3.214 \(\int x^{3/2} (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac{16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]

[Out]

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

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Rubi [A]  time = 0.440401, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac{16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.8455, size = 204, normalized size = 0.99 \[ \frac{2 B x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{7}{2}}}{17 c} + \frac{256 b^{4} \left (17 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{765765 c^{6} x^{\frac{7}{2}}} - \frac{128 b^{3} \left (17 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{109395 c^{5} x^{\frac{5}{2}}} + \frac{32 b^{2} \left (17 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{12155 c^{4} x^{\frac{3}{2}}} - \frac{16 b \left (17 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{3315 c^{3} \sqrt{x}} + \frac{2 \sqrt{x} \left (17 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{255 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.125908, size = 120, normalized size = 0.58 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (128 b^4 c (17 A+35 B x)-224 b^3 c^2 x (34 A+45 B x)+336 b^2 c^3 x^2 (51 A+55 B x)-462 b c^4 x^3 (68 A+65 B x)+3003 c^5 x^4 (17 A+15 B x)-1280 b^5 B\right )}{765765 c^6 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(-1280*b^5*B + 3003*c^5*x^4*(17*A + 15*B*x) + 1
28*b^4*c*(17*A + 35*B*x) - 224*b^3*c^2*x*(34*A + 45*B*x) + 336*b^2*c^3*x^2*(51*A
+ 55*B*x) - 462*b*c^4*x^3*(68*A + 65*B*x)))/(765765*c^6*Sqrt[x])

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Maple [A]  time = 0.009, size = 131, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 45045\,B{x}^{5}{c}^{5}+51051\,A{c}^{5}{x}^{4}-30030\,Bb{c}^{4}{x}^{4}-31416\,Ab{c}^{4}{x}^{3}+18480\,B{b}^{2}{c}^{3}{x}^{3}+17136\,A{b}^{2}{c}^{3}{x}^{2}-10080\,B{b}^{3}{c}^{2}{x}^{2}-7616\,A{b}^{3}{c}^{2}x+4480\,B{b}^{4}cx+2176\,A{b}^{4}c-1280\,B{b}^{5} \right ) }{765765\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)*(c*x^2+b*x)^(5/2),x)

[Out]

2/765765*(c*x+b)*(45045*B*c^5*x^5+51051*A*c^5*x^4-30030*B*b*c^4*x^4-31416*A*b*c^
4*x^3+18480*B*b^2*c^3*x^3+17136*A*b^2*c^3*x^2-10080*B*b^3*c^2*x^2-7616*A*b^3*c^2
*x+4480*B*b^4*c*x+2176*A*b^4*c-1280*B*b^5)*(c*x^2+b*x)^(5/2)/c^6/x^(5/2)

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Maxima [A]  time = 0.717596, size = 684, normalized size = 3.3 \[ \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 10 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5} + 13 \,{\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt{c x + b} A}{45045 \, c^{5} x^{6}} + \frac{2 \,{\left (7 \,{\left (6435 \, c^{8} x^{8} + 429 \, b c^{7} x^{7} - 462 \, b^{2} c^{6} x^{6} + 504 \, b^{3} c^{5} x^{5} - 560 \, b^{4} c^{4} x^{4} + 640 \, b^{5} c^{3} x^{3} - 768 \, b^{6} c^{2} x^{2} + 1024 \, b^{7} c x - 2048 \, b^{8}\right )} x^{7} + 34 \,{\left (3003 \, b c^{7} x^{8} + 231 \, b^{2} c^{6} x^{7} - 252 \, b^{3} c^{5} x^{6} + 280 \, b^{4} c^{4} x^{5} - 320 \, b^{5} c^{3} x^{4} + 384 \, b^{6} c^{2} x^{3} - 512 \, b^{7} c x^{2} + 1024 \, b^{8} x\right )} x^{6} + 85 \,{\left (693 \, b^{2} c^{6} x^{8} + 63 \, b^{3} c^{5} x^{7} - 70 \, b^{4} c^{4} x^{6} + 80 \, b^{5} c^{3} x^{5} - 96 \, b^{6} c^{2} x^{4} + 128 \, b^{7} c x^{3} - 256 \, b^{8} x^{2}\right )} x^{5}\right )} \sqrt{c x + b} B}{765765 \, c^{6} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="maxima")

