3.2184 \(\int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)
Optimal. Leaf size=413 \[ \frac{(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2048 c^{11/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \]
[Out]
((2*c*d - b*e)^4*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(1024*c^5*e) + ((2*c*d - b*e)^2*(14*c*e*f + 4*c*d*g - 9*b
*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(384*c^4*e) - ((2
*c*d - b*e)*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(120*c^3*e^2) - ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(d + e*x)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(5/2))/(84*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2) + ((2*c*d - b*e)^6*(14*c*e*f + 4*c*d*g - 9*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
])])/(2048*c^(11/2)*e^2)
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Rubi [A] time = 1.09985, antiderivative size = 413, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159
\[ \frac{(2 c d-b e)^6 (-9 b e g+4 c d g+14 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2048 c^{11/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+4 c d g+14 c e f)}{1024 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+4 c d g+14 c e f)}{384 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{120 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+4 c d g+14 c e f)}{84 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
[Out]
((2*c*d - b*e)^4*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(1024*c^5*e) + ((2*c*d - b*e)^2*(14*c*e*f + 4*c*d*g - 9*b
*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(384*c^4*e) - ((2
*c*d - b*e)*(14*c*e*f + 4*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(120*c^3*e^2) - ((14*c*e*f + 4*c*d*g - 9*b*e*g)*(d + e*x)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(5/2))/(84*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2)^(5/2))/(7*c*e^2) + ((2*c*d - b*e)^6*(14*c*e*f + 4*c*d*g - 9*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
])])/(2048*c^(11/2)*e^2)
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Rubi in Sympy [A] time = 128.021, size = 398, normalized size = 0.96 \[ - \frac{g \left (d + e x\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{7 c e^{2}} + \frac{\left (d + e x\right ) \left (9 b e g - 4 c d g - 14 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{84 c^{2} e^{2}} - \frac{\left (b e - 2 c d\right ) \left (9 b e g - 4 c d g - 14 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{120 c^{3} e^{2}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{2} \left (9 b e g - 4 c d g - 14 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{384 c^{4} e} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{4} \left (9 b e g - 4 c d g - 14 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{1024 c^{5} e} - \frac{\left (b e - 2 c d\right )^{6} \left (9 b e g - 4 c d g - 14 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{2048 c^{\frac{11}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
-g*(d + e*x)**2*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(7*c*e**2) + (
d + e*x)*(9*b*e*g - 4*c*d*g - 14*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(5/2)/(84*c**2*e**2) - (b*e - 2*c*d)*(9*b*e*g - 4*c*d*g - 14*c*e*f)*(-b*e**2
*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(120*c**3*e**2) - (b + 2*c*x)*(b*e - 2
*c*d)**2*(9*b*e*g - 4*c*d*g - 14*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(3/2)/(384*c**4*e) - (b + 2*c*x)*(b*e - 2*c*d)**4*(9*b*e*g - 4*c*d*g - 14*c*
e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(1024*c**5*e) - (b*e - 2*c*d
