3.947 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx\)
Optimal. Leaf size=219 \[ -\frac{5 \left (b^2-4 a c\right )^3 (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2048 a^{9/2}}+\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{1024 a^4 x^2}-\frac{5 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{384 a^3 x^4}+\frac{(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{5/2}}{24 a^2 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7} \]
[Out]
(5*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(1024*a^4*x^
2) - (5*(A*b - 2*a*B)*(b^2 - 4*a*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(384*a^
3*x^4) + ((A*b - 2*a*B)*(2*a + b*x)*(a + b*x + c*x^2)^(5/2))/(24*a^2*x^6) - (A*(
a + b*x + c*x^2)^(7/2))/(7*a*x^7) - (5*(A*b - 2*a*B)*(b^2 - 4*a*c)^3*ArcTanh[(2*
a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2048*a^(9/2))
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Rubi [A] time = 0.334288, antiderivative size = 219, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174
\[ -\frac{5 \left (b^2-4 a c\right )^3 (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2048 a^{9/2}}+\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{1024 a^4 x^2}-\frac{5 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{384 a^3 x^4}+\frac{(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{5/2}}{24 a^2 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8,x]
[Out]
(5*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(1024*a^4*x^
2) - (5*(A*b - 2*a*B)*(b^2 - 4*a*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(384*a^
3*x^4) + ((A*b - 2*a*B)*(2*a + b*x)*(a + b*x + c*x^2)^(5/2))/(24*a^2*x^6) - (A*(
a + b*x + c*x^2)^(7/2))/(7*a*x^7) - (5*(A*b - 2*a*B)*(b^2 - 4*a*c)^3*ArcTanh[(2*
a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2048*a^(9/2))
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Rubi in Sympy [A] time = 40.0476, size = 211, normalized size = 0.96 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} + \frac{\left (2 a + b x\right ) \left (A b - 2 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{24 a^{2} x^{6}} - \frac{5 \left (2 a + b x\right ) \left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{384 a^{3} x^{4}} + \frac{5 \left (2 a + b x\right ) \left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{1024 a^{4} x^{2}} - \frac{5 \left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right )^{3} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2048 a^{\frac{9}{2}}} \]
