3.933 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx\)
Optimal. Leaf size=230 \[ -\frac{\left (b^2-4 a c\right )^2 \left (-4 a A c-12 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{1024 a^{9/2}}+\frac{\left (b^2-4 a c\right ) (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-12 a b B+7 A b^2\right )}{512 a^4 x^2}-\frac{(2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-12 a b B+7 A b^2\right )}{192 a^3 x^4}+\frac{(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac{A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6} \]
[Out]
((b^2 - 4*a*c)*(7*A*b^2 - 12*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])
/(512*a^4*x^2) - ((7*A*b^2 - 12*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(
3/2))/(192*a^3*x^4) - (A*(a + b*x + c*x^2)^(5/2))/(6*a*x^6) + ((7*A*b - 12*a*B)*
(a + b*x + c*x^2)^(5/2))/(60*a^2*x^5) - ((b^2 - 4*a*c)^2*(7*A*b^2 - 12*a*b*B - 4
*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(1024*a^(9/2))
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Rubi [A] time = 0.469279, antiderivative size = 230, 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
\[ -\frac{\left (b^2-4 a c\right )^2 \left (-4 a A c-12 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{1024 a^{9/2}}+\frac{\left (b^2-4 a c\right ) (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-12 a b B+7 A b^2\right )}{512 a^4 x^2}-\frac{(2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-12 a b B+7 A b^2\right )}{192 a^3 x^4}+\frac{(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac{A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]
[Out]
((b^2 - 4*a*c)*(7*A*b^2 - 12*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])
/(512*a^4*x^2) - ((7*A*b^2 - 12*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(
3/2))/(192*a^3*x^4) - (A*(a + b*x + c*x^2)^(5/2))/(6*a*x^6) + ((7*A*b - 12*a*B)*
(a + b*x + c*x^2)^(5/2))/(60*a^2*x^5) - ((b^2 - 4*a*c)^2*(7*A*b^2 - 12*a*b*B - 4
*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(1024*a^(9/2))
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Rubi in Sympy [A] time = 48.0077, size = 223, normalized size = 0.97 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{\left (7 A b - 12 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{60 a^{2} x^{5}} - \frac{\left (2 a + b x\right ) \left (- 4 A a c + b \left (7 A b - 12 B a\right )\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{192 a^{3} x^{4}} + \frac{\left (2 a + b x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (- 4 A a c + 7 A b^{2} - 12 B a b\right )}{512 a^{4} x^{2}} - \frac{\left (- 4 a c + b^{2}\right )^{2} \left (- 4 A a c + b \left (7 A b - 12 B a\right )\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{1024 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**7,x)
[Out]
-A*(a + b*x + c*x**2)**(5/2)/(6*a*x**6) + (7*A*b - 12*B*a)*(a + b*x + c*x**2)**(
