3.82 \(\int x^{7/2} \left (b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

[Out]

(-512*b^5*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (256*b^4*(b*x + c*x^2)^(5/2
))/(9009*c^5*x^(3/2)) - (64*b^3*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (32*b^
2*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (4*b*x^(3/2)*(b*x + c*x^2)^(5/2))/(39
*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi [A]  time = 0.211741, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac{64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac{4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(b*x + c*x^2)^(3/2),x]

[Out]

(-512*b^5*(b*x + c*x^2)^(5/2))/(45045*c^6*x^(5/2)) + (256*b^4*(b*x + c*x^2)^(5/2
))/(9009*c^5*x^(3/2)) - (64*b^3*(b*x + c*x^2)^(5/2))/(1287*c^4*Sqrt[x]) + (32*b^
2*Sqrt[x]*(b*x + c*x^2)^(5/2))/(429*c^3) - (4*b*x^(3/2)*(b*x + c*x^2)^(5/2))/(39
*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(5/2))/(15*c)

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Rubi in Sympy [A]  time = 23.6539, size = 155, normalized size = 0.95 \[ - \frac{512 b^{5} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{45045 c^{6} x^{\frac{5}{2}}} + \frac{256 b^{4} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{9009 c^{5} x^{\frac{3}{2}}} - \frac{64 b^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{1287 c^{4} \sqrt{x}} + \frac{32 b^{2} \sqrt{x} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{429 c^{3}} - \frac{4 b x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{39 c^{2}} + \frac{2 x^{\frac{5}{2}} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(c*x**2+b*x)**(3/2),x)

[Out]

-512*b**5*(b*x + c*x**2)**(5/2)/(45045*c**6*x**(5/2)) + 256*b**4*(b*x + c*x**2)*
*(5/2)/(9009*c**5*x**(3/2)) - 64*b**3*(b*x + c*x**2)**(5/2)/(1287*c**4*sqrt(x))
+ 32*b**2*sqrt(x)*(b*x + c*x**2)**(5/2)/(429*c**3) - 4*b*x**(3/2)*(b*x + c*x**2)
**(5/2)/(39*c**2) + 2*x**(5/2)*(b*x + c*x**2)**(5/2)/(15*c)

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Mathematica [A]  time = 0.0519755, size = 75, normalized size = 0.46 \[ \frac{2 (x (b+c x))^{5/2} \left (-256 b^5+640 b^4 c x-1120 b^3 c^2 x^2+1680 b^2 c^3 x^3-2310 b c^4 x^4+3003 c^5 x^5\right )}{45045 c^6 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(b*x + c*x^2)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-256*b^5 + 640*b^4*c*x - 1120*b^3*c^2*x^2 + 1680*b^2*c^3
*x^3 - 2310*b*c^4*x^4 + 3003*c^5*x^5))/(45045*c^6*x^(5/2))

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Maple [A]  time = 0.007, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3003\,{x}^{5}{c}^{5}+2310\,b{x}^{4}{c}^{4}-1680\,{b}^{2}{x}^{3}{c}^{3}+1120\,{b}^{3}{x}^{2}{c}^{2}-640\,{b}^{4}xc+256\,{b}^{5} \right ) }{45045\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(c*x^2+b*x)^(3/2),x)

[Out]

-2/45045*(c*x+b)*(-3003*c^5*x^5+2310*b*c^4*x^4-1680*b^2*c^3*x^3+1120*b^3*c^2*x^2
-640*b^4*c*x+256*b^5)*(c*x^2+b*x)^(3/2)/c^6/x^(3/2)

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Maxima [A]  time = 0.715546, size = 227, normalized size = 1.38 \[ \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 5 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5}\right )} \sqrt{c x + b}}{45045 \, c^{6} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(7/2),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 + 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)/(c^6*x^6)

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Fricas [A]  time = 0.219893, size = 144, normalized size = 0.88 \[ \frac{2 \,{\left (3003 \, c^{8} x^{9} + 6699 \, b c^{7} x^{8} + 3759 \, b^{2} c^{6} x^{7} - 7 \, b^{3} c^{5} x^{6} + 10 \, b^{4} c^{4} x^{5} - 16 \, b^{5} c^{3} x^{4} + 32 \, b^{6} c^{2} x^{3} - 128 \, b^{7} c x^{2} - 256 \, b^{8} x\right )}}{45045 \, \sqrt{c x^{2} + b x} c^{6} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(7/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^8*x^9 + 6699*b*c^7*x^8 + 3759*b^2*c^6*x^7 - 7*b^3*c^5*x^6 + 10*b
^4*c^4*x^5 - 16*b^5*c^3*x^4 + 32*b^6*c^2*x^3 - 128*b^7*c*x^2 - 256*b^8*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)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21773, size = 246, normalized size = 1.5 \[ -\frac{2}{45045} \, c{\left (\frac{1024 \, b^{\frac{15}{2}}}{c^{7}} - \frac{3003 \,{\left (c x + b\right )}^{\frac{15}{2}} - 20790 \,{\left (c x + b\right )}^{\frac{13}{2}} b + 61425 \,{\left (c x + b\right )}^{\frac{11}{2}} b^{2} - 100100 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{3} + 96525 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{4} - 54054 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{5} + 15015 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{6}}{c^{7}}\right )} + \frac{2}{9009} \, b{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(7/2),x, algorithm="giac")

[Out]

-2/45045*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
*(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)