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**8,x)
[Out]
-A*(a + b*x + c*x**2)**(7/2)/(7*a*x**7) + (2*a + b*x)*(A*b - 2*B*a)*(a + b*x + c
*x**2)**(5/2)/(24*a**2*x**6) - 5*(2*a + b*x)*(A*b - 2*B*a)*(-4*a*c + b**2)*(a +
b*x + c*x**2)**(3/2)/(384*a**3*x**4) + 5*(2*a + b*x)*(A*b - 2*B*a)*(-4*a*c + b**
2)**2*sqrt(a + b*x + c*x**2)/(1024*a**4*x**2) - 5*(A*b - 2*B*a)*(-4*a*c + b**2)*
*3*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(2048*a**(9/2))
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Mathematica [A] time = 0.746439, size = 342, normalized size = 1.56 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (512 a^6 (6 A+7 B x)+128 a^5 x \left (58 A b+72 A c x+70 b B x+91 B c x^2\right )+32 a^4 x^2 \left (2 A \left (74 b^2+197 b c x+144 c^2 x^2\right )+21 B x \left (9 b^2+26 b c x+22 c^2 x^2\right )\right )+16 a^3 x^3 \left (3 A \left (b^3+10 b^2 c x+38 b c^2 x^2+64 c^3 x^3\right )+7 b B x \left (b^2+12 b c x+66 c^2 x^2\right )\right )-28 a^2 b^2 x^4 \left (2 A \left (b^2+12 b c x+66 c^2 x^2\right )+5 b B x (b+16 c x)\right )+70 a b^4 x^5 (A (b+16 c x)+3 b B x)-105 A b^6 x^6\right )+105 x^7 \log (x) \left (b^2-4 a c\right )^3 (A b-2 a B)-105 x^7 \left (b^2-4 a c\right )^3 (A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{43008 a^{9/2} x^7} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^8,x]
[Out]
(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-105*A*b^6*x^6 + 512*a^6*(6*A + 7*B*x) + 128*
a^5*x*(58*A*b + 70*b*B*x + 72*A*c*x + 91*B*c*x^2) + 70*a*b^4*x^5*(3*b*B*x + A*(b
+ 16*c*x)) - 28*a^2*b^2*x^4*(5*b*B*x*(b + 16*c*x) + 2*A*(b^2 + 12*b*c*x + 66*c^
2*x^2)) + 32*a^4*x^2*(21*B*x*(9*b^2 + 26*b*c*x + 22*c^2*x^2) + 2*A*(74*b^2 + 197
*b*c*x + 144*c^2*x^2)) + 16*a^3*x^3*(7*b*B*x*(b^2 + 12*b*c*x + 66*c^2*x^2) + 3*A
*(b^3 + 10*b^2*c*x + 38*b*c^2*x^2 + 64*c^3*x^3))) + 105*(A*b - 2*a*B)*(b^2 - 4*a
*c)^3*x^7*Log[x] - 105*(A*b - 2*a*B)*(b^2 - 4*a*c)^3*x^7*Log[2*a + b*x + 2*Sqrt[
a]*Sqrt[a + x*(b + c*x)]])/(43008*a^(9/2)*x^7)
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Maple [B] time = 0.055, size = 1874, normalized size = 8.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^8,x)
[Out]
1/512*B*b^5/a^6/x*(c*x^2+b*x+a)^(7/2)-5/32*B*b^2/a^3*c^2*(c*x^2+b*x+a)^(3/2)-1/8
*B*b^2/a^4*c^2*(c*x^2+b*x+a)^(5/2)-5/16*B*b^2/a^2*c^2*(c*x^2+b*x+a)^(1/2)-1/32*B
*b^2/a^3/x^4*(c*x^2+b*x+a)^(7/2)-1/7*A*(c*x^2+b*x+a)^(7/2)/a/x^7-5/96*A*b/a^3*c^
3*(c*x^2+b*x+a)^(3/2)-5/1536*B*b^5/a^5*c*(c*x^2+b*x+a)^(3/2)*x-1/512*B*b^5/a^6*c
