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

Optimal. Leaf size=273 \[ -\frac{5 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2}}-\frac{5}{8} \sqrt{a} \left (4 a A c+4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{5 \sqrt{a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{64 c}-\frac{(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{24 x} \]

[Out]

(5*(b^3*B + 40*A*b^2*c + 44*a*b*B*c + 32*a*A*c^2 + 2*c*(b^2*B + 16*A*b*c + 12*a*
B*c)*x)*Sqrt[a + b*x + c*x^2])/(64*c) - (5*(6*(A*b + a*B) - (b*B + 4*A*c)*x)*(a
+ b*x + c*x^2)^(3/2))/(24*x) - ((2*A - B*x)*(a + b*x + c*x^2)^(5/2))/(4*x^2) - (
5*Sqrt[a]*(3*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 - (5*(b^4*B - 8*A*b^3*c - 24*a*b^2*B*c - 96*a*A*b*c^2 - 48*a^2
*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3/2))

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Rubi [A]  time = 0.666977, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{5 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2}}-\frac{5}{8} \sqrt{a} \left (4 a A c+4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{5 \sqrt{a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{64 c}-\frac{(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{24 x} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b^3*B + 40*A*b^2*c + 44*a*b*B*c + 32*a*A*c^2 + 2*c*(b^2*B + 16*A*b*c + 12*a*
B*c)*x)*Sqrt[a + b*x + c*x^2])/(64*c) - (5*(6*(A*b + a*B) - (b*B + 4*A*c)*x)*(a
+ b*x + c*x^2)^(3/2))/(24*x) - ((2*A - B*x)*(a + b*x + c*x^2)^(5/2))/(4*x^2) - (
5*Sqrt[a]*(3*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 - (5*(b^4*B - 8*A*b^3*c - 24*a*b^2*B*c - 96*a*A*b*c^2 - 48*a^2
*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3/2))

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Rubi in Sympy [A]  time = 85.5504, size = 279, normalized size = 1.02 \[ - \frac{5 \sqrt{a} \left (4 A a c + 3 A b^{2} + 4 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (12 A b + 12 B a - x \left (8 A c + 2 B b\right )\right )}{48 x} - \frac{\left (4 A - 2 B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{8 x^{2}} + \frac{5 \sqrt{a + b x + c x^{2}} \left (32 A a c^{2} + 40 A b^{2} c + 44 B a b c + B b^{3} + 2 c x \left (16 A b c + 12 B a c + B b^{2}\right )\right )}{64 c} - \frac{5 \left (- 96 A a b c^{2} - 8 A b^{3} c - 48 B a^{2} c^{2} - 24 B a b^{2} c + B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(a)*(4*A*a*c + 3*A*b**2 + 4*B*a*b)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a +
b*x + c*x**2)))/8 - 5*(a + b*x + c*x**2)**(3/2)*(12*A*b + 12*B*a - x*(8*A*c + 2*
B*b))/(48*x) - (4*A - 2*B*x)*(a + b*x + c*x**2)**(5/2)/(8*x**2) + 5*sqrt(a + b*x
+ c*x**2)*(32*A*a*c**2 + 40*A*b**2*c + 44*B*a*b*c + B*b**3 + 2*c*x*(16*A*b*c +
12*B*a*c + B*b**2))/(64*c) - 5*(-96*A*a*b*c**2 - 8*A*b**3*c - 48*B*a**2*c**2 - 2
4*B*a*b**2*c + B*b**4)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(12
8*c**(3/2))

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Mathematica [A]  time = 0.890323, size = 278, normalized size = 1.02 \[ \frac{\sqrt{a+x (b+c x)} \left (-96 a^2 c (A+2 B x)+4 a c x (B x (139 b+54 c x)-4 A (27 b-28 c x))+x^2 \left (2 b^2 c (132 A+59 B x)+8 b c^2 x (26 A+17 B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )\right )}{192 c x^2}+\frac{5 \left (48 a^2 B c^2+96 a A b c^2+24 a b^2 B c+8 A b^3 c+b^4 (-B)\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{128 c^{3/2}}+\frac{5}{8} \sqrt{a} \log (x) \left (4 a A c+4 a b B+3 A b^2\right )-\frac{5}{8} \sqrt{a} \left (4 a A c+4 a b B+3 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-96*a^2*c*(A + 2*B*x) + x^2*(15*b^3*B + 16*c^3*x^2*(4*A
+ 3*B*x) + 8*b*c^2*x*(26*A + 17*B*x) + 2*b^2*c*(132*A + 59*B*x)) + 4*a*c*x*(-4*A
*(27*b - 28*c*x) + B*x*(139*b + 54*c*x))))/(192*c*x^2) + (5*Sqrt[a]*(3*A*b^2 + 4
*a*b*B + 4*a*A*c)*Log[x])/8 - (5*Sqrt[a]*(3*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 + (5*(-(b^4*B) + 8*A*b^3*c + 24*a*b^2
*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*
x)]])/(128*c^(3/2))

