3.941 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx\)
Optimal. Leaf size=310 \[ -\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )-\frac{\sqrt{a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{128 c^2}+\frac{\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{48 c}-\frac{(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x} \]
[Out]
-((3*b^4*B - 10*A*b^3*c - 28*a*b^2*B*c - 440*a*A*b*c^2 - 128*a^2*B*c^2 + 2*c*(3*
b^3*B - 10*A*b^2*c - 28*a*b*B*c - 120*a*A*c^2)*x)*Sqrt[a + b*x + c*x^2])/(128*c^
2) + ((3*b^2*B + 70*A*b*c + 16*a*B*c + 6*c*(b*B + 10*A*c)*x)*(a + b*x + c*x^2)^(
3/2))/(48*c) - ((5*A - B*x)*(a + b*x + c*x^2)^(5/2))/(5*x) - (a^(3/2)*(5*A*b + 2
*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/2 + ((3*b^5*B - 10
*A*b^4*c - 40*a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*c^2 + 480*a^2*A*c^3)*Arc
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(5/2))
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Rubi [A] time = 0.769897, antiderivative size = 310, 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
\[ -\frac{1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )-\frac{\sqrt{a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{128 c^2}+\frac{\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{48 c}-\frac{(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2,x]
[Out]
-((3*b^4*B - 10*A*b^3*c - 28*a*b^2*B*c - 440*a*A*b*c^2 - 128*a^2*B*c^2 + 2*c*(3*
b^3*B - 10*A*b^2*c - 28*a*b*B*c - 120*a*A*c^2)*x)*Sqrt[a + b*x + c*x^2])/(128*c^
2) + ((3*b^2*B + 70*A*b*c + 16*a*B*c + 6*c*(b*B + 10*A*c)*x)*(a + b*x + c*x^2)^(
3/2))/(48*c) - ((5*A - B*x)*(a + b*x + c*x^2)^(5/2))/(5*x) - (a^(3/2)*(5*A*b + 2
*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/2 + ((3*b^5*B - 10
*A*b^4*c - 40*a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*c^2 + 480*a^2*A*c^3)*Arc
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(5/2))
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Rubi in Sympy [A] time = 97.8172, size = 321, normalized size = 1.04 \[ - \frac{a^{\frac{3}{2}} \left (5 A b + 2 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2} - \frac{\left (5 A - B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{5 x} + \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (35 A b c + 8 B a c + \frac{3 B b^{2}}{2} + 3 c x \left (10 A c + B b\right )\right )}{24 c} - \frac{\sqrt{a + b x + c x^{2}} \left (- 110 A a b c^{2} - \frac{5 A b^{3} c}{2} - 32 B a^{2} c^{2} - 7 B a b^{2} c + \frac{3 B b^{4}}{4} + \frac{c x \left (- 120 A a c^{2} - 10 A b^{2} c - 28 B a b c + 3 B b^{3}\right )}{2}\right )}{32 c^{2}} + \frac{\left (480 A a^{2} c^{3} + 240 A a b^{2} c^{2} - 10 A b^{4} c + 240 B a^{2} b c^{2} - 40 B a b^{3} c + 3 B b^{5}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**2,x)
[Out]
-a**(3/2)*(5*A*b + 2*B*a)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/
