3.932 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=170 \[ \frac{3 \left (b^2-4 a c\right )^2 (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{7/2}}-\frac{3 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{128 a^3 x^2}+\frac{(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac{A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5} \]

[Out]

(-3*(A*b - 2*a*B)*(b^2 - 4*a*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(128*a^3*x^2)
+ ((A*b - 2*a*B)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(16*a^2*x^4) - (A*(a + b*
x + c*x^2)^(5/2))/(5*a*x^5) + (3*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*
x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(7/2))

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Rubi [A]  time = 0.259504, antiderivative size = 170, 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 \[ \frac{3 \left (b^2-4 a c\right )^2 (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{7/2}}-\frac{3 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{128 a^3 x^2}+\frac{(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac{A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6,x]

[Out]

(-3*(A*b - 2*a*B)*(b^2 - 4*a*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(128*a^3*x^2)
+ ((A*b - 2*a*B)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(16*a^2*x^4) - (A*(a + b*
x + c*x^2)^(5/2))/(5*a*x^5) + (3*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*
x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(7/2))

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Rubi in Sympy [A]  time = 28.7809, size = 162, normalized size = 0.95 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} + \frac{\left (2 a + b x\right ) \left (A b - 2 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{16 a^{2} x^{4}} - \frac{3 \left (2 a + b x\right ) \left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{128 a^{3} x^{2}} + \frac{3 \left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{256 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**6,x)

[Out]

-A*(a + b*x + c*x**2)**(5/2)/(5*a*x**5) + (2*a + b*x)*(A*b - 2*B*a)*(a + b*x + c
*x**2)**(3/2)/(16*a**2*x**4) - 3*(2*a + b*x)*(A*b - 2*B*a)*(-4*a*c + b**2)*sqrt(
a + b*x + c*x**2)/(128*a**3*x**2) + 3*(A*b - 2*B*a)*(-4*a*c + b**2)**2*atanh((2*
a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(256*a**(7/2))

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Mathematica [A]  time = 0.379856, size = 224, normalized size = 1.32 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (32 a^4 (4 A+5 B x)+16 a^3 x (A (11 b+16 c x)+5 B x (3 b+5 c x))+4 a^2 x^2 \left (2 A \left (b^2+7 b c x+16 c^2 x^2\right )+5 b B x (b+10 c x)\right )-10 a b^2 x^3 (A (b+10 c x)+3 b B x)+15 A b^4 x^4\right )-15 x^5 \log (x) \left (b^2-4 a c\right )^2 (A b-2 a B)+15 x^5 \left (b^2-4 a c\right )^2 (A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{1280 a^{7/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^6,x]

[Out]

(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(15*A*b^4*x^4 + 32*a^4*(4*A + 5*B*x) - 10*a*b^
2*x^3*(3*b*B*x + A*(b + 10*c*x)) + 16*a^3*x*(5*B*x*(3*b + 5*c*x) + A*(11*b + 16*
c*x)) + 4*a^2*x^2*(5*b*B*x*(b + 10*c*x) + 2*A*(b^2 + 7*b*c*x + 16*c^2*x^2))) - 1
5*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*x^5*Log[x] + 15*(A*b - 2*a*B)*(b^2 - 4*a*c)^2*x^
5*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(1280*a^(7/2)*x^5)

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Maple [B]  time = 0.025, size = 978, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^6,x)

[Out]

