3.78 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^8} \, dx\)

Optimal. Leaf size=195 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{45045 b^6 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{15015 b^5 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{3003 b^4 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{1287 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{143 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(13*b*x^8) - (2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2)
)/(143*b^2*x^7) + (16*c*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(1287*b^3*x^6) -
(32*c^2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(3003*b^4*x^5) + (128*c^3*(13*b*B
- 10*A*c)*(b*x + c*x^2)^(3/2))/(15015*b^5*x^4) - (256*c^4*(13*b*B - 10*A*c)*(b*
x + c*x^2)^(3/2))/(45045*b^6*x^3)

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Rubi [A]  time = 0.437185, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{45045 b^6 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{15015 b^5 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{3003 b^4 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{1287 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{143 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^8,x]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(13*b*x^8) - (2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2)
)/(143*b^2*x^7) + (16*c*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(1287*b^3*x^6) -
(32*c^2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(3003*b^4*x^5) + (128*c^3*(13*b*B
- 10*A*c)*(b*x + c*x^2)^(3/2))/(15015*b^5*x^4) - (256*c^4*(13*b*B - 10*A*c)*(b*
x + c*x^2)^(3/2))/(45045*b^6*x^3)

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Rubi in Sympy [A]  time = 27.6353, size = 194, normalized size = 0.99 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{3}{2}}}{13 b x^{8}} + \frac{2 \left (10 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{143 b^{2} x^{7}} - \frac{16 c \left (10 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1287 b^{3} x^{6}} + \frac{32 c^{2} \left (10 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3003 b^{4} x^{5}} - \frac{128 c^{3} \left (10 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15015 b^{5} x^{4}} + \frac{256 c^{4} \left (10 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{45045 b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**8,x)

[Out]

-2*A*(b*x + c*x**2)**(3/2)/(13*b*x**8) + 2*(10*A*c - 13*B*b)*(b*x + c*x**2)**(3/
2)/(143*b**2*x**7) - 16*c*(10*A*c - 13*B*b)*(b*x + c*x**2)**(3/2)/(1287*b**3*x**
6) + 32*c**2*(10*A*c - 13*B*b)*(b*x + c*x**2)**(3/2)/(3003*b**4*x**5) - 128*c**3
*(10*A*c - 13*B*b)*(b*x + c*x**2)**(3/2)/(15015*b**5*x**4) + 256*c**4*(10*A*c -
13*B*b)*(b*x + c*x**2)**(3/2)/(45045*b**6*x**3)

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Mathematica [A]  time = 0.13018, size = 123, normalized size = 0.63 \[ -\frac{2 (x (b+c x))^{3/2} \left (5 A \left (693 b^5-630 b^4 c x+560 b^3 c^2 x^2-480 b^2 c^3 x^3+384 b c^4 x^4-256 c^5 x^5\right )+13 b B x \left (315 b^4-280 b^3 c x+240 b^2 c^2 x^2-192 b c^3 x^3+128 c^4 x^4\right )\right )}{45045 b^6 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^8,x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(13*b*B*x*(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) + 5*A*(693*b^5 - 630*b^4*c*x + 560*b^3*c^2*x^2 - 480*b
^2*c^3*x^3 + 384*b*c^4*x^4 - 256*c^5*x^5)))/(45045*b^6*x^8)

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Maple [A]  time = 0.01, size = 134, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1280\,A{c}^{5}{x}^{5}+1664\,Bb{c}^{4}{x}^{5}+1920\,Ab{c}^{4}{x}^{4}-2496\,B{b}^{2}{c}^{3}{x}^{4}-2400\,A{b}^{2}{c}^{3}{x}^{3}+3120\,B{b}^{3}{c}^{2}{x}^{3}+2800\,A{b}^{3}{c}^{2}{x}^{2}-3640\,B{b}^{4}c{x}^{2}-3150\,A{b}^{4}cx+4095\,B{b}^{5}x+3465\,A{b}^{5} \right ) }{45045\,{x}^{7}{b}^{6}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(1/2)/x^8,x)

[Out]

-2/45045*(c*x+b)*(-1280*A*c^5*x^5+1664*B*b*c^4*x^5+1920*A*b*c^4*x^4-2496*B*b^2*c
^3*x^4-2400*A*b^2*c^3*x^3+3120*B*b^3*c^2*x^3+2800*A*b^3*c^2*x^2-3640*B*b^4*c*x^2
-3150*A*b^4*c*x+4095*B*b^5*x+3465*A*b^5)*(c*x^2+b*x)^(1/2)/x^7/b^6

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2779, size = 207, normalized size = 1.06 \[ -\frac{2 \,{\left (3465 \, A b^{6} + 128 \,{\left (13 \, B b c^{5} - 10 \, A c^{6}\right )} x^{6} - 64 \,{\left (13 \, B b^{2} c^{4} - 10 \, A b c^{5}\right )} x^{5} + 48 \,{\left (13 \, B b^{3} c^{3} - 10 \, A b^{2} c^{4}\right )} x^{4} - 40 \,{\left (13 \, B b^{4} c^{2} - 10 \, A b^{3} c^{3}\right )} x^{3} + 35 \,{\left (13 \, B b^{5} c - 10 \, A b^{4} c^{2}\right )} x^{2} + 315 \,{\left (13 \, B b^{6} + A b^{5} c\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, b^{6} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

-2/45045*(3465*A*b^6 + 128*(13*B*b*c^5 - 10*A*c^6)*x^6 - 64*(13*B*b^2*c^4 - 10*A
*b*c^5)*x^5 + 48*(13*B*b^3*c^3 - 10*A*b^2*c^4)*x^4 - 40*(13*B*b^4*c^2 - 10*A*b^3
*c^3)*x^3 + 35*(13*B*b^5*c - 10*A*b^4*c^2)*x^2 + 315*(13*B*b^6 + A*b^5*c)*x)*sqr
t(c*x^2 + b*x)/(b^6*x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**8,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**8, x)

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GIAC/XCAS [A]  time = 0.278719, size = 582, normalized size = 2.98 \[ \frac{2 \,{\left (144144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B c^{3} + 480480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b c^{\frac{5}{2}} + 240240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A c^{\frac{7}{2}} + 669240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 926640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b c^{3} + 495495 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac{3}{2}} + 1531530 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac{5}{2}} + 205205 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{4} c + 1401400 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{5} \sqrt{c} + 765765 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac{3}{2}} + 4095 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{6} + 249795 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{5} c + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{6} \sqrt{c} + 3465 \, A b^{7}\right )}}{45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)/x^8,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^3 + 480480*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^7*B*b*c^(5/2) + 240240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^(7
/2) + 669240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^2 + 926640*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^6*A*b*c^3 + 495495*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^(
3/2) + 1531530*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(5/2) + 205205*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*B*b^4*c + 1401400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*
b^3*c^2 + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*sqrt(c) + 765765*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^(3/2) + 4095*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
2*B*b^6 + 249795*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^5*c + 45045*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*A*b^6*sqrt(c) + 3465*A*b^7)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^1
3