3.945 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=346 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{192 a^2 x^3}+\frac{\sqrt{a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{128 a^2 x}+\frac{\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac{1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5} \]

[Out]

((10*a*b*B*(b^2 - 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(
b^2 + 12*a*c) - A*(3*b^3 - 28*a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(128*a^2*x) + ((
4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c) - A*(3*b^3 - 20*a*
b*c))*x)*(a + b*x + c*x^2)^(3/2))/(192*a^2*x^3) - ((8*a*A + 5*(A*b + 2*a*B)*x)*(
a + b*x + c*x^2)^(5/2))/(40*a*x^5) + ((10*a*B*(b^4 - 24*a*b^2*c - 48*a^2*c^2) -
A*(3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(256*a^(5/2)) + (c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*
Sqrt[c]*Sqrt[a + b*x + c*x^2])])/2

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Rubi [A]  time = 1.1506, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{192 a^2 x^3}+\frac{\sqrt{a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{128 a^2 x}+\frac{\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac{1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

((10*a*b*B*(b^2 - 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(
b^2 + 12*a*c) - A*(3*b^3 - 28*a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(128*a^2*x) + ((
4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c) - A*(3*b^3 - 20*a*
b*c))*x)*(a + b*x + c*x^2)^(3/2))/(192*a^2*x^3) - ((8*a*A + 5*(A*b + 2*a*B)*x)*(
a + b*x + c*x^2)^(5/2))/(40*a*x^5) + ((10*a*B*(b^4 - 24*a*b^2*c - 48*a^2*c^2) -
A*(3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(256*a^(5/2)) + (c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*
Sqrt[c]*Sqrt[a + b*x + c*x^2])])/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)*(c*x**2+b*x+a)**(5/2)/x**6,x)

[Out]

Timed out

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Mathematica [A]  time = 1.37412, size = 353, normalized size = 1.02 \[ \frac{15 x^5 \log (x) \left (A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )+10 a B \left (48 a^2 c^2+24 a b^2 c-b^4\right )\right )-15 x^5 \left (A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )+10 a B \left (48 a^2 c^2+24 a b^2 c-b^4\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} \left (96 a^4 (4 A+5 B x)+16 a^3 x (A (63 b+88 c x)+5 B x (17 b+27 c x))+4 a^2 x^2 \left (2 A \left (93 b^2+311 b c x+368 c^2 x^2\right )+5 B x \left (59 b^2+278 b c x-96 c^2 x^2\right )\right )+30 a b^2 x^3 (A (b+18 c x)+5 b B x)-45 A b^4 x^4\right )-960 a^2 c^{3/2} x^5 (2 A c+5 b B) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{3840 a^{5/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

(15*(10*a*B*(-b^4 + 24*a*b^2*c + 48*a^2*c^2) + A*(3*b^5 - 40*a*b^3*c + 240*a^2*b
*c^2))*x^5*Log[x] - 15*(10*a*B*(-b^4 + 24*a*b^2*c + 48*a^2*c^2) + A*(3*b^5 - 40*
a*b^3*c + 240*a^2*b*c^2))*x^5*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]] -
2*Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(-45*A*b^4*x^4 + 96*a^4*(4*A + 5*B*x) + 30*a*b
^2*x^3*(5*b*B*x + A*(b + 18*c*x)) + 16*a^3*x*(5*B*x*(17*b + 27*c*x) + A*(63*b +
88*c*x)) + 4*a^2*x^2*(5*B*x*(59*b^2 + 278*b*c*x - 96*c^2*x^2) + 2*A*(93*b^2 + 31
1*b*c*x + 368*c^2*x^2))) - 960*a^2*c^(3/2)*(5*b*B + 2*A*c)*x^5*Log[b + 2*c*x + 2
*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(3840*a^(5/2)*x^5)

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Maple [B]  time = 0.034, size = 1371, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x)

[Out]

