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

Optimal. Leaf size=184 \[ -\frac{A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{a^{5/2}}+\frac{2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^(3/2)) + (2*(8*a^2*b*B*c + A*(3*b^4 - 22*a*b^2*c + 24*a^2*c^2) + c*(3*A*b
^3 - 20*a*A*b*c + 16*a^2*B*c)*x))/(3*a^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
- (A*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/a^(5/2)

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Rubi [A]  time = 0.357202, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{a^{5/2}}+\frac{2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \left (-A \left (b^2-2 a c\right )-c x (A b-2 a B)+a b B\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^(3/2)) + (2*(8*a^2*b*B*c + A*(3*b^4 - 22*a*b^2*c + 24*a^2*c^2) + c*(3*A*
b^3 - 20*a*A*b*c + 16*a^2*B*c)*x))/(3*a^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
- (A*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/a^(5/2)

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Rubi in Sympy [A]  time = 42.834, size = 189, normalized size = 1.03 \[ - \frac{A \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{a^{\frac{5}{2}}} + \frac{2 \left (- 2 A a c + A b^{2} - B a b + c x \left (A b - 2 B a\right )\right )}{3 a \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (12 A a^{2} c^{2} - 11 A a b^{2} c + \frac{3 A b^{4}}{2} + 4 B a^{2} b c + \frac{c x \left (- 20 A a b c + 3 A b^{3} + 16 B a^{2} c\right )}{2}\right )}{3 a^{2} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/a**(5/2) + 2*(-2*A*a*c
+ A*b**2 - B*a*b + c*x*(A*b - 2*B*a))/(3*a*(-4*a*c + b**2)*(a + b*x + c*x**2)**(
3/2)) + 4*(12*A*a**2*c**2 - 11*A*a*b**2*c + 3*A*b**4/2 + 4*B*a**2*b*c + c*x*(-20
*A*a*b*c + 3*A*b**3 + 16*B*a**2*c)/2)/(3*a**2*(-4*a*c + b**2)**2*sqrt(a + b*x +
c*x**2))

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Mathematica [A]  time = 0.560408, size = 183, normalized size = 0.99 \[ \frac{-\frac{2 a^{3/2} \left (a B (b+2 c x)-A \left (-2 a c+b^2+b c x\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}}+\frac{2 \sqrt{a} \left (A \left (24 a^2 c^2-22 a b^2 c-20 a b c^2 x+3 b^4+3 b^3 c x\right )+8 a^2 B c (b+2 c x)\right )}{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}}-3 A \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+3 A \log (x)}{3 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a^(3/2)*(a*B*(b + 2*c*x) - A*(b^2 - 2*a*c + b*c*x)))/((b^2 - 4*a*c)*(a + x*
(b + c*x))^(3/2)) + (2*Sqrt[a]*(8*a^2*B*c*(b + 2*c*x) + A*(3*b^4 - 22*a*b^2*c +
24*a^2*c^2 + 3*b^3*c*x - 20*a*b*c^2*x)))/((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)])
+ 3*A*Log[x] - 3*A*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(3*a^(5/2)
)

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Maple [B]  time = 0.012, size = 390, normalized size = 2.1 \[{\frac{4\,Bcx}{12\,ac-3\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bb}{12\,ac-3\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{32\,B{c}^{2}x}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{16\,Bbc}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{A}{3\,a} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,Abcx}{3\, \left ( 4\,ac-{b}^{2} \right ) a} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-{\frac{{b}^{2}A}{3\, \left ( 4\,ac-{b}^{2} \right ) a} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,Axb{c}^{2}}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2}a}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{8\,A{b}^{2}c}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2}a}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{A}{{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{Abcx}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}A}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*B/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+2/3*B/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*
b+32/3*B*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*B*c/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^(1/2)*b+1/3*A/a/(c*x^2+b*x+a)^(3/2)-2/3*A*b/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3
/2)*c*x-1/3*A*b^2/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-16/3*A*b/a*c^2/(4*a*c-b^2)^2
/(c*x^2+b*x+a)^(1/2)*x-8/3*A*b^2/a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+A/a^2/(c*
x^2+b*x+a)^(1/2)-2*A/a^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-A/a^2*b^2/(4*a*c-
b^2)/(c*x^2+b*x+a)^(1/2)-A/a^(5/2)*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)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.505198, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x + a)^(5/2)*x),x, algorithm="fricas")

