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

Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{45045 c^6 x^{5/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{9009 c^5 x^{3/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{1287 c^4 \sqrt{x}}+\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{429 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

[Out]

(-256*b^4*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (128*b^3*(2
*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(9009*c^5*x^(3/2)) - (32*b^2*(2*b*B - 3*A*c)*
(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (16*b*(2*b*B - 3*A*c)*Sqrt[x]*(b*x + c
*x^2)^(5/2))/(429*c^3) - (2*(2*b*B - 3*A*c)*x^(3/2)*(b*x + c*x^2)^(5/2))/(39*c^2
) + (2*B*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi [A]  time = 0.36629, 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 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{45045 c^6 x^{5/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{9009 c^5 x^{3/2}}-\frac{32 b^2 \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{1287 c^4 \sqrt{x}}+\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{429 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2} (2 b B-3 A c)}{39 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

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

[Out]

(-256*b^4*(2*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (128*b^3*(2
*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(9009*c^5*x^(3/2)) - (32*b^2*(2*b*B - 3*A*c)*
(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (16*b*(2*b*B - 3*A*c)*Sqrt[x]*(b*x + c
*x^2)^(5/2))/(429*c^3) - (2*(2*b*B - 3*A*c)*x^(3/2)*(b*x + c*x^2)^(5/2))/(39*c^2
) + (2*B*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi in Sympy [A]  time = 27.4465, size = 204, normalized size = 0.99 \[ \frac{2 B x^{\frac{5}{2}} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15 c} + \frac{512 b^{4} \left (\frac{3 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{45045 c^{6} x^{\frac{5}{2}}} - \frac{256 b^{3} \left (\frac{3 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{9009 c^{5} x^{\frac{3}{2}}} + \frac{64 b^{2} \left (\frac{3 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{1287 c^{4} \sqrt{x}} - \frac{16 b \sqrt{x} \left (3 A c - 2 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{429 c^{3}} + \frac{4 x^{\frac{3}{2}} \left (\frac{3 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{39 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**(5/2)*(b*x + c*x**2)**(5/2)/(15*c) + 512*b**4*(3*A*c/2 - B*b)*(b*x + c*x*
*2)**(5/2)/(45045*c**6*x**(5/2)) - 256*b**3*(3*A*c/2 - B*b)*(b*x + c*x**2)**(5/2
)/(9009*c**5*x**(3/2)) + 64*b**2*(3*A*c/2 - B*b)*(b*x + c*x**2)**(5/2)/(1287*c**
4*sqrt(x)) - 16*b*sqrt(x)*(3*A*c - 2*B*b)*(b*x + c*x**2)**(5/2)/(429*c**3) + 4*x
**(3/2)*(3*A*c/2 - B*b)*(b*x + c*x**2)**(5/2)/(39*c**2)

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Mathematica [A]  time = 0.125578, size = 110, normalized size = 0.53 \[ \frac{2 (x (b+c x))^{5/2} \left (128 b^4 c (3 A+5 B x)-160 b^3 c^2 x (6 A+7 B x)+1680 b^2 c^3 x^2 (A+B x)-210 b c^4 x^3 (12 A+11 B x)+231 c^5 x^4 (15 A+13 B x)-256 b^5 B\right )}{45045 c^6 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(-256*b^5*B + 1680*b^2*c^3*x^2*(A + B*x) + 128*b^4*c*(3*A
+ 5*B*x) - 160*b^3*c^2*x*(6*A + 7*B*x) - 210*b*c^4*x^3*(12*A + 11*B*x) + 231*c^
5*x^4*(15*A + 13*B*x)))/(45045*c^6*x^(5/2))