2 - (5*A - B*x)*(a + b*x + c*x**2)**(5/2)/(5*x) + (a + b*x + c*x**2)**(3/2)*(35*
A*b*c + 8*B*a*c + 3*B*b**2/2 + 3*c*x*(10*A*c + B*b))/(24*c) - sqrt(a + b*x + c*x
**2)*(-110*A*a*b*c**2 - 5*A*b**3*c/2 - 32*B*a**2*c**2 - 7*B*a*b**2*c + 3*B*b**4/
4 + c*x*(-120*A*a*c**2 - 10*A*b**2*c - 28*B*a*b*c + 3*B*b**3)/2)/(32*c**2) + (48
0*A*a**2*c**3 + 240*A*a*b**2*c**2 - 10*A*b**4*c + 240*B*a**2*b*c**2 - 40*B*a*b**
3*c + 3*B*b**5)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(256*c**(5
/2))
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Mathematica [A] time = 1.21359, size = 304, normalized size = 0.98 \[ -\frac{1}{2} a^{3/2} (2 a B+5 A b) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+\frac{1}{2} a^{3/2} \log (x) (2 a B+5 A b)+\frac{\sqrt{a+x (b+c x)} \left (-128 a^2 c^2 (15 A-23 B x)+4 a c x \left (2 b c (695 A+311 B x)+4 c^2 x (135 A+88 B x)+135 b^2 B\right )+x \left (30 b^3 c (5 A+B x)+4 b^2 c^2 x (295 A+186 B x)+16 b c^3 x^2 (85 A+63 B x)+96 c^4 x^3 (5 A+4 B x)-45 b^4 B\right )\right )}{1920 c^2 x}+\frac{\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{256 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^2,x]
[Out]
(Sqrt[a + x*(b + c*x)]*(-128*a^2*c^2*(15*A - 23*B*x) + x*(-45*b^4*B + 30*b^3*c*(
5*A + B*x) + 96*c^4*x^3*(5*A + 4*B*x) + 16*b*c^3*x^2*(85*A + 63*B*x) + 4*b^2*c^2
*x*(295*A + 186*B*x)) + 4*a*c*x*(135*b^2*B + 4*c^2*x*(135*A + 88*B*x) + 2*b*c*(6
95*A + 311*B*x))))/(1920*c^2*x) + (a^(3/2)*(5*A*b + 2*a*B)*Log[x])/2 - (a^(3/2)*
(5*A*b + 2*a*B)*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/2 + ((3*b^5*B
- 10*A*b^4*c - 40*a*b^3*B*c + 240*a*A*b^2*c^2 + 240*a^2*b*B*c^2 + 480*a^2*A*c^3)
*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(256*c^(5/2))
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Maple [B] time = 0.016, size = 615, normalized size = 2. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^2,x)
[Out]
35/24*A*b*(c*x^2+b*x+a)^(3/2)+1/3*B*a*(c*x^2+b*x+a)^(3/2)+B*a^2*(c*x^2+b*x+a)^(1
/2)-B*a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+7/16*B*b*(c*x^2+b*x+
a)^(1/2)*x*a+A/a*c*(c*x^2+b*x+a)^(5/2)*x+15/8*A*(c*x^2+b*x+a)^(1/2)*x*a*c+15/16*
A/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*b^2+7/32*B/c*(c*x^2+b*x+
a)^(1/2)*b^2*a+15/16*B*b/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2
-5/32*B/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^3*a+1/5*B*(c*x^2+b
*x+a)^(5/2)+1/8*B*b*(c*x^2+b*x+a)^(3/2)*x+1/16*B/c*(c*x^2+b*x+a)^(3/2)*b^2-3/128
*B/c^2*(c*x^2+b*x+a)^(1/2)*b^4+3/256*B/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*b^5+5/4*A*c*(c*x^2+b*x+a)^(3/2)*x+55/16*A*(c*x^2+b*x+a)^(1/2)*b*a+15/
8*A*c^(1/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-A/a/x*(c*x^2+b*x+a)^
(7/2)+A*b/a*(c*x^2+b*x+a)^(5/2)+5/32*A*(c*x^2+b*x+a)^(1/2)*x*b^2+5/64*A/c*(c*x^2
+b*x+a)^(1/2)*b^3-5/128*A/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^
4-5/2*A*b*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-3/64*B/c*(c*x^2+