3/256*A*b^5/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/128*A*b^5/a^
5*(c*x^2+b*x+a)^(3/2)-3/128*A*b^5/a^4*(c*x^2+b*x+a)^(1/2)-3/8*B/a^(1/2)*c^2*ln((
2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-3/128*B*b^4/a^(5/2)*ln((2*a+b*x+2*a^(1
/2)*(c*x^2+b*x+a)^(1/2))/x)-1/4*B/a/x^4*(c*x^2+b*x+a)^(5/2)+1/64*B*b^4/a^4*(c*x^
2+b*x+a)^(3/2)+3/64*B*b^4/a^3*(c*x^2+b*x+a)^(1/2)+1/8*B/a^2*c^2*(c*x^2+b*x+a)^(3
/2)+3/8*B/a*c^2*(c*x^2+b*x+a)^(1/2)+5/64*A*b^3/a^4*c*(c*x^2+b*x+a)^(3/2)+9/64*A*
b^3/a^3*c*(c*x^2+b*x+a)^(1/2)-1/16*A*b^2/a^3/x^3*(c*x^2+b*x+a)^(5/2)+1/64*A*b^3/
a^4/x^2*(c*x^2+b*x+a)^(5/2)+1/128*A*b^4/a^5/x*(c*x^2+b*x+a)^(5/2)+1/8*A*b/a^2/x^
4*(c*x^2+b*x+a)^(5/2)-1/16*A*b/a^3*c^2*(c*x^2+b*x+a)^(3/2)-3/16*A*b/a^2*c^2*(c*x
^2+b*x+a)^(1/2)-5/32*B*b^2/a^3*c*(c*x^2+b*x+a)^(3/2)-9/32*B*b^2/a^2*c*(c*x^2+b*x
+a)^(1/2)+1/8*B*b/a^2/x^3*(c*x^2+b*x+a)^(5/2)-1/32*B*b^2/a^3/x^2*(c*x^2+b*x+a)^(
5/2)-1/64*B*b^3/a^4/x*(c*x^2+b*x+a)^(5/2)-1/8*B/a^2*c/x^2*(c*x^2+b*x+a)^(5/2)-1/
5*A*(c*x^2+b*x+a)^(5/2)/a/x^5+3/64*B*b^3/a^3*c*(c*x^2+b*x+a)^(1/2)*x+3/16*B*b/a^
3*c/x*(c*x^2+b*x+a)^(5/2)-3/16*B*b/a^3*c^2*(c*x^2+b*x+a)^(3/2)*x-3/16*B*b/a^2*c^
2*(c*x^2+b*x+a)^(1/2)*x-3/128*A*b^4/a^4*c*(c*x^2+b*x+a)^(1/2)*x-3/32*A*b^2/a^4*c
/x*(c*x^2+b*x+a)^(5/2)+3/32*A*b^2/a^4*c^2*(c*x^2+b*x+a)^(3/2)*x+3/32*A*b^2/a^3*c
^2*(c*x^2+b*x+a)^(1/2)*x+1/16*A*b/a^3*c/x^2*(c*x^2+b*x+a)^(5/2)-1/128*A*b^4/a^5*
c*(c*x^2+b*x+a)^(3/2)*x+1/64*B*b^3/a^4*c*(c*x^2+b*x+a)^(3/2)*x+3/16*B*b^2/a^(3/2
)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-3/32*A*b^3/a^(5/2)*c*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+3/16*A*b/a^(3/2)*c^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((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.364288, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (2 \, B a b^{4} - A b^{5} + 16 \,{\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{5} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) - 4 \,{\left (128 \, A a^{4} -{\left (30 \, B a b^{3} - 15 \, A b^{4} - 128 \, A a^{2} c^{2} - 100 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} c\right )} x^{4} + 2 \,{\left (10 \, B a^{2} b^{2} - 5 \, A a b^{3} + 4 \,{\left (50 \, B a^{3} + 7 \, A a^{2} b\right )} c\right )} x^{3} + 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2} + 32 \, A a^{3} c\right )} x^{2} + 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a}}{2560 \, a^{\frac{7}{2}} x^{5}}, -\frac{15 \,{\left (2 \, B a b^{4} - A b^{5} + 16 \,{\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} x^{5} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) + 2 \,{\left (128 \, A a^{4} -{\left (30 \, B a b^{3} - 15 \, A b^{4} - 128 \, A a^{2} c^{2} - 100 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} c\right )} x^{4} + 2 \,{\left (10 \, B a^{2} b^{2} - 5 \, A a b^{3} + 4 \,{\left (50 \, B a^{3} + 7 \, A a^{2} b\right )} c\right )} x^{3} + 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2} + 32 \, A a^{3} c\right )} x^{2} + 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a}}{1280 \, \sqrt{-a} a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

[1/2560*(15*(2*B*a*b^4 - A*b^5 + 16*(2*B*a^3 - A*a^2*b)*c^2 - 8*(2*B*a^2*b^2 - A
*a*b^3)*c)*x^5*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*(128*A*a^4 - (30*B*a*b^3 - 15*A*b^4 - 128*
A*a^2*c^2 - 100*(2*B*a^2*b - A*a*b^2)*c)*x^4 + 2*(10*B*a^2*b^2 - 5*A*a*b^3 + 4*(
50*B*a^3 + 7*A*a^2*b)*c)*x^3 + 8*(30*B*a^3*b + A*a^2*b^2 + 32*A*a^3*c)*x^2 + 16*
(10*B*a^4 + 11*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(7/2)*x^5), -1/1280
*(15*(2*B*a*b^4 - A*b^5 + 16*(2*B*a^3 - A*a^2*b)*c^2 - 8*(2*B*a^2*b^2 - A*a*b^3)
*c)*x^5*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) + 2*(128*A*a^
4 - (30*B*a*b^3 - 15*A*b^4 - 128*A*a^2*c^2 - 100*(2*B*a^2*b - A*a*b^2)*c)*x^4 +
2*(10*B*a^2*b^2 - 5*A*a*b^3 + 4*(50*B*a^3 + 7*A*a^2*b)*c)*x^3 + 8*(30*B*a^3*b +
A*a^2*b^2 + 32*A*a^3*c)*x^2 + 16*(10*B*a^4 + 11*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(-a))/(sqrt(-a)*a^3*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/x**6,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.29012, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

Done