*(c*x^2+b*x+a)^(5/2)*x-5/512*B*b^5/a^4*(c*x^2+b*x+a)^(1/2)*x*c+5/64*B*b^3/a^3*c^
2*(c*x^2+b*x+a)^(1/2)*x-5/32*B*b/a^3*c^3*(c*x^2+b*x+a)^(3/2)*x-5/32*B*b/a^4*c^3*
(c*x^2+b*x+a)^(5/2)*x+5/32*B*b/a^4*c^2/x*(c*x^2+b*x+a)^(7/2)+1/16*B*b/a^3*c/x^3*
(c*x^2+b*x+a)^(7/2)-5/32*B*b/a^2*c^3*(c*x^2+b*x+a)^(1/2)*x-1/32*B*b^2/a^4*c/x^2*
(c*x^2+b*x+a)^(7/2)+5/96*B*b^3/a^4*c^2*(c*x^2+b*x+a)^(3/2)*x+5/192*B*b^3/a^5*c^2
*(c*x^2+b*x+a)^(5/2)*x-5/192*B*b^3/a^5*c/x*(c*x^2+b*x+a)^(7/2)-5/192*A*b^4/a^5*c
^2*(c*x^2+b*x+a)^(3/2)*x-5/384*A*b^4/a^6*c^2*(c*x^2+b*x+a)^(5/2)*x+5/384*A*b^4/a
^6*c/x*(c*x^2+b*x+a)^(7/2)+5/3072*A*b^6/a^6*c*(c*x^2+b*x+a)^(3/2)*x+1/1024*A*b^6
/a^7*c*(c*x^2+b*x+a)^(5/2)*x+5/1024*A*b^6/a^5*(c*x^2+b*x+a)^(1/2)*x*c-5/128*A*b^
4/a^4*c^2*(c*x^2+b*x+a)^(1/2)*x+5/64*A*b^2/a^4*c^3*(c*x^2+b*x+a)^(3/2)*x+5/64*A*
b^2/a^5*c^3*(c*x^2+b*x+a)^(5/2)*x-5/64*A*b^2/a^5*c^2/x*(c*x^2+b*x+a)^(7/2)-1/32*
A*b^2/a^4*c/x^3*(c*x^2+b*x+a)^(7/2)+5/64*A*b^2/a^3*c^3*(c*x^2+b*x+a)^(1/2)*x+1/3
2*A*b/a^4*c^2/x^2*(c*x^2+b*x+a)^(7/2)+1/48*A*b/a^3*c/x^4*(c*x^2+b*x+a)^(7/2)+1/6
4*A*b^3/a^5*c/x^2*(c*x^2+b*x+a)^(7/2)-5/32*A*b/a^2*c^3*(c*x^2+b*x+a)^(1/2)+1/12*
A*b/a^2/x^6*(c*x^2+b*x+a)^(7/2)-1/1024*A*b^6/a^7/x*(c*x^2+b*x+a)^(7/2)+5/64*A*b^
3/a^4*c^2*(c*x^2+b*x+a)^(3/2)+1/16*A*b^3/a^5*c^2*(c*x^2+b*x+a)^(5/2)+5/32*A*b^3/
a^3*c^2*(c*x^2+b*x+a)^(1/2)+1/64*A*b^3/a^4/x^4*(c*x^2+b*x+a)^(7/2)-35/1536*A*b^5
/a^5*c*(c*x^2+b*x+a)^(3/2)-19/1536*A*b^5/a^6*c*(c*x^2+b*x+a)^(5/2)-1/1536*A*b^5/
a^6/x^2*(c*x^2+b*x+a)^(7/2)-1/384*A*b^4/a^5/x^3*(c*x^2+b*x+a)^(7/2)-25/512*A*b^5
/a^4*c*(c*x^2+b*x+a)^(1/2)-1/24*A*b^2/a^3/x^5*(c*x^2+b*x+a)^(7/2)+15/512*A*b^5/a
^(7/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/128*A*b^3/a^(5/2)*c^2*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/32*A*b/a^(3/2)*c^3*ln((2*a+b*x+2
*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/16*B/a^3*c^2/x^2*(c*x^2+b*x+a)^(7/2)-1/24*B/a
^2*c/x^4*(c*x^2+b*x+a)^(7/2)-15/256*B*b^4/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2
+b*x+a)^(1/2))/x)+15/64*B*b^2/a^(3/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x)+35/768*B*b^4/a^4*c*(c*x^2+b*x+a)^(3/2)+19/768*B*b^4/a^5*c*(c*x^2+b*x+a)^
(5/2)+1/768*B*b^4/a^5/x^2*(c*x^2+b*x+a)^(7/2)+1/192*B*b^3/a^4/x^3*(c*x^2+b*x+a)^
(7/2)+25/256*B*b^4/a^3*c*(c*x^2+b*x+a)^(1/2)+1/12*B*b/a^2/x^5*(c*x^2+b*x+a)^(7/2
)+5/48*B/a^2*c^3*(c*x^2+b*x+a)^(3/2)+1/16*B/a^3*c^3*(c*x^2+b*x+a)^(5/2)+5/16*B/a
*c^3*(c*x^2+b*x+a)^(1/2)-1/6*B/a/x^6*(c*x^2+b*x+a)^(7/2)-5/16*B/a^(1/2)*c^3*ln((