5/2)/(60*a**2*x**5) - (2*a + b*x)*(-4*A*a*c + b*(7*A*b - 12*B*a))*(a + b*x + c*x
**2)**(3/2)/(192*a**3*x**4) + (2*a + b*x)*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)
*(-4*A*a*c + 7*A*b**2 - 12*B*a*b)/(512*a**4*x**2) - (-4*a*c + b**2)**2*(-4*A*a*c
+ b*(7*A*b - 12*B*a))*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(10
24*a**(9/2))
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Mathematica [A] time = 0.483856, size = 290, normalized size = 1.26 \[ \frac{\frac{\log (x) \left (b^2-4 a c\right )^2 \left (-4 a A c-12 a b B+7 A b^2\right )}{a^{9/2}}+\frac{\left (b^2-4 a c\right )^2 \left (4 a A c+12 a b B-7 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{a^{9/2}}-\frac{2 \sqrt{a+x (b+c x)} \left (256 a^5 (5 A+6 B x)+64 a^4 x \left (26 A b+35 A c x+33 b B x+48 B c x^2\right )+48 a^3 x^2 \left (A \left (b^2+6 b c x+10 c^2 x^2\right )+2 B x \left (b^2+7 b c x+16 c^2 x^2\right )\right )-8 a^2 b x^3 \left (A \left (7 b^2+54 b c x+162 c^2 x^2\right )+15 b B x (b+10 c x)\right )+10 a b^3 x^4 (7 A b+76 A c x+18 b B x)-105 A b^5 x^5\right )}{15 a^4 x^6}}{1024} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]
[Out]
((-2*Sqrt[a + x*(b + c*x)]*(-105*A*b^5*x^5 + 256*a^5*(5*A + 6*B*x) + 10*a*b^3*x^
4*(7*A*b + 18*b*B*x + 76*A*c*x) + 64*a^4*x*(26*A*b + 33*b*B*x + 35*A*c*x + 48*B*
c*x^2) + 48*a^3*x^2*(A*(b^2 + 6*b*c*x + 10*c^2*x^2) + 2*B*x*(b^2 + 7*b*c*x + 16*
c^2*x^2)) - 8*a^2*b*x^3*(15*b*B*x*(b + 10*c*x) + A*(7*b^2 + 54*b*c*x + 162*c^2*x
^2))))/(15*a^4*x^6) + ((b^2 - 4*a*c)^2*(7*A*b^2 - 12*a*b*B - 4*a*A*c)*Log[x])/a^
(9/2) + ((b^2 - 4*a*c)^2*(-7*A*b^2 + 12*a*b*B + 4*a*A*c)*Log[2*a + b*x + 2*Sqrt[
a]*Sqrt[a + x*(b + c*x)]])/a^(9/2))/1024
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Maple [B] time = 0.031, size = 1264, normalized size = 5.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^7,x)
[Out]
3/16*B*b/a^(3/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+7/192*A*b^3/a
^4/x^3*(c*x^2+b*x+a)^(5/2)-7/768*A*b^4/a^5/x^2*(c*x^2+b*x+a)^(5/2)-7/1536*A*b^5/
a^6/x*(c*x^2+b*x+a)^(5/2)-37/768*A*b^4/a^5*c*(c*x^2+b*x+a)^(3/2)-23/256*A*b^4/a^
4*c*(c*x^2+b*x+a)^(1/2)+1/24*A/a^2*c/x^4*(c*x^2+b*x+a)^(5/2)+1/48*A/a^3*c^2/x^2*
(c*x^2+b*x+a)^(5/2)-1/16*B*b^2/a^3/x^3*(c*x^2+b*x+a)^(5/2)+1/64*B*b^3/a^4/x^2*(c
*x^2+b*x+a)^(5/2)+1/128*B*b^4/a^5/x*(c*x^2+b*x+a)^(5/2)+5/64*B*b^3/a^4*c*(c*x^2+
b*x+a)^(3/2)+9/64*B*b^3/a^3*c*(c*x^2+b*x+a)^(1/2)+15/256*A*b^4/a^(7/2)*c*ln((2*a
+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-9/64*A*b^2/a^(5/2)*c^2*ln((2*a+b*x+2*a^(1
/2)*(c*x^2+b*x+a)^(1/2))/x)+7/60*A*b/a^2/x^5*(c*x^2+b*x+a)^(5/2)+1/16*A*b^2/a^4*
c^2*(c*x^2+b*x+a)^(3/2)+5/32*A*b^2/a^3*c^2*(c*x^2+b*x+a)^(1/2)-3/32*B*b^3/a^(5/2
)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/16*B*b/a^3*c^2*(c*x^2+b*x+a)
^(3/2)-3/16*B*b/a^2*c^2*(c*x^2+b*x+a)^(1/2)+1/8*B*b/a^2/x^4*(c*x^2+b*x+a)^(5/2)-
1/6*A*(c*x^2+b*x+a)^(5/2)/a/x^6+7/512*A*b^5/a^5*c*(c*x^2+b*x+a)^(1/2)*x-1/16*A*b
