3.108 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{765765 b^6 x^7}+\frac{128 c^3 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{109395 b^5 x^8}-\frac{32 c^2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{12155 b^4 x^9}+\frac{16 c \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{3315 b^3 x^{10}}-\frac{2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{255 b^2 x^{11}}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}} \]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(17*b*x^12) - (2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2
))/(255*b^2*x^11) + (16*c*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(3315*b^3*x^10)
- (32*c^2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(12155*b^4*x^9) + (128*c^3*(17
*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(109395*b^5*x^8) - (256*c^4*(17*b*B - 10*A*c
)*(b*x + c*x^2)^(7/2))/(765765*b^6*x^7)

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Rubi [A]  time = 0.44448, 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 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{765765 b^6 x^7}+\frac{128 c^3 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{109395 b^5 x^8}-\frac{32 c^2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{12155 b^4 x^9}+\frac{16 c \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{3315 b^3 x^{10}}-\frac{2 \left (b x+c x^2\right )^{7/2} (17 b B-10 A c)}{255 b^2 x^{11}}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{17 b x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^12,x]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(17*b*x^12) - (2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2
))/(255*b^2*x^11) + (16*c*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(3315*b^3*x^10)
- (32*c^2*(17*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(12155*b^4*x^9) + (128*c^3*(17
*b*B - 10*A*c)*(b*x + c*x^2)^(7/2))/(109395*b^5*x^8) - (256*c^4*(17*b*B - 10*A*c
)*(b*x + c*x^2)^(7/2))/(765765*b^6*x^7)

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Rubi in Sympy [A]  time = 27.3627, size = 194, normalized size = 0.99 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{17 b x^{12}} + \frac{2 \left (10 A c - 17 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{255 b^{2} x^{11}} - \frac{16 c \left (10 A c - 17 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{3315 b^{3} x^{10}} + \frac{32 c^{2} \left (10 A c - 17 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{12155 b^{4} x^{9}} - \frac{128 c^{3} \left (10 A c - 17 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{109395 b^{5} x^{8}} + \frac{256 c^{4} \left (10 A c - 17 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{765765 b^{6} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**12,x)

[Out]

-2*A*(b*x + c*x**2)**(7/2)/(17*b*x**12) + 2*(10*A*c - 17*B*b)*(b*x + c*x**2)**(7
/2)/(255*b**2*x**11) - 16*c*(10*A*c - 17*B*b)*(b*x + c*x**2)**(7/2)/(3315*b**3*x
**10) + 32*c**2*(10*A*c - 17*B*b)*(b*x + c*x**2)**(7/2)/(12155*b**4*x**9) - 128*
c**3*(10*A*c - 17*B*b)*(b*x + c*x**2)**(7/2)/(109395*b**5*x**8) + 256*c**4*(10*A
*c - 17*B*b)*(b*x + c*x**2)**(7/2)/(765765*b**6*x**7)

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Mathematica [A]  time = 0.152578, size = 130, normalized size = 0.67 \[ -\frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (5 A \left (9009 b^5-6006 b^4 c x+3696 b^3 c^2 x^2-2016 b^2 c^3 x^3+896 b c^4 x^4-256 c^5 x^5\right )+17 b B x \left (3003 b^4-1848 b^3 c x+1008 b^2 c^2 x^2-448 b c^3 x^3+128 c^4 x^4\right )\right )}{765765 b^6 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^12,x]

[Out]

(-2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(17*b*B*x*(3003*b^4 - 1848*b^3*c*x + 1008*b^2*
c^2*x^2 - 448*b*c^3*x^3 + 128*c^4*x^4) + 5*A*(9009*b^5 - 6006*b^4*c*x + 3696*b^3
*c^2*x^2 - 2016*b^2*c^3*x^3 + 896*b*c^4*x^4 - 256*c^5*x^5)))/(765765*b^6*x^9)

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Maple [A]  time = 0.009, size = 134, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1280\,A{c}^{5}{x}^{5}+2176\,Bb{c}^{4}{x}^{5}+4480\,Ab{c}^{4}{x}^{4}-7616\,B{b}^{2}{c}^{3}{x}^{4}-10080\,A{b}^{2}{c}^{3}{x}^{3}+17136\,B{b}^{3}{c}^{2}{x}^{3}+18480\,A{b}^{3}{c}^{2}{x}^{2}-31416\,B{b}^{4}c{x}^{2}-30030\,A{b}^{4}cx+51051\,B{b}^{5}x+45045\,A{b}^{5} \right ) }{765765\,{x}^{11}{b}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^12,x)

[Out]

