3.940 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x} \, dx\)
Optimal. Leaf size=350 \[ a^{5/2} (-A) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{\sqrt{a+b x+c x^2} \left (512 a^2 A c^3+2 c x \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )+b \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )\right )}{512 c^3}+\frac{\left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (80 a^2 B c^2+112 a A b c^2-40 a b^2 B c-12 A b^3 c+5 b^4 B\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (2 c x \left (-20 a B c-12 A b c+5 b^2 B\right )-64 a A c^2-20 a b B c-12 A b^2 c+5 b^3 B\right )}{192 c^2}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 A c+5 b B+10 B c x)}{60 c} \]
[Out]
((512*a^2*A*c^3 + b*(64*a*A*b*c^2 + (b^2 - 4*a*c)*(5*b^2*B - 12*A*b*c - 20*a*B*c
)) + 2*c*(64*a*A*b*c^2 + (b^2 - 4*a*c)*(5*b^2*B - 12*A*b*c - 20*a*B*c))*x)*Sqrt[
a + b*x + c*x^2])/(512*c^3) - ((5*b^3*B - 12*A*b^2*c - 20*a*b*B*c - 64*a*A*c^2 +
2*c*(5*b^2*B - 12*A*b*c - 20*a*B*c)*x)*(a + b*x + c*x^2)^(3/2))/(192*c^2) + ((5
*b*B + 12*A*c + 10*B*c*x)*(a + b*x + c*x^2)^(5/2))/(60*c) - a^(5/2)*A*ArcTanh[(2
*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])] + ((512*a^2*A*b*c^3 - (b^2 - 4*a*c)
*(5*b^4*B - 12*A*b^3*c - 40*a*b^2*B*c + 112*a*A*b*c^2 + 80*a^2*B*c^2))*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(7/2))
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Rubi [A] time = 0.969127, antiderivative size = 350, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217
\[ a^{5/2} (-A) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{\sqrt{a+b x+c x^2} \left (512 a^2 A c^3+2 c x \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )+b \left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b^2 c^2\right )}{512 c^3}+\frac{\left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (80 a^2 B c^2+112 a A b c^2-40 a b^2 B c-12 A b^3 c+5 b^4 B\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (2 c x \left (-20 a B c-12 A b c+5 b^2 B\right )-64 a A c^2-20 a b B c-12 A b^2 c+5 b^3 B\right )}{192 c^2}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 A c+5 b B+10 B c x)}{60 c} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x,x]
[Out]
((64*a*A*b^2*c^2 + 512*a^2*A*c^3 + b*(b^2 - 4*a*c)*(5*b^2*B - 12*A*b*c - 20*a*B*
c) + 2*c*(64*a*A*b*c^2 + (b^2 - 4*a*c)*(5*b^2*B - 12*A*b*c - 20*a*B*c))*x)*Sqrt[
a + b*x + c*x^2])/(512*c^3) - ((5*b^3*B - 12*A*b^2*c - 20*a*b*B*c - 64*a*A*c^2 +
2*c*(5*b^2*B - 12*A*b*c - 20*a*B*c)*x)*(a + b*x + c*x^2)^(3/2))/(192*c^2) + ((5
*b*B + 12*A*c + 10*B*c*x)*(a + b*x + c*x^2)^(5/2))/(60*c) - a^(5/2)*A*ArcTanh[(2
*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])] + ((512*a^2*A*b*c^3 - (b^2 - 4*a*c)
*(5*b^4*B - 12*A*b^3*c - 40*a*b^2*B*c + 112*a*A*b*c^2 + 80*a^2*B*c^2))*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(7/2))
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Rubi in Sympy [A] time = 99.589, size = 364, normalized size = 1.04 \[ - A a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )} + \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (6 A c + \frac{5 B b}{2} + 5 B c x\right )}{30 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 48 A a c^{2} - 9 A b^{2} c - 15 B a b c + \frac{15 B b^{3}}{4} + \frac{3 c x \left (- 12 A b c - 20 B a c + 5 B b^{2}\right )}{2}\right )}{144 c^{2}} + \frac{\sqrt{a + b x + c x^{2}} \left (192 A a^{2} c^{3} + \frac{3 b \left (64 A a b c^{2} + \left (- 4 a c + b^{2}\right ) \left (- 12 A b c - 20 B a c + 5 B b^{2}\right )\right )}{8} + \frac{3 c x \left (64 A a b c^{2} + \left (- 4 a c + b^{2}\right ) \left (- 12 A b c - 20 B a c + 5 B b^{2}\right )\right )}{4}\right )}{192 c^{3}} - \frac{\left (- 512 A a^{2} b c^{3} + \left (- 4 a c + b^{2}\right ) \left (64 A a b c^{2} + \left (- 4 a c + b^{2}\right ) \left (- 12 A b c - 20 B a c + 5 B b^{2}\right )\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{7}{2}}} \]
