3.76 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^6} \, dx\)

Optimal. Leaf size=125 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{315 b^4 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{105 b^3 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{21 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(9*b*x^6) - (2*(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(
21*b^2*x^5) + (8*c*(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(105*b^3*x^4) - (16*c^2*
(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(315*b^4*x^3)

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Rubi [A]  time = 0.261791, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{315 b^4 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{105 b^3 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2} (3 b B-2 A c)}{21 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(9*b*x^6) - (2*(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(
21*b^2*x^5) + (8*c*(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(105*b^3*x^4) - (16*c^2*
(3*b*B - 2*A*c)*(b*x + c*x^2)^(3/2))/(315*b^4*x^3)

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Rubi in Sympy [A]  time = 16.9226, size = 122, normalized size = 0.98 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{9 b x^{6}} + \frac{4 \left (A c - \frac{3 B b}{2}\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{21 b^{2} x^{5}} - \frac{8 c \left (2 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 b^{3} x^{4}} + \frac{32 c^{2} \left (A c - \frac{3 B b}{2}\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{315 b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**6,x)

[Out]

-2*A*(b*x + c*x**2)**(3/2)/(9*b*x**6) + 4*(A*c - 3*B*b/2)*(b*x + c*x**2)**(3/2)/
(21*b**2*x**5) - 8*c*(2*A*c - 3*B*b)*(b*x + c*x**2)**(3/2)/(105*b**3*x**4) + 32*
c**2*(A*c - 3*B*b/2)*(b*x + c*x**2)**(3/2)/(315*b**4*x**3)

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Mathematica [A]  time = 0.094131, size = 78, normalized size = 0.62 \[ -\frac{2 (x (b+c x))^{3/2} \left (A \left (35 b^3-30 b^2 c x+24 b c^2 x^2-16 c^3 x^3\right )+3 b B x \left (15 b^2-12 b c x+8 c^2 x^2\right )\right )}{315 b^4 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(3*b*B*x*(15*b^2 - 12*b*c*x + 8*c^2*x^2) + A*(35*b^3 - 3
0*b^2*c*x + 24*b*c^2*x^2 - 16*c^3*x^3)))/(315*b^4*x^6)

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Maple [A]  time = 0.008, size = 86, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,A{c}^{3}{x}^{3}+24\,B{x}^{3}b{c}^{2}+24\,Ab{c}^{2}{x}^{2}-36\,B{x}^{2}{b}^{2}c-30\,A{b}^{2}cx+45\,Bx{b}^{3}+35\,A{b}^{3} \right ) }{315\,{x}^{5}{b}^{4}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x)

[Out]

-2/315*(c*x+b)*(-16*A*c^3*x^3+24*B*b*c^2*x^3+24*A*b*c^2*x^2-36*B*b^2*c*x^2-30*A*
b^2*c*x+45*B*b^3*x+35*A*b^3)*(c*x^2+b*x)^(1/2)/x^5/b^4

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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(sqrt(c*x^2 + b*x)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266177, size = 142, normalized size = 1.14 \[ -\frac{2 \,{\left (35 \, A b^{4} + 8 \,{\left (3 \, B b c^{3} - 2 \, A c^{4}\right )} x^{4} - 4 \,{\left (3 \, B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 3 \,{\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 5 \,{\left (9 \, B b^{4} + A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, b^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(35*A*b^4 + 8*(3*B*b*c^3 - 2*A*c^4)*x^4 - 4*(3*B*b^2*c^2 - 2*A*b*c^3)*x^3
+ 3*(3*B*b^3*c - 2*A*b^2*c^2)*x^2 + 5*(9*B*b^4 + A*b^3*c)*x)*sqrt(c*x^2 + b*x)/
(b^4*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**6, x)

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GIAC/XCAS [A]  time = 0.27504, size = 420, normalized size = 3.36 \[ \frac{2 \,{\left (420 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B c^{2} + 945 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b c^{\frac{3}{2}} + 630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A c^{\frac{5}{2}} + 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{2} c + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{3} \sqrt{c} + 1995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac{3}{2}} + 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{4} + 1125 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{3} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{4} \sqrt{c} + 35 \, A b^{5}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^2 + 945*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*B*b*c^(3/2) + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^(5/2) + 819*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c + 1764*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
4*A*b*c^2 + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*sqrt(c) + 1995*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2) + 45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*
b^4 + 1125*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c + 315*(sqrt(c)*x - sqrt(c*x
^2 + b*x))*A*b^4*sqrt(c) + 35*A*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^9