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

Optimal. Leaf size=108 \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]

[Out]

(-32*b^3*Sqrt[b*x + c*x^2])/(35*c^4*Sqrt[x]) + (16*b^2*Sqrt[x]*Sqrt[b*x + c*x^2]
)/(35*c^3) - (12*b*x^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*x^(5/2)*Sqrt[b*x + c
*x^2])/(7*c)

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Rubi [A]  time = 0.129986, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-32*b^3*Sqrt[b*x + c*x^2])/(35*c^4*Sqrt[x]) + (16*b^2*Sqrt[x]*Sqrt[b*x + c*x^2]
)/(35*c^3) - (12*b*x^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*x^(5/2)*Sqrt[b*x + c
*x^2])/(7*c)

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Rubi in Sympy [A]  time = 13.6826, size = 100, normalized size = 0.93 \[ - \frac{32 b^{3} \sqrt{b x + c x^{2}}}{35 c^{4} \sqrt{x}} + \frac{16 b^{2} \sqrt{x} \sqrt{b x + c x^{2}}}{35 c^{3}} - \frac{12 b x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{35 c^{2}} + \frac{2 x^{\frac{5}{2}} \sqrt{b x + c x^{2}}}{7 c} \]

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]

-32*b**3*sqrt(b*x + c*x**2)/(35*c**4*sqrt(x)) + 16*b**2*sqrt(x)*sqrt(b*x + c*x**
2)/(35*c**3) - 12*b*x**(3/2)*sqrt(b*x + c*x**2)/(35*c**2) + 2*x**(5/2)*sqrt(b*x
+ c*x**2)/(7*c)

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Mathematica [A]  time = 0.0370991, size = 53, normalized size = 0.49 \[ \frac{2 \sqrt{x (b+c x)} \left (-16 b^3+8 b^2 c x-6 b c^2 x^2+5 c^3 x^3\right )}{35 c^4 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x*(b + c*x)]*(-16*b^3 + 8*b^2*c*x - 6*b*c^2*x^2 + 5*c^3*x^3))/(35*c^4*Sq
rt[x])

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Maple [A]  time = 0.009, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{3}{c}^{3}+6\,b{x}^{2}{c}^{2}-8\,{b}^{2}xc+16\,{b}^{3} \right ) }{35\,{c}^{4}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*x+b)*(-5*c^3*x^3+6*b*c^2*x^2-8*b^2*c*x+16*b^3)*x^(1/2)/c^4/(c*x^2+b*x)^
(1/2)

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Maxima [A]  time = 0.707242, size = 72, normalized size = 0.67 \[ \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )}}{35 \, \sqrt{c x + b} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221352, size = 85, normalized size = 0.79 \[ \frac{2 \,{\left (5 \, c^{4} x^{5} - b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} - 8 \, b^{3} c x^{2} - 16 \, b^{4} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*c^4*x^5 - b*c^3*x^4 + 2*b^2*c^2*x^3 - 8*b^3*c*x^2 - 16*b^4*x)/(sqrt(c*x^
2 + b*x)*c^4*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 = 0.210632, size = 78, normalized size = 0.72 \[ \frac{32 \, b^{\frac{7}{2}}}{35 \, c^{4}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 21 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 35 \, \sqrt{c x + b} b^{3}\right )}}{35 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/35*b^(7/2)/c^4 + 2/35*(5*(c*x + b)^(7/2) - 21*(c*x + b)^(5/2)*b + 35*(c*x + b
)^(3/2)*b^2 - 35*sqrt(c*x + b)*b^3)/c^4