3.943 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=255 \[ -\frac{5 \sqrt{a+b x+c x^2} \left (-x \left (4 a B c+4 A b c+b^2 B\right )+A \left (4 a c+b^2\right )+4 a b B\right )}{8 x}+\frac{5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{c}}-\frac{5 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{a}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (2 a B-x (2 A c+b B)+A b)}{12 x^2}-\frac{(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3} \]

[Out]

(-5*(4*a*b*B + A*(b^2 + 4*a*c) - (b^2*B + 4*A*b*c + 4*a*B*c)*x)*Sqrt[a + b*x + c
*x^2])/(8*x) - (5*(A*b + 2*a*B - (b*B + 2*A*c)*x)*(a + b*x + c*x^2)^(3/2))/(12*x
^2) - ((A - B*x)*(a + b*x + c*x^2)^(5/2))/(3*x^3) - (5*(2*a*B*(3*b^2 + 4*a*c) +
A*(b^3 + 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(16*
Sqrt[a]) + (5*(b^3*B + 6*A*b^2*c + 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*Sqrt[c])

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Rubi [A]  time = 0.779713, antiderivative size = 255, 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 \[ -\frac{5 \sqrt{a+b x+c x^2} \left (-x \left (4 a B c+4 A b c+b^2 B\right )+A \left (4 a c+b^2\right )+4 a b B\right )}{8 x}+\frac{5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{c}}-\frac{5 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{a}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (2 a B-x (2 A c+b B)+A b)}{12 x^2}-\frac{(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^4,x]

[Out]

(-5*(4*a*b*B + A*(b^2 + 4*a*c) - (b^2*B + 4*A*b*c + 4*a*B*c)*x)*Sqrt[a + b*x + c
*x^2])/(8*x) - (5*(A*b + 2*a*B - (b*B + 2*A*c)*x)*(a + b*x + c*x^2)^(3/2))/(12*x
^2) - ((A - B*x)*(a + b*x + c*x^2)^(5/2))/(3*x^3) - (5*(2*a*B*(3*b^2 + 4*a*c) +
A*(b^3 + 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(16*
Sqrt[a]) + (5*(b^3*B + 6*A*b^2*c + 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*Sqrt[c])

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Rubi in Sympy [A]  time = 102.165, size = 265, normalized size = 1.04 \[ - \frac{5 \sqrt{a + b x + c x^{2}} \left (24 A a c + 6 A b^{2} + 24 B a b - x \left (24 A b c + 24 B a c + 6 B b^{2}\right )\right )}{48 x} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (6 A b + 12 B a - x \left (12 A c + 6 B b\right )\right )}{72 x^{2}} - \frac{\left (3 A - 3 B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{9 x^{3}} + \frac{5 \left (8 A a c^{2} + 6 A b^{2} c + 12 B a b c + B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 \sqrt{c}} - \frac{5 \left (12 A a b c + A b^{3} + 8 B a^{2} c + 6 B a b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 \sqrt{a}} \]

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**4,x)

[Out]

-5*sqrt(a + b*x + c*x**2)*(24*A*a*c + 6*A*b**2 + 24*B*a*b - x*(24*A*b*c + 24*B*a
*c + 6*B*b**2))/(48*x) - 5*(a + b*x + c*x**2)**(3/2)*(6*A*b + 12*B*a - x*(12*A*c
+ 6*B*b))/(72*x**2) - (3*A - 3*B*x)*(a + b*x + c*x**2)**(5/2)/(9*x**3) + 5*(8*A
*a*c**2 + 6*A*b**2*c + 12*B*a*b*c + B*b**3)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a
+ b*x + c*x**2)))/(16*sqrt(c)) - 5*(12*A*a*b*c + A*b**3 + 8*B*a**2*c + 6*B*a*b**
2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(16*sqrt(a))

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Mathematica [A]  time = 0.441536, size = 270, normalized size = 1.06 \[ \frac{1}{16} \left (-\frac{2 \sqrt{a+x (b+c x)} \left (4 a^2 (2 A+3 B x)+2 a x (A (13 b+28 c x)+B x (27 b-28 c x))-x^2 \left (A \left (-33 b^2+54 b c x+12 c^2 x^2\right )+B x \left (33 b^2+26 b c x+8 c^2 x^2\right )\right )\right )}{3 x^3}+\frac{5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+\frac{5 \log (x) \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right )}{\sqrt{a}}-\frac{5 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^4,x]

[Out]

((-2*Sqrt[a + x*(b + c*x)]*(4*a^2*(2*A + 3*B*x) + 2*a*x*(B*x*(27*b - 28*c*x) + A
*(13*b + 28*c*x)) - x^2*(B*x*(33*b^2 + 26*b*c*x + 8*c^2*x^2) + A*(-33*b^2 + 54*b
*c*x + 12*c^2*x^2))))/(3*x^3) + (5*(2*a*B*(3*b^2 + 4*a*c) + A*(b^3 + 12*a*b*c))*
Log[x])/Sqrt[a] - (5*(2*a*B*(3*b^2 + 4*a*c) + A*(b^3 + 12*a*b*c))*Log[2*a + b*x
+ 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/Sqrt[a] + (5*(b^3*B + 6*A*b^2*c + 12*a*b*B*c
+ 8*a*A*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c])/16

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Maple [B]  time = 0.022, size = 840, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^4,x)

[Out]