-2/765765*(c*x+b)*(-1280*A*c^5*x^5+2176*B*b*c^4*x^5+4480*A*b*c^4*x^4-7616*B*b^2*
c^3*x^4-10080*A*b^2*c^3*x^3+17136*B*b^3*c^2*x^3+18480*A*b^3*c^2*x^2-31416*B*b^4*
c*x^2-30030*A*b^4*c*x+51051*B*b^5*x+45045*A*b^5)*(c*x^2+b*x)^(5/2)/x^11/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((c*x^2 + b*x)^(5/2)*(B*x + A)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293019, size = 273, normalized size = 1.4 \[ -\frac{2 \,{\left (45045 \, A b^{8} + 128 \,{\left (17 \, B b c^{7} - 10 \, A c^{8}\right )} x^{8} - 64 \,{\left (17 \, B b^{2} c^{6} - 10 \, A b c^{7}\right )} x^{7} + 48 \,{\left (17 \, B b^{3} c^{5} - 10 \, A b^{2} c^{6}\right )} x^{6} - 40 \,{\left (17 \, B b^{4} c^{4} - 10 \, A b^{3} c^{5}\right )} x^{5} + 35 \,{\left (17 \, B b^{5} c^{3} - 10 \, A b^{4} c^{4}\right )} x^{4} + 63 \,{\left (1207 \, B b^{6} c^{2} + 5 \, A b^{5} c^{3}\right )} x^{3} + 231 \,{\left (527 \, B b^{7} c + 275 \, A b^{6} c^{2}\right )} x^{2} + 3003 \,{\left (17 \, B b^{8} + 35 \, A b^{7} c\right )} x\right )} \sqrt{c x^{2} + b x}}{765765 \, b^{6} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^12,x, algorithm="fricas")

[Out]

-2/765765*(45045*A*b^8 + 128*(17*B*b*c^7 - 10*A*c^8)*x^8 - 64*(17*B*b^2*c^6 - 10
*A*b*c^7)*x^7 + 48*(17*B*b^3*c^5 - 10*A*b^2*c^6)*x^6 - 40*(17*B*b^4*c^4 - 10*A*b
^3*c^5)*x^5 + 35*(17*B*b^5*c^3 - 10*A*b^4*c^4)*x^4 + 63*(1207*B*b^6*c^2 + 5*A*b^
5*c^3)*x^3 + 231*(527*B*b^7*c + 275*A*b^6*c^2)*x^2 + 3003*(17*B*b^8 + 35*A*b^7*c
)*x)*sqrt(c*x^2 + b*x)/(b^6*x^9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{12}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**12,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**12, x)

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GIAC/XCAS [A]  time = 0.286112, size = 906, normalized size = 4.65 \[ \frac{2 \,{\left (2450448 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{12} B c^{5} + 16336320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11} B b c^{\frac{9}{2}} + 4084080 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11} A c^{\frac{11}{2}} + 49884120 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{10} B b^{2} c^{4} + 29755440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{10} A b c^{5} + 91126035 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} B b^{3} c^{\frac{7}{2}} + 99549450 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9} A b^{2} c^{\frac{9}{2}} + 109674565 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B b^{4} c^{3} + 200800600 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} A b^{3} c^{4} + 90513423 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b^{5} c^{\frac{5}{2}} + 270315045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A b^{4} c^{\frac{7}{2}} + 51723945 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{6} c^{2} + 254303595 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b^{5} c^{3} + 20165145 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{7} c^{\frac{3}{2}} + 170255085 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{6} c^{\frac{5}{2}} + 5124735 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{8} c + 80994375 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{7} c^{2} + 765765 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{9} \sqrt{c} + 26801775 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{8} c^{\frac{3}{2}} + 51051 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{10} + 5870865 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{9} c + 765765 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{10} \sqrt{c} + 45045 \, A b^{11}\right )}}{765765 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^12,x, algorithm="giac")

[Out]

2/765765*(2450448*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*c^5 + 16336320*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^11*B*b*c^(9/2) + 4084080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^1
1*A*c^(11/2) + 49884120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^4 + 29755440*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b*c^5 + 91126035*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^9*B*b^3*c^(7/2) + 99549450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2*c^(9/2)
+ 109674565*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^3 + 200800600*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^8*A*b^3*c^4 + 90513423*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*
b^5*c^(5/2) + 270315045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^(7/2) + 517239
45*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^6*c^2 + 254303595*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^6*A*b^5*c^3 + 20165145*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^7*c^(3/2
) + 170255085*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c^(5/2) + 5124735*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*B*b^8*c + 80994375*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A
*b^7*c^2 + 765765*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^9*sqrt(c) + 26801775*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*c^(3/2) + 51051*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^2*B*b^10 + 5870865*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9*c + 765765*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*A*b^10*sqrt(c) + 45045*A*b^11)/(sqrt(c)*x - sqrt(c*x^
2 + b*x))^17