3.944 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^5} \, dx\)
Optimal. Leaf size=284 \[ -\frac{5 \left (8 a b B \left (12 a c+b^2\right )-A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{3/2}}+\frac{5}{8} \sqrt{c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 x \left (A \left (4 a c+b^2\right )+8 a b B\right )+4 a (4 a B+A b)\right )}{96 a x^3}-\frac{5 \sqrt{a+b x+c x^2} \left (-A \left (b^3-20 a b c\right )-2 c x \left (A \left (12 a c+b^2\right )+16 a b B\right )+8 a B \left (4 a c+b^2\right )\right )}{64 a x}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4} \]
[Out]
(-5*(8*a*B*(b^2 + 4*a*c) - A*(b^3 - 20*a*b*c) - 2*c*(16*a*b*B + A*(b^2 + 12*a*c)
)*x)*Sqrt[a + b*x + c*x^2])/(64*a*x) - (5*(4*a*(A*b + 4*a*B) + 3*(8*a*b*B + A*(b
^2 + 4*a*c))*x)*(a + b*x + c*x^2)^(3/2))/(96*a*x^3) - ((A - 2*B*x)*(a + b*x + c*
x^2)^(5/2))/(4*x^4) - (5*(8*a*b*B*(b^2 + 12*a*c) - A*(b^4 - 24*a*b^2*c - 48*a^2*
c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(3/2)) + (5
*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/8
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Rubi [A] time = 0.911748, antiderivative size = 284, 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
\[ -\frac{5 \left (8 a b B \left (12 a c+b^2\right )-A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{3/2}}+\frac{5}{8} \sqrt{c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 x \left (A \left (4 a c+b^2\right )+8 a b B\right )+4 a (4 a B+A b)\right )}{96 a x^3}-\frac{5 \sqrt{a+b x+c x^2} \left (-A \left (b^3-20 a b c\right )-2 c x \left (A \left (12 a c+b^2\right )+16 a b B\right )+8 a B \left (4 a c+b^2\right )\right )}{64 a x}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^5,x]
[Out]
(-5*(8*a*B*(b^2 + 4*a*c) - A*(b^3 - 20*a*b*c) - 2*c*(16*a*b*B + A*(b^2 + 12*a*c)
)*x)*Sqrt[a + b*x + c*x^2])/(64*a*x) - (5*(4*a*(A*b + 4*a*B) + 3*(8*a*b*B + A*(b
^2 + 4*a*c))*x)*(a + b*x + c*x^2)^(3/2))/(96*a*x^3) - ((A - 2*B*x)*(a + b*x + c*
x^2)^(5/2))/(4*x^4) - (5*(8*a*b*B*(b^2 + 12*a*c) - A*(b^4 - 24*a*b^2*c - 48*a^2*
c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(3/2)) + (5
*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/8
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Rubi in Sympy [A] time = 140.265, size = 296, normalized size = 1.04 \[ \frac{5 \sqrt{c} \left (4 A b c + 4 B a c + 3 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8} - \frac{\left (2 A - 4 B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{8 x^{4}} + \frac{5 \sqrt{a + b x + c x^{2}} \left (- 20 A a b c + A b^{3} - 32 B a^{2} c - 8 B a b^{2} + 2 c x \left (12 A a c + A b^{2} + 16 B a b\right )\right )}{64 a x} - \frac{5 \left (4 a \left (A b + 4 B a\right ) + x \left (12 A a c + 3 A b^{2} + 24 B a b\right )\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{96 a x^{3}} + \frac{5 \left (- 48 A a^{2} c^{2} - 24 A a b^{2} c + A b^{4} - 96 B a^{2} b c - 8 B a b^{3}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{3}{2}}} \]
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**5,x)
[Out]
5*sqrt(c)*(4*A*b*c + 4*B*a*c + 3*B*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b
*x + c*x**2)))/8 - (2*A - 4*B*x)*(a + b*x + c*x**2)**(5/2)/(8*x**4) + 5*sqrt(a +
b*x + c*x**2)*(-20*A*a*b*c + A*b**3 - 32*B*a**2*c - 8*B*a*b**2 + 2*c*x*(12*A*a*