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,x)
[Out]
-A*a**(5/2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2))) + (a + b*x + c
*x**2)**(5/2)*(6*A*c + 5*B*b/2 + 5*B*c*x)/(30*c) - (a + b*x + c*x**2)**(3/2)*(-4
8*A*a*c**2 - 9*A*b**2*c - 15*B*a*b*c + 15*B*b**3/4 + 3*c*x*(-12*A*b*c - 20*B*a*c
+ 5*B*b**2)/2)/(144*c**2) + sqrt(a + b*x + c*x**2)*(192*A*a**2*c**3 + 3*b*(64*A
*a*b*c**2 + (-4*a*c + b**2)*(-12*A*b*c - 20*B*a*c + 5*B*b**2))/8 + 3*c*x*(64*A*a
*b*c**2 + (-4*a*c + b**2)*(-12*A*b*c - 20*B*a*c + 5*B*b**2))/4)/(192*c**3) - (-5
12*A*a**2*b*c**3 + (-4*a*c + b**2)*(64*A*a*b*c**2 + (-4*a*c + b**2)*(-12*A*b*c -
20*B*a*c + 5*B*b**2)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1
024*c**(7/2))
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Mathematica [A] time = 0.940163, size = 331, normalized size = 0.95 \[ -a^{5/2} A \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+a^{5/2} A \log (x)+\frac{\sqrt{a+x (b+c x)} \left (16 b c^2 \left (165 a^2 B+2 a c x (311 A+195 B x)+4 c^2 x^3 (63 A+50 B x)\right )+32 c^3 \left (a^2 (368 A+165 B x)+2 a c x^2 (88 A+65 B x)+8 c^2 x^4 (6 A+5 B x)\right )+40 b^3 c (c x (3 A+B x)-20 a B)+48 b^2 c^2 \left (5 a (9 A+2 B x)+c x^2 (62 A+45 B x)\right )-10 b^4 c (18 A+5 B x)+75 b^5 B\right )}{7680 c^3}+\frac{\left (320 a^3 B c^3+960 a^2 A b c^3-240 a^2 b^2 B c^2-160 a A b^3 c^2+60 a b^4 B c+12 A b^5 c-5 b^6 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{1024 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x,x]
[Out]
(Sqrt[a + x*(b + c*x)]*(75*b^5*B - 10*b^4*c*(18*A + 5*B*x) + 40*b^3*c*(-20*a*B +
c*x*(3*A + B*x)) + 48*b^2*c^2*(5*a*(9*A + 2*B*x) + c*x^2*(62*A + 45*B*x)) + 32*
c^3*(8*c^2*x^4*(6*A + 5*B*x) + 2*a*c*x^2*(88*A + 65*B*x) + a^2*(368*A + 165*B*x)
) + 16*b*c^2*(165*a^2*B + 4*c^2*x^3*(63*A + 50*B*x) + 2*a*c*x*(311*A + 195*B*x))
))/(7680*c^3) + a^(5/2)*A*Log[x] - a^(5/2)*A*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a +
x*(b + c*x)]] + ((-5*b^6*B + 12*A*b^5*c + 60*a*b^4*B*c - 160*a*A*b^3*c^2 - 240*a
^2*b^2*B*c^2 + 960*a^2*A*b*c^3 + 320*a^3*B*c^3)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a
+ x*(b + c*x)]])/(1024*c^(7/2))
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Maple [B] time = 0.013, size = 694, 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,x)
[Out]
5/256*B/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*B/c*(c*x^2+b*x+a)^(1/2)*b*a^2-5/64*B/
c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+7/16*A*b*(c*x^2+b*x+a)^(1/2)*x*a-3/64*A/c*(c*x^2+b
*x+a)^(1/2)*x*b^3+7/32*A/c*(c*x^2+b*x+a)^(1/2)*b^2*a+15/16*A*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*A/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*b^3*a-15/64*B/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*b^2*a^2+15/256*B/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-5
/32*B/c*(c*x^2+b*x+a)^(1/2)*x*a*b^2+1/8*A*b*(c*x^2+b*x+a)^(3/2)*x+1/16*A/c*(c*x^
2+b*x+a)^(3/2)*b^2-3/128*A/c^2*(c*x^2+b*x+a)^(1/2)*b^4+3/256*A/c^(5/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^5+1/3*A*a*(c*x^2+b*x+a)^(3/2)+A*a^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)+1/6*B*x*(c*
x^2+b*x+a)^(5/2)+5/48*B/c*(c*x^2+b*x+a)^(3/2)*b*a-5/96*B/c*(c*x^2+b*x+a)^(3/2)*x