)**6*(9*b*e*g - 4*c*d*g - 14*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2
*x - c*e**2*x**2 + d*(-b*e + c*d))))/(2048*c**(11/2)*e**2)
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Mathematica [C] time = 2.9362, size = 599, normalized size = 1.45 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{2 \left (945 b^6 e^6 g-210 b^5 c e^5 (50 d g+7 e f+3 e g x)+28 b^4 c^2 e^4 \left (1708 d^2 g+d e (560 f+226 g x)+e^2 x (35 f+18 g x)\right )-16 b^3 c^3 e^3 \left (7090 d^3 g+2 d^2 e (2107 f+786 g x)+4 d e^2 x (147 f+71 g x)+e^3 x^2 (49 f+27 g x)\right )+48 b^2 c^4 e^2 \left (3037 d^4 g+4 d^3 e (763 f+255 g x)+14 d^2 e^2 x (52 f+23 g x)+4 d e^3 x^2 (35 f+18 g x)+2 e^4 x^3 (7 f+4 g x)\right )+32 b c^5 e \left (-3054 d^5 g-123 d^4 e (35 f+11 g x)+12 d^3 e^2 x (91 f+75 g x)+2 d^2 e^3 x^2 (1911 f+1409 g x)+4 d e^4 x^3 (707 f+556 g x)+8 e^5 x^4 (91 f+75 g x)\right )+64 c^6 \left (432 d^6 g+42 d^5 e (16 f+5 g x)-3 d^4 e^2 x (315 f+208 g x)-28 d^3 e^3 x^2 (48 f+35 g x)-2 d^2 e^4 x^3 (35 f+24 g x)+112 d e^5 x^4 (6 f+5 g x)+40 e^6 x^5 (7 f+6 g x)\right )\right )}{105 c^5 e^2 (d+e x) (b e-c d+c e x)}+\frac{i (b e-2 c d)^6 (-9 b e g+4 c d g+14 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{11/2} e^2 (d+e x)^{3/2} (c (d-e x)-b e)^{3/2}}\right )}{2048} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
[Out]
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((2*(945*b^6*e^6*g - 210*b^5*c*e^5*(7*
e*f + 50*d*g + 3*e*g*x) + 64*c^6*(432*d^6*g + 112*d*e^5*x^4*(6*f + 5*g*x) + 42*d
^5*e*(16*f + 5*g*x) + 40*e^6*x^5*(7*f + 6*g*x) - 2*d^2*e^4*x^3*(35*f + 24*g*x) -
28*d^3*e^3*x^2*(48*f + 35*g*x) - 3*d^4*e^2*x*(315*f + 208*g*x)) + 28*b^4*c^2*e^
4*(1708*d^2*g + e^2*x*(35*f + 18*g*x) + d*e*(560*f + 226*g*x)) + 48*b^2*c^4*e^2*
(3037*d^4*g + 2*e^4*x^3*(7*f + 4*g*x) + 4*d*e^3*x^2*(35*f + 18*g*x) + 14*d^2*e^2
*x*(52*f + 23*g*x) + 4*d^3*e*(763*f + 255*g*x)) - 16*b^3*c^3*e^3*(7090*d^3*g + e
^3*x^2*(49*f + 27*g*x) + 4*d*e^2*x*(147*f + 71*g*x) + 2*d^2*e*(2107*f + 786*g*x)
) + 32*b*c^5*e*(-3054*d^5*g - 123*d^4*e*(35*f + 11*g*x) + 12*d^3*e^2*x*(91*f + 7
5*g*x) + 8*e^5*x^4*(91*f + 75*g*x) + 4*d*e^4*x^3*(707*f + 556*g*x) + 2*d^2*e^3*x
^2*(1911*f + 1409*g*x))))/(105*c^5*e^2*(d + e*x)*(-(c*d) + b*e + c*e*x)) + (I*(-
2*c*d + b*e)^6*(14*c*e*f + 4*c*d*g - 9*b*e*g)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] +
2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(c^(11/2)*e^2*(d + e*x)^(3/2)*(-(b
*e) + c*(d - e*x))^(3/2))))/2048
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Maple [B] time = 0.021, size = 2799, normalized size = 6.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
[Out]
7/16*d^6*f*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2))+3/28*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-9/10
24*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3/128*e^2*g*b^4/c^4*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+7/60*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
5/2)*f-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*f-3/40*g*b^2/c^3*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-1/7*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c
+7/24*d^2*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+7/32*d^4*f*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*b-7/16*d^3*f/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b
^2+61/210*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/e*d*g+7/96*b^2/c^2*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^2*f+19/256*b^5/c^4*e^3*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)*d*g+3/8*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*e*
