3.969 \(\int \frac{A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=317 \[ -\frac{\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{9/2}}-\frac{\sqrt{a+b x+c x^2} \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{\sqrt{a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
[Out]
(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^3*Sqrt[a +
b*x + c*x^2]) - ((7*A*b^2 - 6*a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(3*a^2*(b
^2 - 4*a*c)*x^3) - ((6*a*B*(5*b^2 - 12*a*c) - A*(35*b^3 - 116*a*b*c))*Sqrt[a + b
*x + c*x^2])/(12*a^3*(b^2 - 4*a*c)*x^2) + ((6*a*b*B*(15*b^2 - 52*a*c) - A*(105*b
^4 - 460*a*b^2*c + 256*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(24*a^4*(b^2 - 4*a*c)*x)
- ((6*a*B*(5*b^2 - 4*a*c) - A*(35*b^3 - 60*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[
a]*Sqrt[a + b*x + c*x^2])])/(16*a^(9/2))
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Rubi [A] time = 0.959863, antiderivative size = 317, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217
\[ -\frac{\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{9/2}}-\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{\sqrt{a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}+\frac{\sqrt{a+b x+c x^2} \left (72 a^2 B c-116 a A b c-30 a b^2 B+35 A b^3\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac{2 \left (-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^3*Sqrt[a +
b*x + c*x^2]) - ((7*A*b^2 - 6*a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(3*a^2*(
b^2 - 4*a*c)*x^3) + ((35*A*b^3 - 30*a*b^2*B - 116*a*A*b*c + 72*a^2*B*c)*Sqrt[a +
b*x + c*x^2])/(12*a^3*(b^2 - 4*a*c)*x^2) + ((6*a*b*B*(15*b^2 - 52*a*c) - A*(105
*b^4 - 460*a*b^2*c + 256*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(24*a^4*(b^2 - 4*a*c)*
x) - ((6*a*B*(5*b^2 - 4*a*c) - A*(35*b^3 - 60*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqr
t[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(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((B*x+A)/x**4/(c*x**2+b*x+a)**(3/2),x)
[Out]
Timed out
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Mathematica [A] time = 1.59701, size = 278, normalized size = 0.88 \[ \frac{1}{16} \left (\frac{\log (x) \left (A \left (60 a b c-35 b^3\right )+6 a B \left (5 b^2-4 a c\right )\right )}{a^{9/2}}+\frac{\left (5 A \left (7 b^3-12 a b c\right )+6 a B \left (4 a c-5 b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{a^{9/2}}+\frac{2 \sqrt{a+x (b+c x)} \left (\frac{48 \left (a B \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )-A \left (5 a^2 b c^2+2 a^2 c^3 x-5 a b^3 c-4 a b^2 c^2 x+b^5+b^4 c x\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{8 a^2 A}{x^3}+\frac{40 a A c+42 a b B-57 A b^2}{x}+\frac{2 a (11 A b-6 a B)}{x^2}\right )}{3 a^4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x]
[Out]
((2*Sqrt[a + x*(b + c*x)]*((-8*a^2*A)/x^3 + (2*a*(11*A*b - 6*a*B))/x^2 + (-57*A*
b^2 + 42*a*b*B + 40*a*A*c)/x + (48*(a*B*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x -
3*a*b*c^2*x) - A*(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + b^4*c*x - 4*a*b^2*c^2*x + 2*a
^2*c^3*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x)))))/(3*a^4) + ((6*a*B*(5*b^2 - 4*a*c
) + A*(-35*b^3 + 60*a*b*c))*Log[x])/a^(9/2) + ((6*a*B*(-5*b^2 + 4*a*c) + 5*A*(7*
b^3 - 12*a*b*c))*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/a^(9/2))/16
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Maple [B] time = 0.02, size = 708, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^4/(c*x^2+b*x+a)^(3/2),x)
[Out]
-1/3*A/a/x^3/(c*x^2+b*x+a)^(1/2)+7/12*A*b/a^2/x^2/(c*x^2+b*x+a)^(1/2)-35/24*A*b^
2/a^3/x/(c*x^2+b*x+a)^(1/2)-35/16*A*b^3/a^4/(c*x^2+b*x+a)^(1/2)+35/8*A*b^4/a^4/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x+35/16*A*b^5/a^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)+35/16*A*b^3/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-115/6*A*b^2
/a^3*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-115/12*A*b^3/a^3*c/(4*a*c-b^2)/(c*x^2
+b*x+a)^(1/2)+15/4*A*b/a^3*c/(c*x^2+b*x+a)^(1/2)-15/4*A*b/a^(7/2)*c*ln((2*a+b*x+
2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+4/3*A/a^2*c/x/(c*x^2+b*x+a)^(1/2)+32/3*A/a^2*c
^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16/3*A/a^2*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*b-1/2*B/a/x^2/(c*x^2+b*x+a)^(1/2)+5/4*B*b/a^2/x/(c*x^2+b*x+a)^(1/2)+15/8*B*b
^2/a^3/(c*x^2+b*x+a)^(1/2)-15/4*B*b^3/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-15
