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

Optimal. Leaf size=158 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

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

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Rubi [A]  time = 0.332165, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}-\frac{2 \left (-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B\right )}{a x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^2*(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*Sqrt[a + b
*x + c*x^2]) - ((3*A*b^2 - 2*a*b*B - 8*a*A*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 -
4*a*c)*x) + ((3*A*b - 2*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^
2])])/(2*a^(5/2))

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Rubi in Sympy [A]  time = 39., size = 151, normalized size = 0.96 \[ \frac{2 \left (- 2 A a c + A b^{2} - B a b + c x \left (A b - 2 B a\right )\right )}{a x \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{a + b x + c x^{2}} \left (- 8 A a c + 3 A b^{2} - 2 B a b\right )}{a^{2} x \left (- 4 a c + b^{2}\right )} + \frac{\left (6 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**2/(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*(-4*a*c + b**2)*sqrt(a +
b*x + c*x**2)) - sqrt(a + b*x + c*x**2)*(-8*A*a*c + 3*A*b**2 - 2*B*a*b)/(a**2*x*
(-4*a*c + b**2)) + (6*A*b - 4*B*a)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c
*x**2)))/(4*a**(5/2))

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Mathematica [A]  time = 0.594968, size = 149, normalized size = 0.94 \[ \frac{-\frac{2 \sqrt{a} \left (-4 a^2 c (A-B x)+a A \left (b^2-10 b c x-8 c^2 x^2\right )-2 a b B x (b+c x)+3 A b^2 x (b+c x)\right )}{x \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}+(3 A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+\log (x) (2 a B-3 A b)}{2 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 330, normalized size = 2.1 \[ -{\frac{A}{ax}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{A{b}^{2}cx}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,A{b}^{3}}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-8\,{\frac{Ax{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) a\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{Abc}{ \left ( 4\,ac-{b}^{2} \right ) a\sqrt{c{x}^{2}+bx+a}}}+{\frac{B}{a}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{Bxbc}{ \left ( 4\,ac-{b}^{2} \right ) a\sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}B}{ \left ( 4\,ac-{b}^{2} \right ) a}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.436932, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (A a b^{2} - 4 \, A a^{2} c -{\left (8 \, A a c^{2} +{\left (2 \, B a b - 3 \, A b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3} - 2 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a} -{\left ({\left (4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} -{\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \,{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \log \left (-\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} +{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right )}{4 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{3} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )} \sqrt{a}}, -\frac{2 \,{\left (A a b^{2} - 4 \, A a^{2} c -{\left (8 \, A a c^{2} +{\left (2 \, B a b - 3 \, A b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3} - 2 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a} -{\left ({\left (4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} -{\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \,{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right )}{2 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{3} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(4*(A*a*b^2 - 4*A*a^2*c - (8*A*a*c^2 + (2*B*a*b - 3*A*b^2)*c)*x^2 - (2*B*a
*b^2 - 3*A*b^3 - 2*(2*B*a^2 - 5*A*a*b)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a) - ((4
*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*x^3 - (2*B*a*b^3 - 3*A*b^4 -
4*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*
a^2*b)*c)*x)*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^2*b^2*c - 4*a^3*c^2)*x^3 + (a^2*b^3 - 4*a
^3*b*c)*x^2 + (a^3*b^2 - 4*a^4*c)*x)*sqrt(a)), -1/2*(2*(A*a*b^2 - 4*A*a^2*c - (8
*A*a*c^2 + (2*B*a*b - 3*A*b^2)*c)*x^2 - (2*B*a*b^2 - 3*A*b^3 - 2*(2*B*a^2 - 5*A*
a*b)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a) - ((4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a
*b^2 - 3*A*b^3)*c)*x^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2
- (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*x)*arctan(1/2*(b*x + 2*
a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)))/(((a^2*b^2*c - 4*a^3*c^2)*x^3 + (a^2*b^3
- 4*a^3*b*c)*x^2 + (a^3*b^2 - 4*a^4*c)*x)*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**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293636, size = 297, normalized size = 1.88 \[ \frac{2 \,{\left (\frac{{\left (B a^{3} b c - A a^{2} b^{2} c + 2 \, A a^{3} c^{2}\right )} x}{a^{4} b^{2} - 4 \, a^{5} c} + \frac{B a^{3} b^{2} - A a^{2} b^{3} - 2 \, B a^{4} c + 3 \, A a^{3} b c}{a^{4} b^{2} - 4 \, a^{5} c}\right )}}{\sqrt{c x^{2} + b x + a}} + \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((B*a^3*b*c - A*a^2*b^2*c + 2*A*a^3*c^2)*x/(a^4*b^2 - 4*a^5*c) + (B*a^3*b^2 -
A*a^2*b^3 - 2*B*a^4*c + 3*A*a^3*b*c)/(a^4*b^2 - 4*a^5*c))/sqrt(c*x^2 + b*x + a)
+ (2*B*a - 3*A*b)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a
)*a^2) + ((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b + 2*A*a*sqrt(c))/(((sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2 - a)*a^2)