c + A*b**2 + 16*B*a*b))/(64*a*x) - 5*(4*a*(A*b + 4*B*a) + x*(12*A*a*c + 3*A*b**2
+ 24*B*a*b))*(a + b*x + c*x**2)**(3/2)/(96*a*x**3) + 5*(-48*A*a**2*c**2 - 24*A*
a*b**2*c + A*b**4 - 96*B*a**2*b*c - 8*B*a*b**3)*atanh((2*a + b*x)/(2*sqrt(a)*sqr
t(a + b*x + c*x**2)))/(128*a**(3/2))
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Mathematica [A] time = 1.14682, size = 302, normalized size = 1.06 \[ \frac{-15 x^4 \log (x) \left (A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-8 a b B \left (12 a c+b^2\right )\right )+15 x^4 \left (A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-8 a b B \left (12 a c+b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} \left (16 a^3 (3 A+4 B x)+8 a^2 x \left (17 A b+27 A c x+26 b B x+56 B c x^2\right )+2 a x^2 \left (A \left (59 b^2+278 b c x-96 c^2 x^2\right )-12 B x \left (-11 b^2+18 b c x+4 c^2 x^2\right )\right )+15 A b^3 x^3\right )-120 a \sqrt{c} x^4 \left (4 a B c+4 A b c+3 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{384 a^{3/2} x^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^5,x]
[Out]
(-15*(-8*a*b*B*(b^2 + 12*a*c) + A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*x^4*Log[x] +
15*(-8*a*b*B*(b^2 + 12*a*c) + A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*x^4*Log[2*a + b
*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]] - 2*Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(15*A*b
^3*x^3 + 16*a^3*(3*A + 4*B*x) + 8*a^2*x*(17*A*b + 26*b*B*x + 27*A*c*x + 56*B*c*x
^2) + 2*a*x^2*(A*(59*b^2 + 278*b*c*x - 96*c^2*x^2) - 12*B*x*(-11*b^2 + 18*b*c*x
+ 4*c^2*x^2))) - 120*a*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*x^4*Log[b + 2*c*x +
2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(384*a^(3/2)*x^4)
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Maple [B] time = 0.025, size = 1094, normalized size = 3.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^5,x)
[Out]
5/8*A/a*c^2*(c*x^2+b*x+a)^(3/2)-15/8*A*a^(1/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+
b*x+a)^(1/2))/x)+5/128*A*b^4/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/
x)-1/4*A/a/x^4*(c*x^2+b*x+a)^(7/2)+5/2*A*b*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-1/64*A*b^4/a^4*(c*x^2+b*x+a)^(5/2)-5/192*A*b^4/a^3*(c*x^2+b*x+a)^
(3/2)-5/64*A*b^4/a^2*(c*x^2+b*x+a)^(1/2)+3/8*A/a^2*c^2*(c*x^2+b*x+a)^(5/2)+1/8*B
*b^3/a^3*(c*x^2+b*x+a)^(5/2)+5/24*B*b^3/a^2*(c*x^2+b*x+a)^(3/2)+5*B*b*c*(c*x^2+b
*x+a)^(1/2)+5/8*B*b^3/a*(c*x^2+b*x+a)^(1/2)-1/3*B/a/x^3*(c*x^2+b*x+a)^(7/2)+15/8
*B*b^2*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/2*B*c^(3/2)*a*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/16*B*b^3/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*
(c*x^2+b*x+a)^(1/2))/x)+5/2*B*c^2*(c*x^2+b*x+a)^(1/2)*x+1/96*A*b^2/a^3/x^2*(c*x^
2+b*x+a)^(7/2)+5/24*B*b^2/a^2*c*(c*x^2+b*x+a)^(3/2)*x+5/8*B*b^2/a*(c*x^2+b*x+a)^
(1/2)*x*c-5/192*A*b^3/a^3*c*(c*x^2+b*x+a)^(3/2)*x-1/64*A*b^3/a^4*c*(c*x^2+b*x+a)
^(5/2)*x-5/64*A*b^3/a^2*(c*x^2+b*x+a)^(1/2)*x*c+25/16*A*b/a*c^2*(c*x^2+b*x+a)^(1
/2)*x+35/48*A*b/a^2*c^2*(c*x^2+b*x+a)^(3/2)*x+19/48*A*b/a^3*c^2*(c*x^2+b*x+a)^(5
/2)*x-19/48*A*b/a^3*c/x*(c*x^2+b*x+a)^(7/2)+1/8*B*b^2/a^3*c*(c*x^2+b*x+a)^(5/2)*
x-15/16*A*b^2/a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+55/32*A*b^