^3/a^4*c^2*(c*x^2+b*x+a)^(1/2)*x+7/1536*A*b^5/a^6*c*(c*x^2+b*x+a)^(3/2)*x-11/192
*A*b^3/a^5*c^2*(c*x^2+b*x+a)^(3/2)*x-1/32*A/a^4*c^2*b/x*(c*x^2+b*x+a)^(5/2)+1/32
*A/a^4*c^3*b*(c*x^2+b*x+a)^(3/2)*x+1/32*A/a^3*c^3*b*(c*x^2+b*x+a)^(1/2)*x-1/48*A
/a^3*c*b/x^3*(c*x^2+b*x+a)^(5/2)+11/192*A*b^3/a^5*c/x*(c*x^2+b*x+a)^(5/2)+1/16*B
*b/a^3*c/x^2*(c*x^2+b*x+a)^(5/2)-7/96*A*b^2/a^3/x^4*(c*x^2+b*x+a)^(5/2)-3/128*B*
b^4/a^4*c*(c*x^2+b*x+a)^(1/2)*x-3/32*B*b^2/a^4*c/x*(c*x^2+b*x+a)^(5/2)+3/32*B*b^
2/a^4*c^2*(c*x^2+b*x+a)^(3/2)*x+3/32*B*b^2/a^3*c^2*(c*x^2+b*x+a)^(1/2)*x-1/128*B
*b^4/a^5*c*(c*x^2+b*x+a)^(3/2)*x-1/32*A*b^2/a^4*c/x^2*(c*x^2+b*x+a)^(5/2)-1/5*B/
a/x^5*(c*x^2+b*x+a)^(5/2)-1/128*B*b^5/a^5*(c*x^2+b*x+a)^(3/2)-3/128*B*b^5/a^4*(c
*x^2+b*x+a)^(1/2)+3/256*B*b^5/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))
/x)-1/48*A/a^3*c^3*(c*x^2+b*x+a)^(3/2)-1/16*A/a^2*c^3*(c*x^2+b*x+a)^(1/2)-7/1024
*A*b^6/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/16*A/a^(3/2)*c^3*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+7/1536*A*b^6/a^6*(c*x^2+b*x+a)^(3/
2)+7/512*A*b^6/a^5*(c*x^2+b*x+a)^(1/2)
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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)^(3/2)*(B*x + A)/x^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.435407, 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)*(B*x + A)/x^7,x, algorithm="fricas")
[Out]
[1/30720*(15*(12*B*a*b^5 - 7*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 3*A*a^2*b^2)
*c^2 - 12*(8*B*a^2*b^3 - 5*A*a*b^4)*c)*x^6*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*(1280*A*a^5 +
(180*B*a*b^4 - 105*A*b^5 + 48*(32*B*a^3 - 27*A*a^2*b)*c^2 - 40*(30*B*a^2*b^2 -
19*A*a*b^3)*c)*x^5 - 2*(60*B*a^2*b^3 - 35*A*a*b^4 - 240*A*a^3*c^2 - 24*(14*B*a^3
*b - 9*A*a^2*b^2)*c)*x^4 + 8*(12*B*a^3*b^2 - 7*A*a^2*b^3 + 12*(32*B*a^4 + 3*A*a^
3*b)*c)*x^3 + 16*(132*B*a^4*b + 3*A*a^3*b^2 + 140*A*a^4*c)*x^2 + 128*(12*B*a^5 +
13*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(9/2)*x^6), 1/15360*(15*(12*B*
a*b^5 - 7*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 3*A*a^2*b^2)*c^2 - 12*(8*B*a^2*
b^3 - 5*A*a*b^4)*c)*x^6*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a
)) - 2*(1280*A*a^5 + (180*B*a*b^4 - 105*A*b^5 + 48*(32*B*a^3 - 27*A*a^2*b)*c^2 -
40*(30*B*a^2*b^2 - 19*A*a*b^3)*c)*x^5 - 2*(60*B*a^2*b^3 - 35*A*a*b^4 - 240*A*a^
3*c^2 - 24*(14*B*a^3*b - 9*A*a^2*b^2)*c)*x^4 + 8*(12*B*a^3*b^2 - 7*A*a^2*b^3 + 1
2*(32*B*a^4 + 3*A*a^3*b)*c)*x^3 + 16*(132*B*a^4*b + 3*A*a^3*b^2 + 140*A*a^4*c)*x
^2 + 128*(12*B*a^5 + 13*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^
4*x^6)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**7,x)
[Out]
Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**7, x)
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GIAC/XCAS [A] time = 0.298371, 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)^(3/2)*(B*x + A)/x^7,x, algorithm="giac")
[Out]
Done