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

Optimal. Leaf size=231 \[ -\frac{3 \left (-4 a A c-4 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{7/2}}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right )}{4 a^3 x \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^2*Sqrt[a +
b*x + c*x^2]) - ((5*A*b^2 - 4*a*b*B - 12*a*A*c)*Sqrt[a + b*x + c*x^2])/(2*a^2*(b
^2 - 4*a*c)*x^2) - ((4*a*B*(3*b^2 - 8*a*c) - A*(15*b^3 - 52*a*b*c))*Sqrt[a + b*x
+ c*x^2])/(4*a^3*(b^2 - 4*a*c)*x) - (3*(5*A*b^2 - 4*a*b*B - 4*a*A*c)*ArcTanh[(2
*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(7/2))

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Rubi [A]  time = 0.553011, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{3 \left (-4 a A c-4 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{7/2}}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (3 b^2-8 a c\right )-A \left (15 b^3-52 a b c\right )\right )}{4 a^3 x \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (-12 a A c-4 a b B+5 A b^2\right )}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac{2 \left (-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x^2*Sqrt[a +
b*x + c*x^2]) - ((5*A*b^2 - 4*a*b*B - 12*a*A*c)*Sqrt[a + b*x + c*x^2])/(2*a^2*(
b^2 - 4*a*c)*x^2) - ((4*a*B*(3*b^2 - 8*a*c) - A*(15*b^3 - 52*a*b*c))*Sqrt[a + b*
x + c*x^2])/(4*a^3*(b^2 - 4*a*c)*x) - (3*(5*A*b^2 - 4*a*b*B - 4*a*A*c)*ArcTanh[(
2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 94.4351, size = 230, normalized size = 1. \[ \frac{2 \left (- 2 A a c + A b^{2} - B a b + c x \left (A b - 2 B a\right )\right )}{a x^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{a + b x + c x^{2}} \left (- 24 A a c + 10 A b^{2} - 8 B a b\right )}{4 a^{2} x^{2} \left (- 4 a c + b^{2}\right )} + \frac{\sqrt{a + b x + c x^{2}} \left (- 104 A a b c + 30 A b^{3} + 64 B a^{2} c - 24 B a b^{2}\right )}{8 a^{3} x \left (- 4 a c + b^{2}\right )} - \frac{3 \left (- 8 A a c + 10 A b^{2} - 8 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*A*a*c + A*b**2 - B*a*b + c*x*(A*b - 2*B*a))/(a*x**2*(-4*a*c + b**2)*sqrt(a
+ b*x + c*x**2)) - sqrt(a + b*x + c*x**2)*(-24*A*a*c + 10*A*b**2 - 8*B*a*b)/(4*
a**2*x**2*(-4*a*c + b**2)) + sqrt(a + b*x + c*x**2)*(-104*A*a*b*c + 30*A*b**3 +
64*B*a**2*c - 24*B*a*b**2)/(8*a**3*x*(-4*a*c + b**2)) - 3*(-8*A*a*c + 10*A*b**2
- 8*B*a*b)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(16*a**(7/2))

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Mathematica [A]  time = 0.642383, size = 223, normalized size = 0.97 \[ \frac{\frac{2 \sqrt{a} \left (8 a^3 c (A+2 B x)+a^2 \left (4 B x \left (-b^2+10 b c x+8 c^2 x^2\right )-2 A \left (b^2+10 b c x-12 c^2 x^2\right )\right )-a b x \left (A \left (-5 b^2+62 b c x+52 c^2 x^2\right )+12 b B x (b+c x)\right )+15 A b^3 x^2 (b+c x)\right )}{x^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}+3 \log (x) \left (-4 a A c-4 a b B+5 A b^2\right )+3 \left (4 a A c+4 a b B-5 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{8 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*(8*a^3*c*(A + 2*B*x) + 15*A*b^3*x^2*(b + c*x) + a^2*(-2*A*(b^2 + 10*
b*c*x - 12*c^2*x^2) + 4*B*x*(-b^2 + 10*b*c*x + 8*c^2*x^2)) - a*b*x*(12*b*B*x*(b
+ c*x) + A*(-5*b^2 + 62*b*c*x + 52*c^2*x^2))))/((b^2 - 4*a*c)*x^2*Sqrt[a + x*(b
+ c*x)]) + 3*(5*A*b^2 - 4*a*b*B - 4*a*A*c)*Log[x] + 3*(-5*A*b^2 + 4*a*b*B + 4*a*
A*c)*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(8*a^(7/2))