-3/4*B*b/a^2/x*(c*x^2+b*x+a)^(7/2)-1/3*A/a/x^3*(c*x^2+b*x+a)^(7/2)+15/8*A*b^2*c^
(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/2*A*c^(3/2)*a*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/16*A*b^3/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*
x+a)^(1/2))/x)+5*A*b*c*(c*x^2+b*x+a)^(1/2)+5/8*A*b^3/a*(c*x^2+b*x+a)^(1/2)+1/8*A
*b^3/a^3*(c*x^2+b*x+a)^(5/2)+5/24*A*b^3/a^2*(c*x^2+b*x+a)^(3/2)+5/2*A*c^2*(c*x^2
+b*x+a)^(1/2)*x-5/2*B*a^(3/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15
/8*B*b^2*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/2*B/a*c*(c*x^2+
b*x+a)^(5/2)+5/2*B*a*c*(c*x^2+b*x+a)^(1/2)+3/4*B*b^2/a^2*(c*x^2+b*x+a)^(5/2)+5/4
*B*b^2/a*(c*x^2+b*x+a)^(3/2)+5/16*B*b^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))-1/2*B/a/x^2*(c*x^2+b*x+a)^(7/2)+5/24*A*b^2/a^2*c*(c*x^2+b*x+a)^(3/2)
*x+1/8*A*b^2/a^3*c*(c*x^2+b*x+a)^(5/2)*x+5/8*A*b^2/a*(c*x^2+b*x+a)^(1/2)*x*c+3/4
*B*b/a^2*c*(c*x^2+b*x+a)^(5/2)*x+5/2*B*b*(c*x^2+b*x+a)^(1/2)*x*c-15/4*A*b*a^(1/2
)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/8*A*b^2/a^3/x*(c*x^2+b*x+a)^
(7/2)+17/12*A*b/a^2*c*(c*x^2+b*x+a)^(5/2)+25/12*A*b/a*c*(c*x^2+b*x+a)^(3/2)+15/4
*B*b*a*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+25/8*B*b^2*(c*x^2+b*x
+a)^(1/2)+5/6*B*c*(c*x^2+b*x+a)^(3/2)+5/4*B*b/a*c*(c*x^2+b*x+a)^(3/2)*x-1/12*A*b
/a^2/x^2*(c*x^2+b*x+a)^(7/2)+5/3*A/a*c^2*(c*x^2+b*x+a)^(3/2)*x+4/3*A/a^2*c^2*(c*
x^2+b*x+a)^(5/2)*x-4/3*A/a^2*c/x*(c*x^2+b*x+a)^(7/2)

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60254, 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^4,x, algorithm="fricas")

[Out]

[1/96*(15*(B*b^3 + 8*A*a*c^2 + 6*(2*B*a*b + A*b^2)*c)*sqrt(a)*x^3*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)) +
15*(6*B*a*b^2 + A*b^3 + 4*(2*B*a^2 + 3*A*a*b)*c)*sqrt(c)*x^3*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*(8*B*c^2*x^5 + 2*(13*B*b*c + 6*A*c^2)*x^4 + (33*B*b^2 + 2*(28*B*a + 27*A*b)*
c)*x^3 - 8*A*a^2 - (54*B*a*b + 33*A*b^2 + 56*A*a*c)*x^2 - 2*(6*B*a^2 + 13*A*a*b)
*x)*sqrt(c*x^2 + b*x + a)*sqrt(a)*sqrt(c))/(sqrt(a)*sqrt(c)*x^3), 1/96*(30*(B*b^
3 + 8*A*a*c^2 + 6*(2*B*a*b + A*b^2)*c)*sqrt(a)*x^3*arctan(1/2*(2*c*x + b)*sqrt(-
c)/(sqrt(c*x^2 + b*x + a)*c)) + 15*(6*B*a*b^2 + A*b^3 + 4*(2*B*a^2 + 3*A*a*b)*c)
*sqrt(-c)*x^3*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*(8*B*c^2*x^5 + 2*(13*B*b*c + 6*A*c^2)*x^4 +
(33*B*b^2 + 2*(28*B*a + 27*A*b)*c)*x^3 - 8*A*a^2 - (54*B*a*b + 33*A*b^2 + 56*A*
a*c)*x^2 - 2*(6*B*a^2 + 13*A*a*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a)*sqrt(-c))/(sq
rt(a)*sqrt(-c)*x^3), -1/96*(30*(6*B*a*b^2 + A*b^3 + 4*(2*B*a^2 + 3*A*a*b)*c)*sqr
t(c)*x^3*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 15*(B*b^3
+ 8*A*a*c^2 + 6*(2*B*a*b + A*b^2)*c)*sqrt(-a)*x^3*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*(8*B*c^2*x^5 +
2*(13*B*b*c + 6*A*c^2)*x^4 + (33*B*b^2 + 2*(28*B*a + 27*A*b)*c)*x^3 - 8*A*a^2 -
(54*B*a*b + 33*A*b^2 + 56*A*a*c)*x^2 - 2*(6*B*a^2 + 13*A*a*b)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(-a)*sqrt(c))/(sqrt(-a)*sqrt(c)*x^3), -1/48*(15*(6*B*a*b^2 + A*b^3 +
4*(2*B*a^2 + 3*A*a*b)*c)*sqrt(-c)*x^3*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x
^2 + b*x + a)*a)) - 15*(B*b^3 + 8*A*a*c^2 + 6*(2*B*a*b + A*b^2)*c)*sqrt(-a)*x^3*
arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) - 2*(8*B*c^2*x^5 + 2*
(13*B*b*c + 6*A*c^2)*x^4 + (33*B*b^2 + 2*(28*B*a + 27*A*b)*c)*x^3 - 8*A*a^2 - (5
4*B*a*b + 33*A*b^2 + 56*A*a*c)*x^2 - 2*(6*B*a^2 + 13*A*a*b)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(-a)*sqrt(-c))/(sqrt(-a)*sqrt(-c)*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**4,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.632427, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

sage0*x