3.930 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=206 \[ -\frac{\left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac{\sqrt{a+b x+c x^2} \left (2 c x (6 a B+A b)-8 a A c-6 a b B+A b^2\right )}{8 a x}+\frac{1}{2} \sqrt{c} (2 A c+3 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{3/2} (3 x (2 a B+A b)+4 a A)}{12 a x^3} \]

[Out]

((A*b^2 - 6*a*b*B - 8*a*A*c + 2*(A*b + 6*a*B)*c*x)*Sqrt[a + b*x + c*x^2])/(8*a*x
) - ((4*a*A + 3*(A*b + 2*a*B)*x)*(a + b*x + c*x^2)^(3/2))/(12*a*x^3) - ((6*a*B*(
b^2 + 4*a*c) - A*(b^3 - 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x +
c*x^2])])/(16*a^(3/2)) + (Sqrt[c]*(3*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 = 0.562497, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{\left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (-A \left (b^2-8 a c\right )-2 c x (6 a B+A b)+6 a b B\right )}{8 a x}+\frac{1}{2} \sqrt{c} (2 A c+3 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{3/2} (3 x (2 a B+A b)+4 a A)}{12 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4,x]

[Out]

-((6*a*b*B - A*(b^2 - 8*a*c) - 2*(A*b + 6*a*B)*c*x)*Sqrt[a + b*x + c*x^2])/(8*a*
x) - ((4*a*A + 3*(A*b + 2*a*B)*x)*(a + b*x + c*x^2)^(3/2))/(12*a*x^3) - ((6*a*B*
(b^2 + 4*a*c) - A*(b^3 - 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x
+ c*x^2])])/(16*a^(3/2)) + (Sqrt[c]*(3*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 [A]  time = 90.3279, size = 201, normalized size = 0.98 \[ \frac{\sqrt{c} \left (2 A c + 3 B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2} + \frac{\sqrt{a + b x + c x^{2}} \left (- 4 A a c + \frac{A b^{2}}{2} - 3 B a b + c x \left (A b + 6 B a\right )\right )}{4 a x} - \frac{\left (2 A a + x \left (\frac{3 A b}{2} + 3 B a\right )\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{6 a x^{3}} + \frac{\left (- 12 A a b c + A b^{3} - 24 B a^{2} c - 6 B a b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 a^{\frac{3}{2}}} \]

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**4,x)

[Out]

sqrt(c)*(2*A*c + 3*B*b)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/2
+ sqrt(a + b*x + c*x**2)*(-4*A*a*c + A*b**2/2 - 3*B*a*b + c*x*(A*b + 6*B*a))/(4*
a*x) - (2*A*a + x*(3*A*b/2 + 3*B*a))*(a + b*x + c*x**2)**(3/2)/(6*a*x**3) + (-12
*A*a*b*c + A*b**3 - 24*B*a**2*c - 6*B*a*b**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(
a + b*x + c*x**2)))/(16*a**(3/2))

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Mathematica [A]  time = 0.7062, size = 221, normalized size = 1.07 \[ \frac{-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} \left (4 a^2 (2 A+3 B x)+2 a x (A (7 b+16 c x)+3 B x (5 b-4 c x))+3 A b^2 x^2\right )-12 a \sqrt{c} x^3 (2 A c+3 b B) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )+3 x^3 \log (x) \left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right )+3 x^3 \left (A \left (b^3-12 a b c\right )-6 a B \left (4 a c+b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{48 a^{3/2} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^4,x]

[Out]

(3*(6*a*B*(b^2 + 4*a*c) - A*(b^3 - 12*a*b*c))*x^3*Log[x] + 3*(-6*a*B*(b^2 + 4*a*
c) + A*(b^3 - 12*a*b*c))*x^3*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]] -
2*Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(3*A*b^2*x^2 + 4*a^2*(2*A + 3*B*x) + 2*a*x*(3*B
*x*(5*b - 4*c*x) + A*(7*b + 16*c*x))) - 12*a*Sqrt[c]*(3*b*B + 2*A*c)*x^3*Log[b +
2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(48*a^(3/2)*x^3)

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Maple [B]  time = 0.017, size = 635, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^4,x)

[Out]

