3.103 \(\int \frac{x^{11/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]

[Out]

(-256*b^4*Sqrt[x])/(35*c^5*Sqrt[b*x + c*x^2]) - (128*b^3*x^(3/2))/(35*c^4*Sqrt[b
*x + c*x^2]) + (32*b^2*x^(5/2))/(35*c^3*Sqrt[b*x + c*x^2]) - (16*b*x^(7/2))/(35*
c^2*Sqrt[b*x + c*x^2]) + (2*x^(9/2))/(7*c*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.170437, 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 \[ -\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(11/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-256*b^4*Sqrt[x])/(35*c^5*Sqrt[b*x + c*x^2]) - (128*b^3*x^(3/2))/(35*c^4*Sqrt[b
*x + c*x^2]) + (32*b^2*x^(5/2))/(35*c^3*Sqrt[b*x + c*x^2]) - (16*b*x^(7/2))/(35*
c^2*Sqrt[b*x + c*x^2]) + (2*x^(9/2))/(7*c*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 18.4632, size = 128, normalized size = 0.94 \[ - \frac{256 b^{4} \sqrt{x}}{35 c^{5} \sqrt{b x + c x^{2}}} - \frac{128 b^{3} x^{\frac{3}{2}}}{35 c^{4} \sqrt{b x + c x^{2}}} + \frac{32 b^{2} x^{\frac{5}{2}}}{35 c^{3} \sqrt{b x + c x^{2}}} - \frac{16 b x^{\frac{7}{2}}}{35 c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{9}{2}}}{7 c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(11/2)/(c*x**2+b*x)**(3/2),x)

[Out]

-256*b**4*sqrt(x)/(35*c**5*sqrt(b*x + c*x**2)) - 128*b**3*x**(3/2)/(35*c**4*sqrt
(b*x + c*x**2)) + 32*b**2*x**(5/2)/(35*c**3*sqrt(b*x + c*x**2)) - 16*b*x**(7/2)/
(35*c**2*sqrt(b*x + c*x**2)) + 2*x**(9/2)/(7*c*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0340494, size = 64, normalized size = 0.47 \[ \frac{2 \sqrt{x} \left (-128 b^4-64 b^3 c x+16 b^2 c^2 x^2-8 b c^3 x^3+5 c^4 x^4\right )}{35 c^5 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(11/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(-128*b^4 - 64*b^3*c*x + 16*b^2*c^2*x^2 - 8*b*c^3*x^3 + 5*c^4*x^4))/(
35*c^5*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.008, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{4}{c}^{4}+8\,b{x}^{3}{c}^{3}-16\,{b}^{2}{x}^{2}{c}^{2}+64\,{b}^{3}xc+128\,{b}^{4} \right ) }{35\,{c}^{5}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2/35*(c*x+b)*(-5*c^4*x^4+8*b*c^3*x^3-16*b^2*c^2*x^2+64*b^3*c*x+128*b^4)*x^(3/2)
/c^5/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.740722, size = 289, normalized size = 2.12 \[ \frac{2 \,{\left (3 \,{\left (5 \, c^{5} x^{4} - b c^{4} x^{3} + 2 \, b^{2} c^{3} x^{2} - 8 \, b^{3} c^{2} x - 16 \, b^{4} c\right )} x^{4} - 2 \,{\left (3 \, b c^{4} x^{4} - 2 \, b^{2} c^{3} x^{3} + 11 \, b^{3} c^{2} x^{2} + 40 \, b^{4} c x + 24 \, b^{5}\right )} x^{3} + 14 \,{\left (b^{2} c^{3} x^{4} - 2 \, b^{3} c^{2} x^{3} - 7 \, b^{4} c x^{2} - 4 \, b^{5} x\right )} x^{2} - 70 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{3} + b^{5} x^{2}\right )} x\right )}}{105 \,{\left (c^{6} x^{4} + b c^{5} x^{3}\right )} \sqrt{c x + b}} - \frac{4 \, b^{4}}{\sqrt{c x + b} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

2/105*(3*(5*c^5*x^4 - b*c^4*x^3 + 2*b^2*c^3*x^2 - 8*b^3*c^2*x - 16*b^4*c)*x^4 -
2*(3*b*c^4*x^4 - 2*b^2*c^3*x^3 + 11*b^3*c^2*x^2 + 40*b^4*c*x + 24*b^5)*x^3 + 14*
(b^2*c^3*x^4 - 2*b^3*c^2*x^3 - 7*b^4*c*x^2 - 4*b^5*x)*x^2 - 70*(b^3*c^2*x^4 + 2*
b^4*c*x^3 + b^5*x^2)*x)/((c^6*x^4 + b*c^5*x^3)*sqrt(c*x + b)) - 4*b^4/(sqrt(c*x
+ b)*c^5)

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Fricas [A]  time = 0.218607, size = 85, normalized size = 0.62 \[ \frac{2 \,{\left (5 \, c^{4} x^{5} - 8 \, b c^{3} x^{4} + 16 \, b^{2} c^{2} x^{3} - 64 \, b^{3} c x^{2} - 128 \, b^{4} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*c^4*x^5 - 8*b*c^3*x^4 + 16*b^2*c^2*x^3 - 64*b^3*c*x^2 - 128*b^4*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**(11/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212533, size = 95, normalized size = 0.7 \[ \frac{256 \, b^{\frac{7}{2}}}{35 \, c^{5}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 28 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 140 \, \sqrt{c x + b} b^{3} - \frac{35 \, b^{4}}{\sqrt{c x + b}}\right )}}{35 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

256/35*b^(7/2)/c^5 + 2/35*(5*(c*x + b)^(7/2) - 28*(c*x + b)^(5/2)*b + 70*(c*x +
b)^(3/2)*b^2 - 140*sqrt(c*x + b)*b^3 - 35*b^4/sqrt(c*x + b))/c^5