3.91 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=160 \[ \frac{32 c^3 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^7}-\frac{2 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^8}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{13 b x^9} \]

[Out]

(-2*A*(b*x + c*x^2)^(5/2))/(13*b*x^9) - (2*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))
/(143*b^2*x^8) + (4*c*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(429*b^3*x^7) - (16*
c^2*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(3003*b^4*x^6) + (32*c^3*(13*b*B - 8*A
*c)*(b*x + c*x^2)^(5/2))/(15015*b^5*x^5)

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Rubi [A]  time = 0.359103, 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 \[ \frac{32 c^3 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{15015 b^5 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{3003 b^4 x^6}+\frac{4 c \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{429 b^3 x^7}-\frac{2 \left (b x+c x^2\right )^{5/2} (13 b B-8 A c)}{143 b^2 x^8}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{13 b x^9} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^9,x]

[Out]

(-2*A*(b*x + c*x^2)^(5/2))/(13*b*x^9) - (2*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))
/(143*b^2*x^8) + (4*c*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(429*b^3*x^7) - (16*
c^2*(13*b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(3003*b^4*x^6) + (32*c^3*(13*b*B - 8*A
*c)*(b*x + c*x^2)^(5/2))/(15015*b^5*x^5)

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Rubi in Sympy [A]  time = 21.4702, size = 158, normalized size = 0.99 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{13 b x^{9}} + \frac{2 \left (8 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{143 b^{2} x^{8}} - \frac{4 c \left (8 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{429 b^{3} x^{7}} + \frac{16 c^{2} \left (8 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3003 b^{4} x^{6}} - \frac{32 c^{3} \left (8 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15015 b^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**9,x)

[Out]

-2*A*(b*x + c*x**2)**(5/2)/(13*b*x**9) + 2*(8*A*c - 13*B*b)*(b*x + c*x**2)**(5/2
)/(143*b**2*x**8) - 4*c*(8*A*c - 13*B*b)*(b*x + c*x**2)**(5/2)/(429*b**3*x**7) +
16*c**2*(8*A*c - 13*B*b)*(b*x + c*x**2)**(5/2)/(3003*b**4*x**6) - 32*c**3*(8*A*
c - 13*B*b)*(b*x + c*x**2)**(5/2)/(15015*b**5*x**5)

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Mathematica [A]  time = 0.142329, size = 100, normalized size = 0.62 \[ \frac{2 (x (b+c x))^{5/2} \left (A \left (-1155 b^4+840 b^3 c x-560 b^2 c^2 x^2+320 b c^3 x^3-128 c^4 x^4\right )+13 b B x \left (-105 b^3+70 b^2 c x-40 b c^2 x^2+16 c^3 x^3\right )\right )}{15015 b^5 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^9,x]

[Out]

(2*(x*(b + c*x))^(5/2)*(13*b*B*x*(-105*b^3 + 70*b^2*c*x - 40*b*c^2*x^2 + 16*c^3*
x^3) + A*(-1155*b^4 + 840*b^3*c*x - 560*b^2*c^2*x^2 + 320*b*c^3*x^3 - 128*c^4*x^
4)))/(15015*b^5*x^9)

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Maple [A]  time = 0.01, size = 110, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-208\,Bb{c}^{3}{x}^{4}-320\,Ab{c}^{3}{x}^{3}+520\,B{b}^{2}{c}^{2}{x}^{3}+560\,A{b}^{2}{c}^{2}{x}^{2}-910\,B{b}^{3}c{x}^{2}-840\,A{b}^{3}cx+1365\,{b}^{4}Bx+1155\,A{b}^{4} \right ) }{15015\,{x}^{8}{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^9,x)

[Out]

-2/15015*(c*x+b)*(128*A*c^4*x^4-208*B*b*c^3*x^4-320*A*b*c^3*x^3+520*B*b^2*c^2*x^
3+560*A*b^2*c^2*x^2-910*B*b^3*c*x^2-840*A*b^3*c*x+1365*B*b^4*x+1155*A*b^4)*(c*x^
2+b*x)^(3/2)/x^8/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((c*x^2 + b*x)^(3/2)*(B*x + A)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274282, size = 207, normalized size = 1.29 \[ -\frac{2 \,{\left (1155 \, A b^{6} - 16 \,{\left (13 \, B b c^{5} - 8 \, A c^{6}\right )} x^{6} + 8 \,{\left (13 \, B b^{2} c^{4} - 8 \, A b c^{5}\right )} x^{5} - 6 \,{\left (13 \, B b^{3} c^{3} - 8 \, A b^{2} c^{4}\right )} x^{4} + 5 \,{\left (13 \, B b^{4} c^{2} - 8 \, A b^{3} c^{3}\right )} x^{3} + 35 \,{\left (52 \, B b^{5} c + A b^{4} c^{2}\right )} x^{2} + 105 \,{\left (13 \, B b^{6} + 14 \, A b^{5} c\right )} x\right )} \sqrt{c x^{2} + b x}}{15015 \, b^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^9,x, algorithm="fricas")

[Out]

-2/15015*(1155*A*b^6 - 16*(13*B*b*c^5 - 8*A*c^6)*x^6 + 8*(13*B*b^2*c^4 - 8*A*b*c
^5)*x^5 - 6*(13*B*b^3*c^3 - 8*A*b^2*c^4)*x^4 + 5*(13*B*b^4*c^2 - 8*A*b^3*c^3)*x^
3 + 35*(52*B*b^5*c + A*b^4*c^2)*x^2 + 105*(13*B*b^6 + 14*A*b^5*c)*x)*sqrt(c*x^2
+ b*x)/(b^5*x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**9,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**9, x)

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GIAC/XCAS [A]  time = 0.27958, size = 663, normalized size = 4.14 \[ \frac{2 \,{\left (30030 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} B c^{\frac{7}{2}} + 132132 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B b c^{3} + 48048 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} A c^{4} + 255255 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b^{2} c^{\frac{5}{2}} + 240240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A b c^{\frac{7}{2}} + 276705 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{3} c^{2} + 531960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b^{2} c^{3} + 180180 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{4} c^{\frac{3}{2}} + 675675 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{3} c^{\frac{5}{2}} + 70070 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{5} c + 535535 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{4} c^{2} + 15015 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{6} \sqrt{c} + 270270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{5} c^{\frac{3}{2}} + 1365 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{7} + 84630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{6} c + 15015 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{7} \sqrt{c} + 1155 \, A b^{8}\right )}}{15015 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^9,x, algorithm="giac")

[Out]

2/15015*(30030*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^(7/2) + 132132*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^8*B*b*c^3 + 48048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^4 +
255255*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^(5/2) + 240240*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^7*A*b*c^(7/2) + 276705*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3*c
^2 + 531960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^3 + 180180*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^5*B*b^4*c^(3/2) + 675675*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^
3*c^(5/2) + 70070*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c + 535535*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^4*A*b^4*c^2 + 15015*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6
*sqrt(c) + 270270*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^(3/2) + 1365*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^2*B*b^7 + 84630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^6
*c + 15015*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^7*sqrt(c) + 1155*A*b^8)/(sqrt(c)*
x - sqrt(c*x^2 + b*x))^13