 - 4*a^3*b*d^4*e
^10 + a^4*d^3*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^8
*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^8 +
2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^9)*x^3 + 15*(c^4*d^12*e^2 - 4*b*c^3*d^11*e^3 - 4*a
^3*b*d^5*e^9 + a^4*d^4*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^4 - 4*(b^3*c + 3*a*
b*c^2)*d^9*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^6 - 4*(a*b^3 + 3*a^2*b*c)*
d^7*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^8)*x^2 + 6*(c^4*d^13*e - 4*b*c^3*d^12*e^
2 - 4*a^3*b*d^6*e^8 + a^4*d^5*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^11*e^3 - 4*(b^3*c
+ 3*a*b*c^2)*d^10*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^9*e^5 - 4*(a*b^3 + 3*a^
2*b*c)*d^8*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^7*e^7)*x)*sqrt(-c*d^2 + b*d*e - a*e^2
))]

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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((c*x**2+b*x+a)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.670358, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

sage0*x