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

Optimal. Leaf size=136 \[ \frac{256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]

[Out]

(256*b^4*(b*x + c*x^2)^(3/2))/(3465*c^5*x^(3/2)) - (128*b^3*(b*x + c*x^2)^(3/2))
/(1155*c^4*Sqrt[x]) + (32*b^2*Sqrt[x]*(b*x + c*x^2)^(3/2))/(231*c^3) - (16*b*x^(
3/2)*(b*x + c*x^2)^(3/2))/(99*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(3/2))/(11*c)

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Rubi [A]  time = 0.16646, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(256*b^4*(b*x + c*x^2)^(3/2))/(3465*c^5*x^(3/2)) - (128*b^3*(b*x + c*x^2)^(3/2))
/(1155*c^4*Sqrt[x]) + (32*b^2*Sqrt[x]*(b*x + c*x^2)^(3/2))/(231*c^3) - (16*b*x^(
3/2)*(b*x + c*x^2)^(3/2))/(99*c^2) + (2*x^(5/2)*(b*x + c*x^2)^(3/2))/(11*c)

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Rubi in Sympy [A]  time = 18.2275, size = 128, normalized size = 0.94 \[ \frac{256 b^{4} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3465 c^{5} x^{\frac{3}{2}}} - \frac{128 b^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1155 c^{4} \sqrt{x}} + \frac{32 b^{2} \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{231 c^{3}} - \frac{16 b x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{99 c^{2}} + \frac{2 x^{\frac{5}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{11 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*b**4*(b*x + c*x**2)**(3/2)/(3465*c**5*x**(3/2)) - 128*b**3*(b*x + c*x**2)**(
3/2)/(1155*c**4*sqrt(x)) + 32*b**2*sqrt(x)*(b*x + c*x**2)**(3/2)/(231*c**3) - 16
*b*x**(3/2)*(b*x + c*x**2)**(3/2)/(99*c**2) + 2*x**(5/2)*(b*x + c*x**2)**(3/2)/(
11*c)

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Mathematica [A]  time = 0.0324837, size = 75, normalized size = 0.55 \[ \frac{2 \sqrt{x (b+c x)} \left (128 b^5-64 b^4 c x+48 b^3 c^2 x^2-40 b^2 c^3 x^3+35 b c^4 x^4+315 c^5 x^5\right )}{3465 c^5 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x*(b + c*x)]*(128*b^5 - 64*b^4*c*x + 48*b^3*c^2*x^2 - 40*b^2*c^3*x^3 + 3
5*b*c^4*x^4 + 315*c^5*x^5))/(3465*c^5*Sqrt[x])

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Maple [A]  time = 0.008, size = 66, normalized size = 0.5 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 315\,{x}^{4}{c}^{4}-280\,b{x}^{3}{c}^{3}+240\,{b}^{2}{x}^{2}{c}^{2}-192\,{b}^{3}xc+128\,{b}^{4} \right ) }{3465\,{c}^{5}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(c*x+b)*(315*c^4*x^4-280*b*c^3*x^3+240*b^2*c^2*x^2-192*b^3*c*x+128*b^4)*(
c*x^2+b*x)^(1/2)/c^5/x^(1/2)

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Maxima [A]  time = 0.716341, size = 86, normalized size = 0.63 \[ \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b}}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*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)/c^5

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Fricas [A]  time = 0.221817, size = 115, normalized size = 0.85 \[ \frac{2 \,{\left (315 \, c^{6} x^{7} + 350 \, b c^{5} x^{6} - 5 \, b^{2} c^{4} x^{5} + 8 \, b^{3} c^{3} x^{4} - 16 \, b^{4} c^{2} x^{3} + 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )}}{3465 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*c^6*x^7 + 350*b*c^5*x^6 - 5*b^2*c^4*x^5 + 8*b^3*c^3*x^4 - 16*b^4*c^2
*x^3 + 64*b^5*c*x^2 + 128*b^6*x)/(sqrt(c*x^2 + b*x)*c^5*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213378, size = 95, normalized size = 0.7 \[ -\frac{256 \, b^{\frac{11}{2}}}{3465 \, c^{5}} + \frac{2 \,{\left (315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}\right )}}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256/3465*b^(11/2)/c^5 + 2/3465*(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