[Out]

[-1/6*(4*(B*a^2*b^3 - 4*A*a*b^4 - 32*A*a^3*c^2 - (3*A*b^3*c^2 + 4*(4*B*a^2 - 5*A
*a*b)*c^3)*x^3 - 6*(A*b^4*c + 4*A*a^2*c^3 + (4*B*a^2*b - 7*A*a*b^2)*c^2)*x^2 - 4
*(3*B*a^3*b - 7*A*a^2*b^2)*c - 3*(A*b^5 + 8*B*a^3*c^2 + 2*(B*a^2*b^2 - 3*A*a*b^3
)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a) - 3*(A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*
c^2 + (A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*x^4 + 2*(A*b^5*c - 8*A*a*b^3*c^
2 + 16*A*a^2*b*c^3)*x^3 + (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*x^2 + 2*(A*a*b^5
- 8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*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^4*b^4 - 8*a^5*b^2*c
+ 16*a^6*c^2 + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^4 + 2*(a^2*b^5*c -
8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^2 + 2
*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x)*sqrt(a)), -1/3*(2*(B*a^2*b^3 - 4*A*a*
b^4 - 32*A*a^3*c^2 - (3*A*b^3*c^2 + 4*(4*B*a^2 - 5*A*a*b)*c^3)*x^3 - 6*(A*b^4*c
+ 4*A*a^2*c^3 + (4*B*a^2*b - 7*A*a*b^2)*c^2)*x^2 - 4*(3*B*a^3*b - 7*A*a^2*b^2)*c
- 3*(A*b^5 + 8*B*a^3*c^2 + 2*(B*a^2*b^2 - 3*A*a*b^3)*c)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(-a) + 3*(A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2 + (A*b^4*c^2 - 8*A*a*b^
2*c^3 + 16*A*a^2*c^4)*x^4 + 2*(A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^3 + (
A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*x^2 + 2*(A*a*b^5 - 8*A*a^2*b^3*c + 16*A*a^3*
b*c^2)*x)*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)))/((a^4*b^4
- 8*a^5*b^2*c + 16*a^6*c^2 + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^4 + 2*
(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5
*c^3)*x^2 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*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/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285988, size = 450, normalized size = 2.45 \[ \frac{{\left ({\left (\frac{{\left (3 \, A a^{5} b^{3} c^{2} + 16 \, B a^{7} c^{3} - 20 \, A a^{6} b c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{6 \,{\left (A a^{5} b^{4} c + 4 \, B a^{7} b c^{2} - 7 \, A a^{6} b^{2} c^{2} + 4 \, A a^{7} c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (A a^{5} b^{5} + 2 \, B a^{7} b^{2} c - 6 \, A a^{6} b^{3} c + 8 \, B a^{8} c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{B a^{7} b^{3} - 4 \, A a^{6} b^{4} - 12 \, B a^{8} b c + 28 \, A a^{7} b^{2} c - 32 \, A a^{8} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} + \frac{2 \, A \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x + a)^(5/2)*x),x, algorithm="giac")

[Out]

1/3*((((3*A*a^5*b^3*c^2 + 16*B*a^7*c^3 - 20*A*a^6*b*c^3)*x/(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4) + 6*(A*a^5*b^4*c + 4*B*a^7*b*c^2 - 7*A*a^6*b^2*c^2 + 4*A*a^7*c^3
)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 3*(A*a^5*b^5 + 2*B*a^7*b^2*c - 6*A*a
^6*b^3*c + 8*B*a^8*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x - (B*a^7*b^3 - 4
*A*a^6*b^4 - 12*B*a^8*b*c + 28*A*a^7*b^2*c - 32*A*a^8*c^2)/(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) + 2*A*arctan(-(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))/sqrt(-a))/(sqrt(-a)*a^2)