3.931 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=217 \[ \frac{\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{64 a^2 x^2}-\frac{\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \]

[Out]

-((2*a*(8*a*b*B - 3*A*(b^2 - 4*a*c)) + (8*a*B*(b^2 + 8*a*c) - 3*A*(b^3 - 4*a*b*c
))*x)*Sqrt[a + b*x + c*x^2])/(64*a^2*x^2) - ((6*a*A + (3*A*b + 8*a*B)*x)*(a + b*
x + c*x^2)^(3/2))/(24*a*x^4) + ((8*a*b*B*(b^2 - 12*a*c) - 3*A*(b^2 - 4*a*c)^2)*A
rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(5/2)) + B*c^(3/2)
*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]

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Rubi [A]  time = 0.601697, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{64 a^2 x^2}-\frac{\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

-((2*a*(8*a*b*B - 3*A*(b^2 - 4*a*c)) + (8*a*B*(b^2 + 8*a*c) - 3*A*(b^3 - 4*a*b*c
))*x)*Sqrt[a + b*x + c*x^2])/(64*a^2*x^2) - ((6*a*A + (3*A*b + 8*a*B)*x)*(a + b*
x + c*x^2)^(3/2))/(24*a*x^4) + ((8*a*b*B*(b^2 - 12*a*c) - 3*A*(b^2 - 4*a*c)^2)*A
rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(5/2)) + B*c^(3/2)
*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]

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Rubi in Sympy [A]  time = 117.574, size = 235, normalized size = 1.08 \[ B c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} - \frac{\left (3 A a + x \left (\frac{3 A b}{2} + 4 B a\right )\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{12 a x^{4}} + \frac{\left (\frac{a \left (- 12 A a c + 3 A b^{2} - 8 B a b\right )}{2} + x \left (- 16 B a^{2} c + \frac{b \left (- 12 A a c + 3 A b^{2} - 8 B a b\right )}{4}\right )\right ) \sqrt{a + b x + c x^{2}}}{16 a^{2} x^{2}} - \frac{\left (48 A a^{2} c^{2} - 24 A a b^{2} c + 3 A b^{4} + 96 B a^{2} b c - 8 B a b^{3}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{5}{2}}} \]

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**5,x)

[Out]

B*c**(3/2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2))) - (3*A*a + x*(3
*A*b/2 + 4*B*a))*(a + b*x + c*x**2)**(3/2)/(12*a*x**4) + (a*(-12*A*a*c + 3*A*b**
2 - 8*B*a*b)/2 + x*(-16*B*a**2*c + b*(-12*A*a*c + 3*A*b**2 - 8*B*a*b)/4))*sqrt(a
+ b*x + c*x**2)/(16*a**2*x**2) - (48*A*a**2*c**2 - 24*A*a*b**2*c + 3*A*b**4 + 9
6*B*a**2*b*c - 8*B*a*b**3)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))
/(128*a**(5/2))

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Mathematica [A]  time = 0.85789, size = 248, normalized size = 1.14 \[ \frac{-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} \left (16 a^3 (3 A+4 B x)+8 a^2 x (3 A (3 b+5 c x)+2 B x (7 b+16 c x))+6 a b x^2 (A (b+10 c x)+4 b B x)-9 A b^3 x^3\right )-192 a^2 B c^{3/2} x^4 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )+3 x^4 \log (x) \left (3 A \left (b^2-4 a c\right )^2+8 a b B \left (12 a c-b^2\right )\right )-3 x^4 \left (3 A \left (b^2-4 a c\right )^2+8 a b B \left (12 a c-b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{384 a^{5/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

(3*(3*A*(b^2 - 4*a*c)^2 + 8*a*b*B*(-b^2 + 12*a*c))*x^4*Log[x] - 3*(3*A*(b^2 - 4*
a*c)^2 + 8*a*b*B*(-b^2 + 12*a*c))*x^4*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b +
c*x)]] - 2*Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(-9*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) +
6*a*b*x^2*(4*b*B*x + A*(b + 10*c*x)) + 8*a^2*x*(3*A*(3*b + 5*c*x) + 2*B*x*(7*b
+ 16*c*x))) - 192*a^2*B*c^(3/2)*x^4*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*
x)]]))/(384*a^(5/2)*x^4)

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Maple [B]  time = 0.022, size = 838, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x)

[Out]