2/a*c*(c*x^2+b*x+a)^(1/2)+1/24*A*b/a^2/x^3*(c*x^2+b*x+a)^(7/2)+37/96*A*b^2/a^3*c
*(c*x^2+b*x+a)^(5/2)+1/64*A*b^3/a^4/x*(c*x^2+b*x+a)^(7/2)+65/96*A*b^2/a^2*c*(c*x
^2+b*x+a)^(3/2)+17/12*B*b/a^2*c*(c*x^2+b*x+a)^(5/2)-1/8*B*b^2/a^3/x*(c*x^2+b*x+a
)^(7/2)+25/12*B*b/a*c*(c*x^2+b*x+a)^(3/2)-1/12*B*b/a^2/x^2*(c*x^2+b*x+a)^(7/2)+5
/3*B/a*c^2*(c*x^2+b*x+a)^(3/2)*x+4/3*B/a^2*c^2*(c*x^2+b*x+a)^(5/2)*x-4/3*B/a^2*c
/x*(c*x^2+b*x+a)^(7/2)-15/4*B*b*a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x)-3/8*A/a^2*c/x^2*(c*x^2+b*x+a)^(7/2)+15/8*A*c^2*(c*x^2+b*x+a)^(1/2)
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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^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 3.44233, 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^5,x, algorithm="fricas")
[Out]
[1/768*(240*(3*B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(a)*sqrt(c)*x^4*log(-8*c^2*x^2
- 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 15*(8*
B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*b^2)*c)*x^4*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) + 4*(96*B*a*c^2*x^5 + 48*(9*B*a*b*c + 4*A*a*c^2)*x^4 - 48*A*a^3 - (264*B*a*b
^2 + 15*A*b^3 + 4*(112*B*a^2 + 139*A*a*b)*c)*x^3 - 2*(104*B*a^2*b + 59*A*a*b^2 +
108*A*a^2*c)*x^2 - 8*(8*B*a^3 + 17*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(
a^(3/2)*x^4), 1/768*(480*(3*B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(a)*sqrt(-c)*x^4*
arctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) + 15*(8*B*a*b^3 - A*b^4
+ 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*b^2)*c)*x^4*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) + 4*(96*B*a
*c^2*x^5 + 48*(9*B*a*b*c + 4*A*a*c^2)*x^4 - 48*A*a^3 - (264*B*a*b^2 + 15*A*b^3 +
4*(112*B*a^2 + 139*A*a*b)*c)*x^3 - 2*(104*B*a^2*b + 59*A*a*b^2 + 108*A*a^2*c)*x
^2 - 8*(8*B*a^3 + 17*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(3/2)*x^4), 1
/384*(120*(3*B*a*b^2 + 4*(B*a^2 + A*a*b)*c)*sqrt(-a)*sqrt(c)*x^4*log(-8*c^2*x^2
- 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 15*(8*B
*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*b^2)*c)*x^4*arctan(1/2*(b*x
+ 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*(96*B*a*c^2*x^5 + 48*(9*B*a*b*c +
4*A*a*c^2)*x^4 - 48*A*a^3 - (264*B*a*b^2 + 15*A*b^3 + 4*(112*B*a^2 + 139*A*a*b)
*c)*x^3 - 2*(104*B*a^2*b + 59*A*a*b^2 + 108*A*a^2*c)*x^2 - 8*(8*B*a^3 + 17*A*a^2
*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^4), 1/384*(240*(3*B*a*b^2 +
4*(B*a^2 + A*a*b)*c)*sqrt(-a)*sqrt(-c)*x^4*arctan(1/2*(2*c*x + b)/(sqrt(c*x^2 +
b*x + a)*sqrt(-c))) - 15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*
a*b^2)*c)*x^4*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*(96
*B*a*c^2*x^5 + 48*(9*B*a*b*c + 4*A*a*c^2)*x^4 - 48*A*a^3 - (264*B*a*b^2 + 15*A*b
^3 + 4*(112*B*a^2 + 139*A*a*b)*c)*x^3 - 2*(104*B*a^2*b + 59*A*a*b^2 + 108*A*a^2*
c)*x^2 - 8*(8*B*a^3 + 17*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a
*x^4)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**5,x)
[Out]
Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**5, x)
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GIAC/XCAS [A] time = 0.644267, 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^5,x, algorithm="giac")
[Out]
sage0*x