g+1/24/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^3*g-31/128*b^4/c^3*e^2*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g+21/16*b^2/(c*e^2)^(1/2)*arctan((c*e^2)
^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^5*e*g+7/256*b^4/c^3
*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f+5/48*b^3/c^3*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(3/2)*d*e*g+1/8*c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*g
+1/8*c^2/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(1/2))*d^7*g-13/48/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d^2*g
-7/8*d^3*f*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b+21/64*d^2*f/c^2*e^2*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^3-3/64*e^2*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(3/2)*x-9/2048*e^6*g*b^7/c^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/
2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-7/64*b^4/c^3*e^3*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*d*f+7/1024*b^6/c^4*e^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)
*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f-21/32*c/(c*e^2)^(1/2)*arc
tan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b*d^6*g+10
5/64*d^4*f*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2))*b^2-7/48/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*
d*f-1/3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*g-9/512*e^4*g*b^5/c^4*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-175/128*b^3/c*e^2/(c*e^2)^(1/2)*arctan((c*
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4*g-31/64*b^3/c
^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*g-7/32*b^3/c^2*e^3*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-147/512*b^5/c^3*e^4/(c*e^2)^(1/2)*arctan((c*
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2*g-21/256*b^5/
c^3*e^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(1/2))*d*f+21/32*d^2*f/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^
2+105/256*d^2*f/c^2*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^4-35/32*d^3*f/c*e^3/(c*e^2)^(1/2)*arctan((c*e^2
)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^3+19/128*b^4/c^3*e
^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*g+3/4*b^2/c*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(1/2)*x*d^3*e*g-21/16*d^5*f*c*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x
+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b+7/128*b^6/c^4*e^5/(c*e^2)^(1
/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d*g
+105/128*b^4/c^2*e^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2))*d^3*g+5/24*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(3/2)*x*d*e*g-7/24/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d*f+7/16*d^4*f
*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+7/48*d^2*f/c*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*b+7/512*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+1/1
2/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*g-13/96/c^2*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(3/2)*b^2*d^2*g-9/35*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2*
d^2*g-2/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e*d*f-17/32*(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2)*x*b*d^4*g-17/64/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b
^2*d^4*g+7/192*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^2*f+1/16/e*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^5*g
