3.20 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(7*b*x^6) + (4*c*(b*x + c*x^2)^(5/2))/(35*b^2*x^5)

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Rubi [A]  time = 0.0605686, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 c \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^6,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(7*b*x^6) + (4*c*(b*x + c*x^2)^(5/2))/(35*b^2*x^5)

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Rubi in Sympy [A]  time = 6.04025, size = 42, normalized size = 0.88 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 b x^{6}} + \frac{4 c \left (b x + c x^{2}\right )^{\frac{5}{2}}}{35 b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**6,x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(7*b*x**6) + 4*c*(b*x + c*x**2)**(5/2)/(35*b**2*x**5)

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Mathematica [A]  time = 0.0324389, size = 29, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{5/2} (2 c x-5 b)}{35 b^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(-5*b + 2*c*x))/(35*b^2*x^6)

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Maple [A]  time = 0.005, size = 33, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+5\,b \right ) }{35\,{b}^{2}{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^6,x)

[Out]

-2/35*(c*x+b)*(-2*c*x+5*b)*(c*x^2+b*x)^(3/2)/b^2/x^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221019, size = 66, normalized size = 1.38 \[ \frac{2 \,{\left (2 \, c^{3} x^{3} - b c^{2} x^{2} - 8 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(b + c*x))**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.21545, size = 223, normalized size = 4.65 \[ \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 98 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 5 \, b^{5}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^6,x, algorithm="giac")

[Out]

2/35*(35*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*c^(5/2) + 105*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^4*b*c^2 + 140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*c^(3/2) + 98*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b^3*c + 35*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*sqrt(
c) + 5*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^7