*b^2+1/5*A*(c*x^2+b*x+a)^(5/2)+1/12*B/c*(c*x^2+b*x+a)^(5/2)*b+5/24*B*(c*x^2+b*x+
a)^(3/2)*x*a-5/192*B/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*B*(c*x^2+b*x+a)^(1/2)*x*a^
2+5/512*B/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16*B/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a^3-5/1024*B/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)*b^6
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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,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 16.4139, 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,x, algorithm="fricas")
[Out]
[1/30720*(15360*A*a^(5/2)*c^(7/2)*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) + 4*(1280*B*c^5*x^5 + 75*B*b^5 +
11776*A*a^2*c^3 + 128*(25*B*b*c^4 + 12*A*c^5)*x^4 + 16*(135*B*b^2*c^3 + 4*(65*B
*a + 63*A*b)*c^4)*x^3 + 240*(11*B*a^2*b + 9*A*a*b^2)*c^2 + 8*(5*B*b^3*c^2 + 704*
A*a*c^4 + 12*(65*B*a*b + 31*A*b^2)*c^3)*x^2 - 20*(40*B*a*b^3 + 9*A*b^4)*c - 2*(2
5*B*b^4*c - 16*(165*B*a^2 + 311*A*a*b)*c^3 - 60*(4*B*a*b^2 + A*b^3)*c^2)*x)*sqrt
(c*x^2 + b*x + a)*sqrt(c) - 15*(5*B*b^6 - 320*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*
a^2*b^2 + 2*A*a*b^3)*c^2 - 12*(5*B*a*b^4 + A*b^5)*c)*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)))/c^(7/2), 1/153
60*(7680*A*a^(5/2)*sqrt(-c)*c^3*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) + 2*(1280*B*c^5*x^5 + 75*B*b^5 + 1
1776*A*a^2*c^3 + 128*(25*B*b*c^4 + 12*A*c^5)*x^4 + 16*(135*B*b^2*c^3 + 4*(65*B*a
+ 63*A*b)*c^4)*x^3 + 240*(11*B*a^2*b + 9*A*a*b^2)*c^2 + 8*(5*B*b^3*c^2 + 704*A*
a*c^4 + 12*(65*B*a*b + 31*A*b^2)*c^3)*x^2 - 20*(40*B*a*b^3 + 9*A*b^4)*c - 2*(25*
B*b^4*c - 16*(165*B*a^2 + 311*A*a*b)*c^3 - 60*(4*B*a*b^2 + A*b^3)*c^2)*x)*sqrt(c
*x^2 + b*x + a)*sqrt(-c) - 15*(5*B*b^6 - 320*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a
^2*b^2 + 2*A*a*b^3)*c^2 - 12*(5*B*a*b^4 + A*b^5)*c)*arctan(1/2*(2*c*x + b)*sqrt(
-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^3), -1/30720*(30720*A*sqrt(-a)*a^2*c
^(7/2)*arctan(1/2*(b*x + 2*a)/(sqrt(c*x^2 + b*x + a)*sqrt(-a))) - 4*(1280*B*c^5*
x^5 + 75*B*b^5 + 11776*A*a^2*c^3 + 128*(25*B*b*c^4 + 12*A*c^5)*x^4 + 16*(135*B*b
^2*c^3 + 4*(65*B*a + 63*A*b)*c^4)*x^3 + 240*(11*B*a^2*b + 9*A*a*b^2)*c^2 + 8*(5*
B*b^3*c^2 + 704*A*a*c^4 + 12*(65*B*a*b + 31*A*b^2)*c^3)*x^2 - 20*(40*B*a*b^3 + 9
*A*b^4)*c - 2*(25*B*b^4*c - 16*(165*B*a^2 + 311*A*a*b)*c^3 - 60*(4*B*a*b^2 + A*b
^3)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 15*(5*B*b^6 - 320*(B*a^3 + 3*A*a^2*b
)*c^3 + 80*(3*B*a^2*b^2 + 2*A*a*b^3)*c^2 - 12*(5*B*a*b^4 + A*b^5)*c)*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))
)/c^(7/2), -1/15360*(15360*A*sqrt(-a)*a^2*sqrt(-c)*c^3*arctan(1/2*(b*x + 2*a)/(s
qrt(c*x^2 + b*x + a)*sqrt(-a))) - 2*(1280*B*c^5*x^5 + 75*B*b^5 + 11776*A*a^2*c^3
+ 128*(25*B*b*c^4 + 12*A*c^5)*x^4 + 16*(135*B*b^2*c^3 + 4*(65*B*a + 63*A*b)*c^4
)*x^3 + 240*(11*B*a^2*b + 9*A*a*b^2)*c^2 + 8*(5*B*b^3*c^2 + 704*A*a*c^4 + 12*(65
*B*a*b + 31*A*b^2)*c^3)*x^2 - 20*(40*B*a*b^3 + 9*A*b^4)*c - 2*(25*B*b^4*c - 16*(
165*B*a^2 + 311*A*a*b)*c^3 - 60*(4*B*a*b^2 + A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(-c) + 15*(5*B*b^6 - 320*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a^2*b^2 + 2*A*a
*b^3)*c^2 - 12*(5*B*a*b^4 + A*b^5)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^
2 + b*x + a)*c)))/(sqrt(-c)*c^3)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x,x)
[Out]
Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x, x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x,x, algorithm="giac")
[Out]
Exception raised: TypeError