b*x+a)^(1/2)*x*b^3
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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^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 8.53776, 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^2,x, algorithm="fricas")
[Out]
[1/7680*(1920*(2*B*a^2 + 5*A*a*b)*sqrt(a)*c^(5/2)*x*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*(3*B*b^5
+ 480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*c)*x*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)*s
qrt(c)) + 4*(384*B*c^4*x^5 - 1920*A*a^2*c^2 + 48*(21*B*b*c^3 + 10*A*c^4)*x^4 + 8
*(93*B*b^2*c^2 + 2*(88*B*a + 85*A*b)*c^3)*x^3 + 2*(15*B*b^3*c + 1080*A*a*c^3 + 2
*(622*B*a*b + 295*A*b^2)*c^2)*x^2 - (45*B*b^4 - 8*(368*B*a^2 + 695*A*a*b)*c^2 -
30*(18*B*a*b^2 + 5*A*b^3)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c))/(c^(5/2)*x), 1/38
40*(960*(2*B*a^2 + 5*A*a*b)*sqrt(a)*sqrt(-c)*c^2*x*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*(3*B*b^5 +
480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*c)*x*arcta
n(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 2*(384*B*c^4*x^5 - 1920*
A*a^2*c^2 + 48*(21*B*b*c^3 + 10*A*c^4)*x^4 + 8*(93*B*b^2*c^2 + 2*(88*B*a + 85*A*
b)*c^3)*x^3 + 2*(15*B*b^3*c + 1080*A*a*c^3 + 2*(622*B*a*b + 295*A*b^2)*c^2)*x^2
- (45*B*b^4 - 8*(368*B*a^2 + 695*A*a*b)*c^2 - 30*(18*B*a*b^2 + 5*A*b^3)*c)*x)*sq
rt(c*x^2 + b*x + a)*sqrt(-c))/(sqrt(-c)*c^2*x), -1/7680*(3840*(2*B*a^2 + 5*A*a*b
)*sqrt(-a)*c^(5/2)*x*arctan(1/2*(b*x + 2*a)/(sqrt(c*x^2 + b*x + a)*sqrt(-a))) -
15*(3*B*b^5 + 480*A*a^2*c^3 + 240*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^
4)*c)*x*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*(384*B*c^4*x^5 - 1920*A*a^2*c^2 + 48*(21*B*b*c^3 + 10*A*
c^4)*x^4 + 8*(93*B*b^2*c^2 + 2*(88*B*a + 85*A*b)*c^3)*x^3 + 2*(15*B*b^3*c + 1080
*A*a*c^3 + 2*(622*B*a*b + 295*A*b^2)*c^2)*x^2 - (45*B*b^4 - 8*(368*B*a^2 + 695*A
*a*b)*c^2 - 30*(18*B*a*b^2 + 5*A*b^3)*c)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c))/(c^(5
/2)*x), -1/3840*(1920*(2*B*a^2 + 5*A*a*b)*sqrt(-a)*sqrt(-c)*c^2*x*arctan(1/2*(b*
x + 2*a)/(sqrt(c*x^2 + b*x + a)*sqrt(-a))) - 15*(3*B*b^5 + 480*A*a^2*c^3 + 240*(
B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*c)*x*arctan(1/2*(2*c*x + b)*sqrt
(-c)/(sqrt(c*x^2 + b*x + a)*c)) - 2*(384*B*c^4*x^5 - 1920*A*a^2*c^2 + 48*(21*B*b
*c^3 + 10*A*c^4)*x^4 + 8*(93*B*b^2*c^2 + 2*(88*B*a + 85*A*b)*c^3)*x^3 + 2*(15*B*
b^3*c + 1080*A*a*c^3 + 2*(622*B*a*b + 295*A*b^2)*c^2)*x^2 - (45*B*b^4 - 8*(368*B
*a^2 + 695*A*a*b)*c^2 - 30*(18*B*a*b^2 + 5*A*b^3)*c)*x)*sqrt(c*x^2 + b*x + a)*sq
rt(-c))/(sqrt(-c)*c^2*x)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**2,x)
[Out]
Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**2, x)
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GIAC/XCAS [A] time = 0.641108, 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^2,x, algorithm="giac")
[Out]
sage0*x