2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/1024*B*b^6/a^(7/2)*ln((2*a+b*x+2*a^(
1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/512*B*b^6/a^6*(c*x^2+b*x+a)^(5/2)-5/1536*B*b^6/a^
5*(c*x^2+b*x+a)^(3/2)-5/512*B*b^6/a^4*(c*x^2+b*x+a)^(1/2)+1/1024*A*b^7/a^7*(c*x^
2+b*x+a)^(5/2)+5/3072*A*b^7/a^6*(c*x^2+b*x+a)^(3/2)+5/1024*A*b^7/a^5*(c*x^2+b*x+
a)^(1/2)-5/2048*A*b^7/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/32
*A*b/a^4*c^3*(c*x^2+b*x+a)^(5/2)
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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^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.450215, 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^8,x, algorithm="fricas")
[Out]
[-1/86016*(105*(2*B*a*b^6 - A*b^7 - 64*(2*B*a^4 - A*a^3*b)*c^3 + 48*(2*B*a^3*b^2
- A*a^2*b^3)*c^2 - 12*(2*B*a^2*b^4 - A*a*b^5)*c)*x^7*log((4*(a*b*x + 2*a^2)*sqr
t(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*(30
72*A*a^6 + (210*B*a*b^5 - 105*A*b^6 + 3072*A*a^3*c^3 + 3696*(2*B*a^3*b - A*a^2*b
^2)*c^2 - 1120*(2*B*a^2*b^3 - A*a*b^4)*c)*x^6 - 2*(70*B*a^2*b^4 - 35*A*a*b^5 - 4
8*(154*B*a^4 + 19*A*a^3*b)*c^2 - 336*(2*B*a^3*b^2 - A*a^2*b^3)*c)*x^5 + 8*(14*B*
a^3*b^3 - 7*A*a^2*b^4 + 1152*A*a^4*c^2 + 12*(182*B*a^4*b + 5*A*a^3*b^2)*c)*x^4 +
16*(378*B*a^4*b^2 + 3*A*a^3*b^3 + 4*(182*B*a^5 + 197*A*a^4*b)*c)*x^3 + 128*(70*
B*a^5*b + 37*A*a^4*b^2 + 72*A*a^5*c)*x^2 + 256*(14*B*a^6 + 29*A*a^5*b)*x)*sqrt(c
*x^2 + b*x + a)*sqrt(a))/(a^(9/2)*x^7), 1/43008*(105*(2*B*a*b^6 - A*b^7 - 64*(2*
B*a^4 - A*a^3*b)*c^3 + 48*(2*B*a^3*b^2 - A*a^2*b^3)*c^2 - 12*(2*B*a^2*b^4 - A*a*
b^5)*c)*x^7*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 2*(3072
*A*a^6 + (210*B*a*b^5 - 105*A*b^6 + 3072*A*a^3*c^3 + 3696*(2*B*a^3*b - A*a^2*b^2
)*c^2 - 1120*(2*B*a^2*b^3 - A*a*b^4)*c)*x^6 - 2*(70*B*a^2*b^4 - 35*A*a*b^5 - 48*
(154*B*a^4 + 19*A*a^3*b)*c^2 - 336*(2*B*a^3*b^2 - A*a^2*b^3)*c)*x^5 + 8*(14*B*a^
3*b^3 - 7*A*a^2*b^4 + 1152*A*a^4*c^2 + 12*(182*B*a^4*b + 5*A*a^3*b^2)*c)*x^4 + 1
6*(378*B*a^4*b^2 + 3*A*a^3*b^3 + 4*(182*B*a^5 + 197*A*a^4*b)*c)*x^3 + 128*(70*B*
a^5*b + 37*A*a^4*b^2 + 72*A*a^5*c)*x^2 + 256*(14*B*a^6 + 29*A*a^5*b)*x)*sqrt(c*x
^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^4*x^7)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**8,x)
[Out]
Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**8, x)
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GIAC/XCAS [A] time = 0.308874, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^8,x, algorithm="giac")
[Out]
Done