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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)^(3/2)*(e*x + d)^2*(g*x + f),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 5.42832, 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)^(3/2)*(e*x + d)^2*(g*x + f),x, algorithm="fricas")
[Out]
[-1/430080*(4*(15360*c^6*e^6*g*x^6 + 1280*(14*c^6*e^6*f + (28*c^6*d*e^5 + 15*b*c
^5*e^6)*g)*x^5 + 128*(14*(24*c^6*d*e^5 + 13*b*c^5*e^6)*f - (24*c^6*d^2*e^4 - 556
*b*c^5*d*e^5 - 3*b^2*c^4*e^6)*g)*x^4 - 16*(14*(20*c^6*d^2*e^4 - 404*b*c^5*d*e^5
- 3*b^2*c^4*e^6)*f + (3920*c^6*d^3*e^3 - 5636*b*c^5*d^2*e^4 - 216*b^2*c^4*d*e^5
+ 27*b^3*c^3*e^6)*g)*x^3 - 8*(14*(768*c^6*d^3*e^3 - 1092*b*c^5*d^2*e^4 - 60*b^2*
c^4*d*e^5 + 7*b^3*c^3*e^6)*f + (4992*c^6*d^4*e^2 - 3600*b*c^5*d^3*e^3 - 1932*b^2
*c^4*d^2*e^4 + 568*b^3*c^3*d*e^5 - 63*b^4*c^2*e^6)*g)*x^2 + 14*(3072*c^6*d^5*e -
9840*b*c^5*d^4*e^2 + 10464*b^2*c^4*d^3*e^3 - 4816*b^3*c^3*d^2*e^4 + 1120*b^4*c^
2*d*e^5 - 105*b^5*c*e^6)*f + (27648*c^6*d^6 - 97728*b*c^5*d^5*e + 145776*b^2*c^4
*d^4*e^2 - 113440*b^3*c^3*d^3*e^3 + 47824*b^4*c^2*d^2*e^4 - 10500*b^5*c*d*e^5 +
945*b^6*e^6)*g - 2*(14*(2160*c^6*d^4*e^2 - 1248*b*c^5*d^3*e^3 - 1248*b^2*c^4*d^2
*e^4 + 336*b^3*c^3*d*e^5 - 35*b^4*c^2*e^6)*f - (6720*c^6*d^5*e - 21648*b*c^5*d^4
*e^2 + 24480*b^2*c^4*d^3*e^3 - 12576*b^3*c^3*d^2*e^4 + 3164*b^4*c^2*d*e^5 - 315*
b^5*c*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 105*(14*(
64*c^7*d^6*e - 192*b*c^6*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*b^3*c^4*d^3*e^4 + 6
0*b^4*c^3*d^2*e^5 - 12*b^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*
d^6*e + 2688*b^2*c^5*d^5*e^2 - 2800*b^3*c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588
*b^5*c^2*d^2*e^5 + 112*b^6*c*d*e^6 - 9*b^7*e^7)*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^5*e^2), -1/215040*(2*(15360*c^
6*e^6*g*x^6 + 1280*(14*c^6*e^6*f + (28*c^6*d*e^5 + 15*b*c^5*e^6)*g)*x^5 + 128*(1
4*(24*c^6*d*e^5 + 13*b*c^5*e^6)*f - (24*c^6*d^2*e^4 - 556*b*c^5*d*e^5 - 3*b^2*c^
4*e^6)*g)*x^4 - 16*(14*(20*c^6*d^2*e^4 - 404*b*c^5*d*e^5 - 3*b^2*c^4*e^6)*f + (3
920*c^6*d^3*e^3 - 5636*b*c^5*d^2*e^4 - 216*b^2*c^4*d*e^5 + 27*b^3*c^3*e^6)*g)*x^
3 - 8*(14*(768*c^6*d^3*e^3 - 1092*b*c^5*d^2*e^4 - 60*b^2*c^4*d*e^5 + 7*b^3*c^3*e
^6)*f + (4992*c^6*d^4*e^2 - 3600*b*c^5*d^3*e^3 - 1932*b^2*c^4*d^2*e^4 + 568*b^3*
c^3*d*e^5 - 63*b^4*c^2*e^6)*g)*x^2 + 14*(3072*c^6*d^5*e - 9840*b*c^5*d^4*e^2 + 1
0464*b^2*c^4*d^3*e^3 - 4816*b^3*c^3*d^2*e^4 + 1120*b^4*c^2*d*e^5 - 105*b^5*c*e^6
)*f + (27648*c^6*d^6 - 97728*b*c^5*d^5*e + 145776*b^2*c^4*d^4*e^2 - 113440*b^3*c
^3*d^3*e^3 + 47824*b^4*c^2*d^2*e^4 - 10500*b^5*c*d*e^5 + 945*b^6*e^6)*g - 2*(14*
(2160*c^6*d^4*e^2 - 1248*b*c^5*d^3*e^3 - 1248*b^2*c^4*d^2*e^4 + 336*b^3*c^3*d*e^
5 - 35*b^4*c^2*e^6)*f - (6720*c^6*d^5*e - 21648*b*c^5*d^4*e^2 + 24480*b^2*c^4*d^
3*e^3 - 12576*b^3*c^3*d^2*e^4 + 3164*b^4*c^2*d*e^5 - 315*b^5*c*e^6)*g)*x)*sqrt(-
c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) - 105*(14*(64*c^7*d^6*e - 192*b*c^6
*d^5*e^2 + 240*b^2*c^5*d^4*e^3 - 160*b^3*c^4*d^3*e^4 + 60*b^4*c^3*d^2*e^5 - 12*b
^5*c^2*d*e^6 + b^6*c*e^7)*f + (256*c^7*d^7 - 1344*b*c^6*d^6*e + 2688*b^2*c^5*d^5
*e^2 - 2800*b^3*c^4*d^4*e^3 + 1680*b^4*c^3*d^3*e^4 - 588*b^5*c^2*d^2*e^5 + 112*b
^6*c*d*e^6 - 9*b^7*e^7)*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^(11/2)*e^2)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2} \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**2*(f + g*x), x)
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GIAC/XCAS [A] time = 0.32217, 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)^(3/2)*(e*x + d)^2*(g*x + f),x, algorithm="giac")
[Out]
Done