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Maple [B]  time = 0.019, size = 663, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*A*b/a^2/x*(c*x^2+b*x+a)^(7/2)+5/2*A*b*(c*x^2+b*x+a)^(1/2)*x*c+15/8*B*(c*x^2
+b*x+a)^(1/2)*x*a*c+5/4*A*b/a*c*(c*x^2+b*x+a)^(3/2)*x+3/4*A*b/a^2*c*(c*x^2+b*x+a
)^(5/2)*x+5/4*B*c*(c*x^2+b*x+a)^(3/2)*x+55/16*B*(c*x^2+b*x+a)^(1/2)*b*a+B*b/a*(c
*x^2+b*x+a)^(5/2)-5/2*B*b*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+
15/8*B*c^(1/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-B/a/x*(c*x^2+b*x+
a)^(7/2)-5/2*A*a^(3/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/8*A*b^
2*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+3/4*A*b^2/a^2*(c*x^2+b*x
+a)^(5/2)+5/4*A*b^2/a*(c*x^2+b*x+a)^(3/2)+5/16*A*b^3/c^(1/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))-1/2*A/a/x^2*(c*x^2+b*x+a)^(7/2)+5/32*B*(c*x^2+b*x+a)^(1
/2)*x*b^2+5/64*B/c*(c*x^2+b*x+a)^(1/2)*b^3-5/128*B/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*b^4+1/2*A/a*c*(c*x^2+b*x+a)^(5/2)+5/2*A*a*c*(c*x^2+b*x+a)
^(1/2)+B/a*c*(c*x^2+b*x+a)^(5/2)*x+15/4*A*b*a*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))+15/16*B/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*
b^2+25/8*A*b^2*(c*x^2+b*x+a)^(1/2)+5/6*A*c*(c*x^2+b*x+a)^(3/2)+35/24*B*b*(c*x^2+
b*x+a)^(3/2)

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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)^(5/2)*(B*x + A)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.0741, 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)^(5/2)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

[1/768*(240*(4*A*a*c^2 + (4*B*a*b + 3*A*b^2)*c)*sqrt(a)*sqrt(c)*x^2*log(-(8*a*b*
x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2
) - 15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*x^2*log(-4*(
2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(
c)) + 4*(48*B*c^3*x^5 + 8*(17*B*b*c^2 + 8*A*c^3)*x^4 - 96*A*a^2*c + 2*(59*B*b^2*
c + 4*(27*B*a + 26*A*b)*c^2)*x^3 - 48*(4*B*a^2 + 9*A*a*b)*c*x + (15*B*b^3 + 448*
A*a*c^2 + 4*(139*B*a*b + 66*A*b^2)*c)*x^2)*sqrt(c*x^2 + b*x + a)*sqrt(c))/(c^(3/
2)*x^2), 1/384*(120*(4*A*a*c^2 + (4*B*a*b + 3*A*b^2)*c)*sqrt(a)*sqrt(-c)*x^2*log
(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8
*a^2)/x^2) - 15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*x^2
*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 2*(48*B*c^3*x^5 +
8*(17*B*b*c^2 + 8*A*c^3)*x^4 - 96*A*a^2*c + 2*(59*B*b^2*c + 4*(27*B*a + 26*A*b)*
c^2)*x^3 - 48*(4*B*a^2 + 9*A*a*b)*c*x + (15*B*b^3 + 448*A*a*c^2 + 4*(139*B*a*b +
66*A*b^2)*c)*x^2)*sqrt(c*x^2 + b*x + a)*sqrt(-c))/(sqrt(-c)*c*x^2), -1/768*(480
*(4*A*a*c^2 + (4*B*a*b + 3*A*b^2)*c)*sqrt(-a)*sqrt(c)*x^2*arctan(1/2*(b*x + 2*a)
/(sqrt(c*x^2 + b*x + a)*sqrt(-a))) + 15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3
*B*a*b^2 + A*b^3)*c)*x^2*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x
^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)) - 4*(48*B*c^3*x^5 + 8*(17*B*b*c^2 + 8*A*c^3
)*x^4 - 96*A*a^2*c + 2*(59*B*b^2*c + 4*(27*B*a + 26*A*b)*c^2)*x^3 - 48*(4*B*a^2
+ 9*A*a*b)*c*x + (15*B*b^3 + 448*A*a*c^2 + 4*(139*B*a*b + 66*A*b^2)*c)*x^2)*sqrt
(c*x^2 + b*x + a)*sqrt(c))/(c^(3/2)*x^2), -1/384*(240*(4*A*a*c^2 + (4*B*a*b + 3*
A*b^2)*c)*sqrt(-a)*sqrt(-c)*x^2*arctan(1/2*(b*x + 2*a)/(sqrt(c*x^2 + b*x + a)*sq
rt(-a))) + 15*(B*b^4 - 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*x^2*a
rctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) - 2*(48*B*c^3*x^5 + 8*
(17*B*b*c^2 + 8*A*c^3)*x^4 - 96*A*a^2*c + 2*(59*B*b^2*c + 4*(27*B*a + 26*A*b)*c^
2)*x^3 - 48*(4*B*a^2 + 9*A*a*b)*c*x + (15*B*b^3 + 448*A*a*c^2 + 4*(139*B*a*b + 6
6*A*b^2)*c)*x^2)*sqrt(c*x^2 + b*x + a)*sqrt(-c))/(sqrt(-c)*c*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.633004, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

sage0*x