1/8*A/a^2*c^2*(c*x^2+b*x+a)^(3/2)+3/8*A/a*c^2*(c*x^2+b*x+a)^(1/2)-3/128*A*b^4/a^
(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/4*A/a/x^4*(c*x^2+b*x+a)^(5
/2)-3/8*A/a^(1/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/3*B/a/x^3*
(c*x^2+b*x+a)^(5/2)-1/24*B*b^3/a^3*(c*x^2+b*x+a)^(3/2)-1/8*B*b^3/a^2*(c*x^2+b*x+
a)^(1/2)+1/16*B*b^3/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/64*A
*b^4/a^4*(c*x^2+b*x+a)^(3/2)+3/64*A*b^4/a^3*(c*x^2+b*x+a)^(1/2)-1/64*A*b^3/a^4/x
*(c*x^2+b*x+a)^(5/2)-5/32*A*b^2/a^3*c*(c*x^2+b*x+a)^(3/2)-9/32*A*b^2/a^2*c*(c*x^
2+b*x+a)^(1/2)+3/16*A*b^2/a^(3/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x
)-1/8*A/a^2*c/x^2*(c*x^2+b*x+a)^(5/2)+1/12*B*b/a^2/x^2*(c*x^2+b*x+a)^(5/2)+1/24*
B*b^2/a^3/x*(c*x^2+b*x+a)^(5/2)-1/24*B*b^2/a^3*c*(c*x^2+b*x+a)^(3/2)*x+1/64*A*b^
3/a^4*c*(c*x^2+b*x+a)^(3/2)*x+B*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))-3/16*A*b/a^3*c^2*(c*x^2+b*x+a)^(3/2)*x-3/16*A*b/a^2*c^2*(c*x^2+b*x+a)^(1/2)*
x-1/8*B*b^2/a^2*c*(c*x^2+b*x+a)^(1/2)*x+3/64*A*b^3/a^3*c*(c*x^2+b*x+a)^(1/2)*x+3
/16*A*b/a^3*c/x*(c*x^2+b*x+a)^(5/2)+7/12*B*b/a^2*c*(c*x^2+b*x+a)^(3/2)+5/4*B*b/a
*c*(c*x^2+b*x+a)^(1/2)-3/4*B*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*B/a^2*c/x*(c*x^2+b*x+a)^(5/2)+2/3*B/a^2*c^2*(c*x^2+b*x+a)^(3/2)*x+B/a
*c^2*(c*x^2+b*x+a)^(1/2)*x+1/8*A*b/a^2/x^3*(c*x^2+b*x+a)^(5/2)-1/32*A*b^2/a^3/x^
2*(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)^(3/2)*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.755, 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^5,x, algorithm="fricas")

[Out]

[1/768*(384*B*a^(5/2)*c^(3/2)*x^4*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*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2
- 24*(4*B*a^2*b - A*a*b^2)*c)*x^4*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*(48*A*a^3 + (24*B*a*b^2
- 9*A*b^3 + 4*(64*B*a^2 + 15*A*a*b)*c)*x^3 + 2*(56*B*a^2*b + 3*A*a*b^2 + 60*A*a
^2*c)*x^2 + 8*(8*B*a^3 + 9*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(5/2)*x
^4), 1/768*(768*B*a^(5/2)*sqrt(-c)*c*x^4*arctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*
x + a)*sqrt(-c))) - 3*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*
b^2)*c)*x^4*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*(48*A*a^3 + (24*B*a*b^2 - 9*A*b^3 + 4*(64*B*a
^2 + 15*A*a*b)*c)*x^3 + 2*(56*B*a^2*b + 3*A*a*b^2 + 60*A*a^2*c)*x^2 + 8*(8*B*a^3
+ 9*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(5/2)*x^4), 1/384*(192*B*sqrt
(-a)*a^2*c^(3/2)*x^4*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*(8*B*a*b^3 - 3*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2
*b - A*a*b^2)*c)*x^4*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a))
- 2*(48*A*a^3 + (24*B*a*b^2 - 9*A*b^3 + 4*(64*B*a^2 + 15*A*a*b)*c)*x^3 + 2*(56*B
*a^2*b + 3*A*a*b^2 + 60*A*a^2*c)*x^2 + 8*(8*B*a^3 + 9*A*a^2*b)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^4), 1/384*(384*B*sqrt(-a)*a^2*sqrt(-c)*c*x^4*a
rctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) + 3*(8*B*a*b^3 - 3*A*b^4
- 48*A*a^2*c^2 - 24*(4*B*a^2*b - A*a*b^2)*c)*x^4*arctan(1/2*(b*x + 2*a)*sqrt(-a
)/(sqrt(c*x^2 + b*x + a)*a)) - 2*(48*A*a^3 + (24*B*a*b^2 - 9*A*b^3 + 4*(64*B*a^2
+ 15*A*a*b)*c)*x^3 + 2*(56*B*a^2*b + 3*A*a*b^2 + 60*A*a^2*c)*x^2 + 8*(8*B*a^3 +
9*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**5,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**5, x)

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GIAC/XCAS [A]  time = 0.660396, 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^5,x, algorithm="giac")

[Out]

sage0*x