3/128*A*b^5/a^3*(c*x^2+b*x+a)^(1/2)+3/640*A*b^5/a^5*(c*x^2+b*x+a)^(5/2)+1/128*A*
b^5/a^4*(c*x^2+b*x+a)^(3/2)-3/256*A*b^5/a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x
+a)^(1/2))/x)-1/5*A/a/x^5*(c*x^2+b*x+a)^(7/2)+5/2*B*b*c^(3/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))-5/64*B*b^4/a^2*(c*x^2+b*x+a)^(1/2)-1/64*B*b^4/a^4*(c*x
^2+b*x+a)^(5/2)-5/192*B*b^4/a^3*(c*x^2+b*x+a)^(3/2)+5/8*B/a*c^2*(c*x^2+b*x+a)^(3
/2)+37/96*B*b^2/a^3*c*(c*x^2+b*x+a)^(5/2)+1/96*B*b^2/a^3/x^2*(c*x^2+b*x+a)^(7/2)
-15/16*B*b^2/a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-3/8*B/a^2*c
/x^2*(c*x^2+b*x+a)^(7/2)+55/32*B*b^2/a*c*(c*x^2+b*x+a)^(1/2)+31/48*A*b/a^2*c^2*(
c*x^2+b*x+a)^(3/2)+65/96*B*b^2/a^2*c*(c*x^2+b*x+a)^(3/2)+15/8*B*c^2*(c*x^2+b*x+a
)^(1/2)+A*c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+109/240*A*b/a^3*c^
2*(c*x^2+b*x+a)^(5/2)+23/16*A*b/a*c^2*(c*x^2+b*x+a)^(1/2)+3/40*A*b/a^2/x^4*(c*x^
2+b*x+a)^(7/2)+5/32*A*b^3/a^(3/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x
)-15/16*A*b/a^(1/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+3/128*A*b^
4/a^3*(c*x^2+b*x+a)^(1/2)*x*c+19/240*A*b/a^3*c/x^2*(c*x^2+b*x+a)^(7/2)-13/96*A*b
^2/a^3*c^2*(c*x^2+b*x+a)^(3/2)*x+25/16*B*b/a*c^2*(c*x^2+b*x+a)^(1/2)*x-5/192*B*b
^3/a^3*c*(c*x^2+b*x+a)^(3/2)*x-1/64*B*b^3/a^4*c*(c*x^2+b*x+a)^(5/2)*x-5/64*B*b^3
/a^2*(c*x^2+b*x+a)^(1/2)*x*c+35/48*B*b/a^2*c^2*(c*x^2+b*x+a)^(3/2)*x+19/48*B*b/a
^3*c^2*(c*x^2+b*x+a)^(5/2)*x-19/48*B*b/a^3*c/x*(c*x^2+b*x+a)^(7/2)-11/160*A*b^2/
a^4*c^2*(c*x^2+b*x+a)^(5/2)*x+11/160*A*b^2/a^4*c/x*(c*x^2+b*x+a)^(7/2)-7/32*A*b^
2/a^2*c^2*(c*x^2+b*x+a)^(1/2)*x+1/128*A*b^4/a^4*c*(c*x^2+b*x+a)^(3/2)*x+3/640*A*
b^4/a^5*c*(c*x^2+b*x+a)^(5/2)*x+1/64*B*b^3/a^4/x*(c*x^2+b*x+a)^(7/2)+3/8*B/a^2*c
^2*(c*x^2+b*x+a)^(5/2)+5/128*B*b^4/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(
1/2))/x)-15/8*B*a^(1/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/4*B/
a/x^4*(c*x^2+b*x+a)^(7/2)-2/15*A/a^2*c/x^3*(c*x^2+b*x+a)^(7/2)+A/a*c^3*(c*x^2+b*
x+a)^(1/2)*x-3/640*A*b^4/a^5/x*(c*x^2+b*x+a)^(7/2)-23/192*A*b^3/a^3*c*(c*x^2+b*x
+a)^(3/2)-21/320*A*b^3/a^4*c*(c*x^2+b*x+a)^(5/2)-1/320*A*b^3/a^4/x^2*(c*x^2+b*x+
a)^(7/2)+1/24*B*b/a^2/x^3*(c*x^2+b*x+a)^(7/2)+2/3*A/a^2*c^3*(c*x^2+b*x+a)^(3/2)*
x+8/15*A/a^3*c^3*(c*x^2+b*x+a)^(5/2)*x-8/15*A/a^3*c^2/x*(c*x^2+b*x+a)^(7/2)-17/6
4*A*b^3/a^2*c*(c*x^2+b*x+a)^(1/2)-1/80*A*b^2/a^3/x^3*(c*x^2+b*x+a)^(7/2)