[Out]

2/45045*((3003*c^7*x^7 + 231*b*c^6*x^6 - 252*b^2*c^5*x^5 + 280*b^3*c^4*x^4 - 320
*b^4*c^3*x^3 + 384*b^5*c^2*x^2 - 512*b^6*c*x + 1024*b^7)*x^6 + 10*(693*b*c^6*x^7
+ 63*b^2*c^5*x^6 - 70*b^3*c^4*x^5 + 80*b^4*c^3*x^4 - 96*b^5*c^2*x^3 + 128*b^6*c
*x^2 - 256*b^7*x)*x^5 + 13*(315*b^2*c^5*x^7 + 35*b^3*c^4*x^6 - 40*b^4*c^3*x^5 +
48*b^5*c^2*x^4 - 64*b^6*c*x^3 + 128*b^7*x^2)*x^4)*sqrt(c*x + b)*A/(c^5*x^6) + 2/
765765*(7*(6435*c^8*x^8 + 429*b*c^7*x^7 - 462*b^2*c^6*x^6 + 504*b^3*c^5*x^5 - 56
0*b^4*c^4*x^4 + 640*b^5*c^3*x^3 - 768*b^6*c^2*x^2 + 1024*b^7*c*x - 2048*b^8)*x^7
+ 34*(3003*b*c^7*x^8 + 231*b^2*c^6*x^7 - 252*b^3*c^5*x^6 + 280*b^4*c^4*x^5 - 32
0*b^5*c^3*x^4 + 384*b^6*c^2*x^3 - 512*b^7*c*x^2 + 1024*b^8*x)*x^6 + 85*(693*b^2*
c^6*x^8 + 63*b^3*c^5*x^7 - 70*b^4*c^4*x^6 + 80*b^5*c^3*x^5 - 96*b^6*c^2*x^4 + 12
8*b^7*c*x^3 - 256*b^8*x^2)*x^5)*sqrt(c*x + b)*B/(c^6*x^7)

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Fricas [A]  time = 0.297265, size = 309, normalized size = 1.49 \[ \frac{2 \,{\left (45045 \, B c^{9} x^{10} + 3003 \,{\left (50 \, B b c^{8} + 17 \, A c^{9}\right )} x^{9} + 462 \,{\left (365 \, B b^{2} c^{7} + 374 \, A b c^{8}\right )} x^{8} + 42 \,{\left (1520 \, B b^{3} c^{6} + 4709 \, A b^{2} c^{7}\right )} x^{7} - 7 \,{\left (5 \, B b^{4} c^{5} - 10948 \, A b^{3} c^{6}\right )} x^{6} + 5 \,{\left (10 \, B b^{5} c^{4} - 17 \, A b^{4} c^{5}\right )} x^{5} - 8 \,{\left (10 \, B b^{6} c^{3} - 17 \, A b^{5} c^{4}\right )} x^{4} + 16 \,{\left (10 \, B b^{7} c^{2} - 17 \, A b^{6} c^{3}\right )} x^{3} - 64 \,{\left (10 \, B b^{8} c - 17 \, A b^{7} c^{2}\right )} x^{2} - 128 \,{\left (10 \, B b^{9} - 17 \, A b^{8} c\right )} x\right )}}{765765 \, \sqrt{c x^{2} + b x} c^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*c^9*x^10 + 3003*(50*B*b*c^8 + 17*A*c^9)*x^9 + 462*(365*B*b^2*c
^7 + 374*A*b*c^8)*x^8 + 42*(1520*B*b^3*c^6 + 4709*A*b^2*c^7)*x^7 - 7*(5*B*b^4*c^
5 - 10948*A*b^3*c^6)*x^6 + 5*(10*B*b^5*c^4 - 17*A*b^4*c^5)*x^5 - 8*(10*B*b^6*c^3
- 17*A*b^5*c^4)*x^4 + 16*(10*B*b^7*c^2 - 17*A*b^6*c^3)*x^3 - 64*(10*B*b^8*c - 1
7*A*b^7*c^2)*x^2 - 128*(10*B*b^9 - 17*A*b^8*c)*x)/(sqrt(c*x^2 + b*x)*c^6*sqrt(x)
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296258, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="giac")

[Out]

Done