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Maple [A]  time = 0.01, size = 131, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3003\,B{x}^{5}{c}^{5}+3465\,A{c}^{5}{x}^{4}-2310\,Bb{c}^{4}{x}^{4}-2520\,Ab{c}^{4}{x}^{3}+1680\,B{b}^{2}{c}^{3}{x}^{3}+1680\,A{b}^{2}{c}^{3}{x}^{2}-1120\,B{b}^{3}{c}^{2}{x}^{2}-960\,A{b}^{3}{c}^{2}x+640\,B{b}^{4}cx+384\,A{b}^{4}c-256\,B{b}^{5} \right ) }{45045\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(c*x+b)*(3003*B*c^5*x^5+3465*A*c^5*x^4-2310*B*b*c^4*x^4-2520*A*b*c^4*x^3
+1680*B*b^2*c^3*x^3+1680*A*b^2*c^3*x^2-1120*B*b^3*c^2*x^2-960*A*b^3*c^2*x+640*B*
b^4*c*x+384*A*b^4*c-256*B*b^5)*(c*x^2+b*x)^(3/2)/c^6/x^(3/2)

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Maxima [A]  time = 0.712577, size = 429, normalized size = 2.07 \[ \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt{c x + b} A}{45045 \, c^{5} x^{5}} + \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 5 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5}\right )} \sqrt{c x + b} B}{45045 \, c^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.308538, size = 275, normalized size = 1.33 \[ \frac{2 \,{\left (3003 \, B c^{8} x^{9} + 231 \,{\left (29 \, B b c^{7} + 15 \, A c^{8}\right )} x^{8} + 21 \,{\left (179 \, B b^{2} c^{6} + 375 \, A b c^{7}\right )} x^{7} - 7 \,{\left (B b^{3} c^{5} - 645 \, A b^{2} c^{6}\right )} x^{6} + 5 \,{\left (2 \, B b^{4} c^{4} - 3 \, A b^{3} c^{5}\right )} x^{5} - 8 \,{\left (2 \, B b^{5} c^{3} - 3 \, A b^{4} c^{4}\right )} x^{4} + 16 \,{\left (2 \, B b^{6} c^{2} - 3 \, A b^{5} c^{3}\right )} x^{3} - 64 \,{\left (2 \, B b^{7} c - 3 \, A b^{6} c^{2}\right )} x^{2} - 128 \,{\left (2 \, B b^{8} - 3 \, A b^{7} c\right )} x\right )}}{45045 \, \sqrt{c x^{2} + b x} c^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*B*c^8*x^9 + 231*(29*B*b*c^7 + 15*A*c^8)*x^8 + 21*(179*B*b^2*c^6 +
375*A*b*c^7)*x^7 - 7*(B*b^3*c^5 - 645*A*b^2*c^6)*x^6 + 5*(2*B*b^4*c^4 - 3*A*b^3*
c^5)*x^5 - 8*(2*B*b^5*c^3 - 3*A*b^4*c^4)*x^4 + 16*(2*B*b^6*c^2 - 3*A*b^5*c^3)*x^
3 - 64*(2*B*b^7*c - 3*A*b^6*c^2)*x^2 - 128*(2*B*b^8 - 3*A*b^7*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**(5/2)*(B*x+A)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286843, size = 463, normalized size = 2.24 \[ -\frac{2}{45045} \, B c{\left (\frac{1024 \, b^{\frac{15}{2}}}{c^{7}} - \frac{3003 \,{\left (c x + b\right )}^{\frac{15}{2}} - 20790 \,{\left (c x + b\right )}^{\frac{13}{2}} b + 61425 \,{\left (c x + b\right )}^{\frac{11}{2}} b^{2} - 100100 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{3} + 96525 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{4} - 54054 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{5} + 15015 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{6}}{c^{7}}\right )} + \frac{2}{9009} \, B b{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} + \frac{2}{9009} \, A c{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, A b{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/45045*B*c*(1024*b^(15/2)/c^7 - (3003*(c*x + b)^(15/2) - 20790*(c*x + b)^(13/2
)*b + 61425*(c*x + b)^(11/2)*b^2 - 100100*(c*x + b)^(9/2)*b^3 + 96525*(c*x + b)^
(7/2)*b^4 - 54054*(c*x + b)^(5/2)*b^5 + 15015*(c*x + b)^(3/2)*b^6)/c^7) + 2/9009
*B*b*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 10010
*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 30
03*(c*x + b)^(3/2)*b^5)/c^6) + 2/9009*A*c*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13
/2) - 4095*(c*x + b)^(11/2)*b + 10010*(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*b*(
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)