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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)^(5/2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.70039, 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)^(5/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(1920*(5*B*a^2*b*c + 2*A*a^2*c^2)*sqrt(a)*sqrt(c)*x^5*log(-8*c^2*x^2 - 8
*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 15*(10*B*a
*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*x^5
*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) + 4*(1920*B*a^2*c^2*x^5 - 384*A*a^4 - (150*B*a*b^3 - 45*A*b^
4 + 2944*A*a^2*c^2 + 20*(278*B*a^2*b + 27*A*a*b^2)*c)*x^4 - 2*(590*B*a^2*b^2 + 1
5*A*a*b^3 + 4*(270*B*a^3 + 311*A*a^2*b)*c)*x^3 - 8*(170*B*a^3*b + 93*A*a^2*b^2 +
176*A*a^3*c)*x^2 - 48*(10*B*a^4 + 21*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))
/(a^(5/2)*x^5), 1/7680*(3840*(5*B*a^2*b*c + 2*A*a^2*c^2)*sqrt(a)*sqrt(-c)*x^5*ar
ctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) - 15*(10*B*a*b^4 - 3*A*b^
5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*x^5*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) + 4*(1920*B*a^2*c^2*x^5 - 384*A*a^4 - (150*B*a*b^3 - 45*A*b^4 + 2944*A*a^
2*c^2 + 20*(278*B*a^2*b + 27*A*a*b^2)*c)*x^4 - 2*(590*B*a^2*b^2 + 15*A*a*b^3 + 4
*(270*B*a^3 + 311*A*a^2*b)*c)*x^3 - 8*(170*B*a^3*b + 93*A*a^2*b^2 + 176*A*a^3*c)
*x^2 - 48*(10*B*a^4 + 21*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(5/2)*x^5
), 1/3840*(960*(5*B*a^2*b*c + 2*A*a^2*c^2)*sqrt(-a)*sqrt(c)*x^5*log(-8*c^2*x^2 -
8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 15*(10*B
*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*x
^5*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*(1920*B*a^2*c^
2*x^5 - 384*A*a^4 - (150*B*a*b^3 - 45*A*b^4 + 2944*A*a^2*c^2 + 20*(278*B*a^2*b +
27*A*a*b^2)*c)*x^4 - 2*(590*B*a^2*b^2 + 15*A*a*b^3 + 4*(270*B*a^3 + 311*A*a^2*b
)*c)*x^3 - 8*(170*B*a^3*b + 93*A*a^2*b^2 + 176*A*a^3*c)*x^2 - 48*(10*B*a^4 + 21*
A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^5), 1/3840*(1920*(5*
B*a^2*b*c + 2*A*a^2*c^2)*sqrt(-a)*sqrt(-c)*x^5*arctan(1/2*(2*c*x + b)/(sqrt(c*x^
2 + b*x + a)*sqrt(-c))) + 15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2
- 40*(6*B*a^2*b^2 - A*a*b^3)*c)*x^5*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2
+ b*x + a)*a)) + 2*(1920*B*a^2*c^2*x^5 - 384*A*a^4 - (150*B*a*b^3 - 45*A*b^4 +
2944*A*a^2*c^2 + 20*(278*B*a^2*b + 27*A*a*b^2)*c)*x^4 - 2*(590*B*a^2*b^2 + 15*A*
a*b^3 + 4*(270*B*a^3 + 311*A*a^2*b)*c)*x^3 - 8*(170*B*a^3*b + 93*A*a^2*b^2 + 176
*A*a^3*c)*x^2 - 48*(10*B*a^4 + 21*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(s
qrt(-a)*a^2*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**6,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.657385, 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)^(5/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

sage0*x