/8*B*b^4/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-15/8*B*b^2/a^(7/2)*ln((2*a+b*x+2*a^
(1/2)*(c*x^2+b*x+a)^(1/2))/x)+13*B*b/a^2*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+1
3/2*B*b^2/a^2*c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3/2*B/a^2*c/(c*x^2+b*x+a)^(1/2)+
3/2*B/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
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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((B*x + A)/((c*x^2 + b*x + a)^(3/2)*x^4),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.54966, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x + a)^(3/2)*x^4),x, algorithm="fricas")
[Out]
[-1/96*(4*(8*A*a^3*b^2 - 32*A*a^4*c + (256*A*a^2*c^3 + 4*(78*B*a^2*b - 115*A*a*b
^2)*c^2 - 15*(6*B*a*b^3 - 7*A*b^4)*c)*x^4 - (90*B*a*b^4 - 105*A*b^5 + 8*(18*B*a^
3 - 61*A*a^2*b)*c^2 - 2*(186*B*a^2*b^2 - 265*A*a*b^3)*c)*x^3 - (30*B*a^2*b^3 - 3
5*A*a*b^4 - 128*A*a^3*c^2 - 4*(30*B*a^3*b - 43*A*a^2*b^2)*c)*x^2 + 2*(6*B*a^3*b^
2 - 7*A*a^2*b^3 - 4*(6*B*a^4 - 7*A*a^3*b)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a) -
3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a*b^3)*c^2 + 5*(6*B*a*
b^4 - 7*A*b^5)*c)*x^5 + (30*B*a*b^5 - 35*A*b^6 + 48*(2*B*a^3*b - 5*A*a^2*b^2)*c^
2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x^4 + (30*B*a^2*b^4 - 35*A*a*b^5 + 48*(2*B*
a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 25*A*a^2*b^3)*c)*x^3)*log((4*(a*b*x + 2
*a^2)*sqrt(c*x^2 + b*x + a) - (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*a^2)*sqrt(a))/x^2
))/(((a^4*b^2*c - 4*a^5*c^2)*x^5 + (a^4*b^3 - 4*a^5*b*c)*x^4 + (a^5*b^2 - 4*a^6*
c)*x^3)*sqrt(a)), -1/48*(2*(8*A*a^3*b^2 - 32*A*a^4*c + (256*A*a^2*c^3 + 4*(78*B*
a^2*b - 115*A*a*b^2)*c^2 - 15*(6*B*a*b^3 - 7*A*b^4)*c)*x^4 - (90*B*a*b^4 - 105*A
*b^5 + 8*(18*B*a^3 - 61*A*a^2*b)*c^2 - 2*(186*B*a^2*b^2 - 265*A*a*b^3)*c)*x^3 -
(30*B*a^2*b^3 - 35*A*a*b^4 - 128*A*a^3*c^2 - 4*(30*B*a^3*b - 43*A*a^2*b^2)*c)*x^
2 + 2*(6*B*a^3*b^2 - 7*A*a^2*b^3 - 4*(6*B*a^4 - 7*A*a^3*b)*c)*x)*sqrt(c*x^2 + b*
x + a)*sqrt(-a) + 3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a*b^
3)*c^2 + 5*(6*B*a*b^4 - 7*A*b^5)*c)*x^5 + (30*B*a*b^5 - 35*A*b^6 + 48*(2*B*a^3*b
- 5*A*a^2*b^2)*c^2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x^4 + (30*B*a^2*b^4 - 35*
A*a*b^5 + 48*(2*B*a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 25*A*a^2*b^3)*c)*x^3)
*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)))/(((a^4*b^2*c - 4*a^
5*c^2)*x^5 + (a^4*b^3 - 4*a^5*b*c)*x^4 + (a^5*b^2 - 4*a^6*c)*x^3)*sqrt(-a))]
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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((B*x+A)/x**4/(c*x**2+b*x+a)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.291141, size = 1077, normalized size = 3.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x + a)^(3/2)*x^4),x, algorithm="giac")
[Out]
2*((B*a^5*b^3*c - A*a^4*b^4*c - 3*B*a^6*b*c^2 + 4*A*a^5*b^2*c^2 - 2*A*a^6*c^3)*x
/(a^8*b^2 - 4*a^9*c) + (B*a^5*b^4 - A*a^4*b^5 - 4*B*a^6*b^2*c + 5*A*a^5*b^3*c +
2*B*a^7*c^2 - 5*A*a^6*b*c^2)/(a^8*b^2 - 4*a^9*c))/sqrt(c*x^2 + b*x + a) + 1/8*(3
0*B*a*b^2 - 35*A*b^3 - 24*B*a^2*c + 60*A*a*b*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) - 1/24*(42*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*B*a*b^2 - 57*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^3 - 24*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*B*a^2*c + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a
*b*c + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^2*b*sqrt(c) - 48*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^4*A*a*b^2*sqrt(c) + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*A*a^2*c^(3/2) - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b^2 + 136*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*A*a^2*b*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c) +
144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^2*b^2*sqrt(c) - 192*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*A*a^3*c^(3/2) + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
B*a^3*b^2 - 87*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 + 24*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*B*a^4*c - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b*c
+ 96*B*a^4*b*sqrt(c) - 144*A*a^3*b^2*sqrt(c) + 80*A*a^4*c^(3/2))/(((sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2 - a)^3*a^4)