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

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]

[Out]

(-256*b^4*(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(45045*c^6*x^(3/2)) + (128*b^3*
(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(15015*c^5*Sqrt[x]) - (32*b^2*(10*b*B - 1
3*A*c)*Sqrt[x]*(b*x + c*x^2)^(3/2))/(3003*c^4) + (16*b*(10*b*B - 13*A*c)*x^(3/2)
*(b*x + c*x^2)^(3/2))/(1287*c^3) - (2*(10*b*B - 13*A*c)*x^(5/2)*(b*x + c*x^2)^(3
/2))/(143*c^2) + (2*B*x^(7/2)*(b*x + c*x^2)^(3/2))/(13*c)

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Rubi [A]  time = 0.444456, 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 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]

Antiderivative was successfully verified.

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

[Out]

(-256*b^4*(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(45045*c^6*x^(3/2)) + (128*b^3*
(10*b*B - 13*A*c)*(b*x + c*x^2)^(3/2))/(15015*c^5*Sqrt[x]) - (32*b^2*(10*b*B - 1
3*A*c)*Sqrt[x]*(b*x + c*x^2)^(3/2))/(3003*c^4) + (16*b*(10*b*B - 13*A*c)*x^(3/2)
*(b*x + c*x^2)^(3/2))/(1287*c^3) - (2*(10*b*B - 13*A*c)*x^(5/2)*(b*x + c*x^2)^(3
/2))/(143*c^2) + (2*B*x^(7/2)*(b*x + c*x^2)^(3/2))/(13*c)

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Rubi in Sympy [A]  time = 27.5173, size = 204, normalized size = 0.99 \[ \frac{2 B x^{\frac{7}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{13 c} + \frac{256 b^{4} \left (13 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{45045 c^{6} x^{\frac{3}{2}}} - \frac{128 b^{3} \left (13 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15015 c^{5} \sqrt{x}} + \frac{32 b^{2} \sqrt{x} \left (13 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3003 c^{4}} - \frac{16 b x^{\frac{3}{2}} \left (13 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1287 c^{3}} + \frac{2 x^{\frac{5}{2}} \left (13 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{143 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.136588, size = 113, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{3/2} \left (128 b^4 c (13 A+15 B x)-96 b^3 c^2 x (26 A+25 B x)+80 b^2 c^3 x^2 (39 A+35 B x)-70 b c^4 x^3 (52 A+45 B x)+315 c^5 x^4 (13 A+11 B x)-1280 b^5 B\right )}{45045 c^6 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(-1280*b^5*B + 315*c^5*x^4*(13*A + 11*B*x) + 128*b^4*c*(1
3*A + 15*B*x) - 96*b^3*c^2*x*(26*A + 25*B*x) + 80*b^2*c^3*x^2*(39*A + 35*B*x) -
70*b*c^4*x^3*(52*A + 45*B*x)))/(45045*c^6*x^(3/2))

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Maple [A]  time = 0.008, size = 131, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3465\,B{x}^{5}{c}^{5}+4095\,A{c}^{5}{x}^{4}-3150\,Bb{c}^{4}{x}^{4}-3640\,Ab{c}^{4}{x}^{3}+2800\,B{b}^{2}{c}^{3}{x}^{3}+3120\,A{b}^{2}{c}^{3}{x}^{2}-2400\,B{b}^{3}{c}^{2}{x}^{2}-2496\,A{b}^{3}{c}^{2}x+1920\,B{b}^{4}cx+1664\,A{b}^{4}c-1280\,B{b}^{5} \right ) }{45045\,{c}^{6}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.700254, size = 192, normalized size = 0.93 \[ \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b} A}{3465 \, c^{5}} + \frac{2 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} \sqrt{c x + b} B}{9009 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2 - 64*b^4*c*
x + 128*b^5)*sqrt(c*x + b)*A/c^5 + 2/9009*(693*c^6*x^6 + 63*b*c^5*x^5 - 70*b^2*c
^4*x^4 + 80*b^3*c^3*x^3 - 96*b^4*c^2*x^2 + 128*b^5*c*x - 256*b^6)*sqrt(c*x + b)*
B/c^6

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Fricas [A]  time = 0.28201, size = 243, normalized size = 1.17 \[ \frac{2 \,{\left (3465 \, B c^{7} x^{8} + 315 \,{\left (12 \, B b c^{6} + 13 \, A c^{7}\right )} x^{7} - 35 \,{\left (B b^{2} c^{5} - 130 \, A b c^{6}\right )} x^{6} + 5 \,{\left (10 \, B b^{3} c^{4} - 13 \, A b^{2} c^{5}\right )} x^{5} - 8 \,{\left (10 \, B b^{4} c^{3} - 13 \, A b^{3} c^{4}\right )} x^{4} + 16 \,{\left (10 \, B b^{5} c^{2} - 13 \, A b^{4} c^{3}\right )} x^{3} - 64 \,{\left (10 \, B b^{6} c - 13 \, A b^{5} c^{2}\right )} x^{2} - 128 \,{\left (10 \, B b^{7} - 13 \, A b^{6} c\right )} x\right )}}{45045 \, \sqrt{c x^{2} + b x} c^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3465*B*c^7*x^8 + 315*(12*B*b*c^6 + 13*A*c^7)*x^7 - 35*(B*b^2*c^5 - 130*
A*b*c^6)*x^6 + 5*(10*B*b^3*c^4 - 13*A*b^2*c^5)*x^5 - 8*(10*B*b^4*c^3 - 13*A*b^3*
c^4)*x^4 + 16*(10*B*b^5*c^2 - 13*A*b^4*c^3)*x^3 - 64*(10*B*b^6*c - 13*A*b^5*c^2)
*x^2 - 128*(10*B*b^7 - 13*A*b^6*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**(7/2)*(B*x+A)*(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27471, size = 213, normalized size = 1.03 \[ \frac{2}{9009} \, B{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, A{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9009*B*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 1
0010*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4
- 3003*(c*x + b)^(3/2)*b^5)/c^6) - 2/3465*A*(128*b^(11/2)/c^5 - (315*(c*x + b)^(
11/2) - 1540*(c*x + b)^(9/2)*b + 2970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)
*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c^5)