-1/4*B*b/a^2/x*(c*x^2+b*x+a)^(5/2)+A*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))+3/2*B*c*(c*x^2+b*x+a)^(1/2)+A/a*c^2*(c*x^2+b*x+a)^(1/2)*x+1/24*A*b^2/a^
3/x*(c*x^2+b*x+a)^(5/2)+7/12*A*b/a^2*c*(c*x^2+b*x+a)^(3/2)+5/4*A*b/a*c*(c*x^2+b*
x+a)^(1/2)-3/4*A*b/a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-2/3*A
/a^2*c/x*(c*x^2+b*x+a)^(5/2)+2/3*A/a^2*c^2*(c*x^2+b*x+a)^(3/2)*x-1/24*A*b^2/a^3*
c*(c*x^2+b*x+a)^(3/2)*x-1/8*A*b^2/a^2*c*(c*x^2+b*x+a)^(1/2)*x+1/4*B*b/a^2*c*(c*x
^2+b*x+a)^(3/2)*x+3/4*B*b/a*c*(c*x^2+b*x+a)^(1/2)*x+1/12*A*b/a^2/x^2*(c*x^2+b*x+
a)^(5/2)-1/2*B/a/x^2*(c*x^2+b*x+a)^(5/2)+1/4*B*b^2/a^2*(c*x^2+b*x+a)^(3/2)+3/4*B
*b^2/a*(c*x^2+b*x+a)^(1/2)-3/8*B*b^2/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)
^(1/2))/x)+3/2*B*b*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*B/a*c
*(c*x^2+b*x+a)^(3/2)-3/2*B*a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/
x)-1/3*A/a/x^3*(c*x^2+b*x+a)^(5/2)-1/24*A*b^3/a^3*(c*x^2+b*x+a)^(3/2)-1/8*A*b^3/
a^2*(c*x^2+b*x+a)^(1/2)+1/16*A*b^3/a^(3/2)*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((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43534, 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^4,x, algorithm="fricas")

[Out]

[1/96*(24*(3*B*a*b + 2*A*a*c)*sqrt(a)*sqrt(c)*x^3*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) + 3*(6*B*a*b^2 - A*b^3 +
12*(2*B*a^2 + A*a*b)*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) + 4*(24*B*a*c*x^3 - 8*A*a^2 - (3
0*B*a*b + 3*A*b^2 + 32*A*a*c)*x^2 - 2*(6*B*a^2 + 7*A*a*b)*x)*sqrt(c*x^2 + b*x +
a)*sqrt(a))/(a^(3/2)*x^3), 1/96*(48*(3*B*a*b + 2*A*a*c)*sqrt(a)*sqrt(-c)*x^3*arc
tan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) + 3*(6*B*a*b^2 - A*b^3 + 1
2*(2*B*a^2 + A*a*b)*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) + 4*(24*B*a*c*x^3 - 8*A*a^2 - (30*
B*a*b + 3*A*b^2 + 32*A*a*c)*x^2 - 2*(6*B*a^2 + 7*A*a*b)*x)*sqrt(c*x^2 + b*x + a)
*sqrt(a))/(a^(3/2)*x^3), 1/48*(12*(3*B*a*b + 2*A*a*c)*sqrt(-a)*sqrt(c)*x^3*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)
- 3*(6*B*a*b^2 - A*b^3 + 12*(2*B*a^2 + A*a*b)*c)*x^3*arctan(1/2*(b*x + 2*a)*sqr
t(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*(24*B*a*c*x^3 - 8*A*a^2 - (30*B*a*b + 3*A*b
^2 + 32*A*a*c)*x^2 - 2*(6*B*a^2 + 7*A*a*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(s
qrt(-a)*a*x^3), 1/48*(24*(3*B*a*b + 2*A*a*c)*sqrt(-a)*sqrt(-c)*x^3*arctan(1/2*(2
*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) - 3*(6*B*a*b^2 - A*b^3 + 12*(2*B*a^2
+ A*a*b)*c)*x^3*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*
(24*B*a*c*x^3 - 8*A*a^2 - (30*B*a*b + 3*A*b^2 + 32*A*a*c)*x^2 - 2*(6*B*a^2 + 7*A
*a*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**4,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.606365, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

sage0*x