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Maple [B]  time = 0.019, size = 506, normalized size = 2.2 \[ -{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{5\,Ab}{4\,{a}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{15\,{b}^{2}A}{8\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{15\,A{b}^{3}cx}{4\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{15\,A{b}^{4}}{8\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{15\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}+13\,{\frac{Axb{c}^{2}}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{13\,A{b}^{2}c}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Ac}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{ax}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Bb}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{Bx{b}^{2}c}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,B{b}^{3}}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-8\,{\frac{B{c}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) a\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{Bbc}{ \left ( 4\,ac-{b}^{2} \right ) a\sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a/x^2/(c*x^2+b*x+a)^(1/2)+5/4*A*b/a^2/x/(c*x^2+b*x+a)^(1/2)+15/8*A*b^2/a^
3/(c*x^2+b*x+a)^(1/2)-15/4*A*b^3/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-15/8*A*
b^4/a^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-15/8*A*b^2/a^(7/2)*ln((2*a+b*x+2*a^(1/2)
*(c*x^2+b*x+a)^(1/2))/x)+13*A*b/a^2*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+13/2*A
*b^2/a^2*c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3/2*A/a^2*c/(c*x^2+b*x+a)^(1/2)+3/2*A
/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-B/a/x/(c*x^2+b*x+a)^(1/
2)-3/2*B*b/a^2/(c*x^2+b*x+a)^(1/2)+3*B*b^2/a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c
*x+3/2*B*b^3/a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3/2*B*b/a^(5/2)*ln((2*a+b*x+2*a
^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-8*B*c^2/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-4*B*c
/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b

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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((B*x + A)/((c*x^2 + b*x + a)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.490889, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(2*A*a^2*b^2 - 8*A*a^3*c - (4*(8*B*a^2 - 13*A*a*b)*c^2 - 3*(4*B*a*b^2
- 5*A*b^3)*c)*x^3 + (12*B*a*b^3 - 15*A*b^4 - 24*A*a^2*c^2 - 2*(20*B*a^2*b - 31*A
*a*b^2)*c)*x^2 + (4*B*a^2*b^2 - 5*A*a*b^3 - 4*(4*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c
*x^2 + b*x + a)*sqrt(a) + 3*((16*A*a^2*c^3 + 8*(2*B*a^2*b - 3*A*a*b^2)*c^2 - (4*
B*a*b^3 - 5*A*b^4)*c)*x^4 - (4*B*a*b^4 - 5*A*b^5 - 16*A*a^2*b*c^2 - 8*(2*B*a^2*b
^2 - 3*A*a*b^3)*c)*x^3 - (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b
- 3*A*a^2*b^2)*c)*x^2)*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))/(((a^3*b^2*c - 4*a^4*c^2)*x^4 + (a^3
*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(a)), -1/8*(2*(2*A*a^2*b^2
- 8*A*a^3*c - (4*(8*B*a^2 - 13*A*a*b)*c^2 - 3*(4*B*a*b^2 - 5*A*b^3)*c)*x^3 + (12
*B*a*b^3 - 15*A*b^4 - 24*A*a^2*c^2 - 2*(20*B*a^2*b - 31*A*a*b^2)*c)*x^2 + (4*B*a
^2*b^2 - 5*A*a*b^3 - 4*(4*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a
) + 3*((16*A*a^2*c^3 + 8*(2*B*a^2*b - 3*A*a*b^2)*c^2 - (4*B*a*b^3 - 5*A*b^4)*c)*
x^4 - (4*B*a*b^4 - 5*A*b^5 - 16*A*a^2*b*c^2 - 8*(2*B*a^2*b^2 - 3*A*a*b^3)*c)*x^3
- (4*B*a^2*b^3 - 5*A*a*b^4 - 16*A*a^3*c^2 - 8*(2*B*a^3*b - 3*A*a^2*b^2)*c)*x^2)
*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)))/(((a^3*b^2*c - 4*a^
4*c^2)*x^4 + (a^3*b^3 - 4*a^4*b*c)*x^3 + (a^4*b^2 - 4*a^5*c)*x^2)*sqrt(-a))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285296, size = 630, normalized size = 2.73 \[ -\frac{2 \,{\left (\frac{{\left (B a^{4} b^{2} c - A a^{3} b^{3} c - 2 \, B a^{5} c^{2} + 3 \, A a^{4} b c^{2}\right )} x}{a^{6} b^{2} - 4 \, a^{7} c} + \frac{B a^{4} b^{3} - A a^{3} b^{4} - 3 \, B a^{5} b c + 4 \, A a^{4} b^{2} c - 2 \, A a^{5} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (4 \, B a b - 5 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} + \frac{4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} B a b - 7 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a c + 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt{c} - 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} A a b \sqrt{c} - 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{2} b + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a b^{2} + 4 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt{c} + 16 \, A a^{2} b \sqrt{c}}{4 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*((B*a^4*b^2*c - A*a^3*b^3*c - 2*B*a^5*c^2 + 3*A*a^4*b*c^2)*x/(a^6*b^2 - 4*a^7
*c) + (B*a^4*b^3 - A*a^3*b^4 - 3*B*a^5*b*c + 4*A*a^4*b^2*c - 2*A*a^5*c^2)/(a^6*b
^2 - 4*a^7*c))/sqrt(c*x^2 + b*x + a) - 3/4*(4*B*a*b - 5*A*b^2 + 4*A*a*c)*arctan(
-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) + 1/4*(4*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*B*a*b - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b
^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*c + 8*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*B*a^2*sqrt(c) - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b*sqrt(
c) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b + 9*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*A*a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c - 8*B*a^3*sqr
t(c) + 16*A*a^2*b*sqrt(c))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2*a^3)