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

Optimal. Leaf size=80 \[ -\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0924031, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{16 b^2 \sqrt{x}}{3 c^3 \sqrt{b x+c x^2}}-\frac{8 b x^{3/2}}{3 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{5/2}}{3 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.25969, size = 73, normalized size = 0.91 \[ - \frac{16 b^{2} \sqrt{x}}{3 c^{3} \sqrt{b x + c x^{2}}} - \frac{8 b x^{\frac{3}{2}}}{3 c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{5}{2}}}{3 c \sqrt{b x + c x^{2}}} \]

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]

-16*b**2*sqrt(x)/(3*c**3*sqrt(b*x + c*x**2)) - 8*b*x**(3/2)/(3*c**2*sqrt(b*x + c
*x**2)) + 2*x**(5/2)/(3*c*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.026972, size = 41, normalized size = 0.51 \[ \frac{2 \sqrt{x} \left (-8 b^2-4 b c x+c^2 x^2\right )}{3 c^3 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(-8*b^2 - 4*b*c*x + c^2*x^2))/(3*c^3*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.008, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -{c}^{2}{x}^{2}+4\,bcx+8\,{b}^{2} \right ) }{3\,{c}^{3}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*x+b)*(-c^2*x^2+4*b*c*x+8*b^2)*x^(3/2)/c^3/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.737561, size = 120, normalized size = 1.5 \[ \frac{2 \,{\left ({\left (c^{3} x^{2} - b c^{2} x - 2 \, b^{2} c\right )} x^{2} - 2 \,{\left (b c^{2} x^{2} + 2 \, b^{2} c x + b^{3}\right )} x\right )}}{3 \,{\left (c^{4} x^{2} + b c^{3} x\right )} \sqrt{c x + b}} - \frac{4 \, b^{2}}{\sqrt{c x + b} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((c^3*x^2 - b*c^2*x - 2*b^2*c)*x^2 - 2*(b*c^2*x^2 + 2*b^2*c*x + b^3)*x)/((c^
4*x^2 + b*c^3*x)*sqrt(c*x + b)) - 4*b^2/(sqrt(c*x + b)*c^3)

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Fricas [A]  time = 0.225242, size = 54, normalized size = 0.68 \[ \frac{2 \,{\left (c^{2} x^{3} - 4 \, b c x^{2} - 8 \, b^{2} x\right )}}{3 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c^2*x^3 - 4*b*c*x^2 - 8*b^2*x)/(sqrt(c*x^2 + b*x)*c^3*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 = 0.211467, size = 59, normalized size = 0.74 \[ \frac{16 \, b^{\frac{3}{2}}}{3 \, c^{3}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 6 \, \sqrt{c x + b} b - \frac{3 \, b^{2}}{\sqrt{c x + b}}\right )}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/3*b^(3/2)/c^3 + 2/3*((c*x + b)^(3/2) - 6*sqrt(c*x + b)*b - 3*b^2/sqrt(c*x + b
))/c^3