3.77 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^7} \, dx\)

Optimal. Leaf size=160 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(11*b*x^7) - (2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))
/(99*b^2*x^6) + (4*c*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) - (16*c
^2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(1155*b^4*x^4) + (32*c^3*(11*b*B - 8*A*
c)*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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Rubi [A]  time = 0.356582, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^7,x]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(11*b*x^7) - (2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))
/(99*b^2*x^6) + (4*c*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) - (16*c
^2*(11*b*B - 8*A*c)*(b*x + c*x^2)^(3/2))/(1155*b^4*x^4) + (32*c^3*(11*b*B - 8*A*
c)*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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Rubi in Sympy [A]  time = 21.6108, size = 158, normalized size = 0.99 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{11 b x^{7}} + \frac{2 \left (8 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{99 b^{2} x^{6}} - \frac{4 c \left (8 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{231 b^{3} x^{5}} + \frac{16 c^{2} \left (8 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1155 b^{4} x^{4}} - \frac{32 c^{3} \left (8 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3465 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**7,x)

[Out]

-2*A*(b*x + c*x**2)**(3/2)/(11*b*x**7) + 2*(8*A*c - 11*B*b)*(b*x + c*x**2)**(3/2
)/(99*b**2*x**6) - 4*c*(8*A*c - 11*B*b)*(b*x + c*x**2)**(3/2)/(231*b**3*x**5) +
16*c**2*(8*A*c - 11*B*b)*(b*x + c*x**2)**(3/2)/(1155*b**4*x**4) - 32*c**3*(8*A*c
- 11*B*b)*(b*x + c*x**2)**(3/2)/(3465*b**5*x**3)

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Mathematica [A]  time = 0.124338, size = 100, normalized size = 0.62 \[ -\frac{2 (x (b+c x))^{3/2} \left (A \left (315 b^4-280 b^3 c x+240 b^2 c^2 x^2-192 b c^3 x^3+128 c^4 x^4\right )+11 b B x \left (35 b^3-30 b^2 c x+24 b c^2 x^2-16 c^3 x^3\right )\right )}{3465 b^5 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^7,x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(11*b*B*x*(35*b^3 - 30*b^2*c*x + 24*b*c^2*x^2 - 16*c^3*x
^3) + A*(315*b^4 - 280*b^3*c*x + 240*b^2*c^2*x^2 - 192*b*c^3*x^3 + 128*c^4*x^4))
)/(3465*b^5*x^7)

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Maple [A]  time = 0.01, size = 110, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-176\,Bb{c}^{3}{x}^{4}-192\,Ab{c}^{3}{x}^{3}+264\,B{b}^{2}{c}^{2}{x}^{3}+240\,A{b}^{2}{c}^{2}{x}^{2}-330\,B{b}^{3}c{x}^{2}-280\,A{b}^{3}cx+385\,{b}^{4}Bx+315\,A{b}^{4} \right ) }{3465\,{x}^{6}{b}^{5}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(1/2)/x^7,x)

[Out]

-2/3465*(c*x+b)*(128*A*c^4*x^4-176*B*b*c^3*x^4-192*A*b*c^3*x^3+264*B*b^2*c^2*x^3
+240*A*b^2*c^2*x^2-330*B*b^3*c*x^2-280*A*b^3*c*x+385*B*b^4*x+315*A*b^4)*(c*x^2+b
*x)^(1/2)/x^6/b^5

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278463, size = 174, normalized size = 1.09 \[ -\frac{2 \,{\left (315 \, A b^{5} - 16 \,{\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{5} + 8 \,{\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{4} - 6 \,{\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{3} + 5 \,{\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{2} + 35 \,{\left (11 \, B b^{5} + A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, b^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

-2/3465*(315*A*b^5 - 16*(11*B*b*c^4 - 8*A*c^5)*x^5 + 8*(11*B*b^2*c^3 - 8*A*b*c^4
)*x^4 - 6*(11*B*b^3*c^2 - 8*A*b^2*c^3)*x^3 + 5*(11*B*b^4*c - 8*A*b^3*c^2)*x^2 +
35*(11*B*b^5 + A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**7,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**7, x)

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GIAC/XCAS [A]  time = 0.277512, size = 501, normalized size = 3.13 \[ \frac{2 \,{\left (6930 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B c^{\frac{5}{2}} + 19404 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b c^{2} + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A c^{3} + 21945 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{2} c^{\frac{3}{2}} + 36960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b c^{\frac{5}{2}} + 12375 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{3} c + 51480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{2} c^{2} + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{4} \sqrt{c} + 38115 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{3} c^{\frac{3}{2}} + 385 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{5} + 15785 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{4} c + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{5} \sqrt{c} + 315 \, A b^{6}\right )}}{3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^7,x, algorithm="giac")

[Out]

2/3465*(6930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^(5/2) + 19404*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^6*B*b*c^2 + 11088*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^3 + 219
45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^(3/2) + 36960*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^5*A*b*c^(5/2) + 12375*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c + 514
80*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*B*b^4*sqrt(c) + 38115*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^(3/2) +
385*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5 + 15785*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^2*A*b^4*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*sqrt(c) + 315*A*b^6)/
(sqrt(c)*x - sqrt(c*x^2 + b*x))^11