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

Optimal. Leaf size=126 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(11*b*x^7) + (16*c*(b*x + c*x^2)^(3/2))/(99*b^2*x^6) -
(32*c^2*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) + (128*c^3*(b*x + c*x^2)^(3/2))/(1155
*b^4*x^4) - (256*c^4*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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Rubi [A]  time = 0.173211, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^7,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(11*b*x^7) + (16*c*(b*x + c*x^2)^(3/2))/(99*b^2*x^6) -
(32*c^2*(b*x + c*x^2)^(3/2))/(231*b^3*x^5) + (128*c^3*(b*x + c*x^2)^(3/2))/(1155
*b^4*x^4) - (256*c^4*(b*x + c*x^2)^(3/2))/(3465*b^5*x^3)

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Rubi in Sympy [A]  time = 18.654, size = 119, normalized size = 0.94 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{11 b x^{7}} + \frac{16 c \left (b x + c x^{2}\right )^{\frac{3}{2}}}{99 b^{2} x^{6}} - \frac{32 c^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{231 b^{3} x^{5}} + \frac{128 c^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1155 b^{4} x^{4}} - \frac{256 c^{4} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3465 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b*x + c*x**2)**(3/2)/(11*b*x**7) + 16*c*(b*x + c*x**2)**(3/2)/(99*b**2*x**6)
- 32*c**2*(b*x + c*x**2)**(3/2)/(231*b**3*x**5) + 128*c**3*(b*x + c*x**2)**(3/2
)/(1155*b**4*x**4) - 256*c**4*(b*x + c*x**2)**(3/2)/(3465*b**5*x**3)

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Mathematica [A]  time = 0.0305571, size = 73, normalized size = 0.58 \[ -\frac{2 \sqrt{x (b+c x)} \left (315 b^5+35 b^4 c x-40 b^3 c^2 x^2+48 b^2 c^3 x^3-64 b c^4 x^4+128 c^5 x^5\right )}{3465 b^5 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x*(b + c*x)]*(315*b^5 + 35*b^4*c*x - 40*b^3*c^2*x^2 + 48*b^2*c^3*x^3 -
64*b*c^4*x^4 + 128*c^5*x^5))/(3465*b^5*x^6)

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Maple [A]  time = 0.009, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,{c}^{4}{x}^{4}-192\,{x}^{3}{c}^{3}b+240\,{c}^{2}{x}^{2}{b}^{2}-280\,cx{b}^{3}+315\,{b}^{4} \right ) }{3465\,{x}^{6}{b}^{5}}\sqrt{c{x}^{2}+bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^7,x)

[Out]

-2/3465*(c*x+b)*(128*c^4*x^4-192*b*c^3*x^3+240*b^2*c^2*x^2-280*b^3*c*x+315*b^4)*
(c*x^2+b*x)^(1/2)/x^6/b^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(sqrt(c*x^2 + b*x)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220348, size = 96, normalized size = 0.76 \[ -\frac{2 \,{\left (128 \, c^{5} x^{5} - 64 \, b c^{4} x^{4} + 48 \, b^{2} c^{3} x^{3} - 40 \, b^{3} c^{2} x^{2} + 35 \, b^{4} c x + 315 \, b^{5}\right )} \sqrt{c x^{2} + b x}}{3465 \, b^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3465*(128*c^5*x^5 - 64*b*c^4*x^4 + 48*b^2*c^3*x^3 - 40*b^3*c^2*x^2 + 35*b^4*c
*x + 315*b^5)*sqrt(c*x^2 + b*x)/(b^5*x^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))/x**7, x)

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GIAC/XCAS [A]  time = 0.215253, size = 262, normalized size = 2.08 \[ \frac{2 \,{\left (11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} c^{3} + 36960 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} b c^{\frac{5}{2}} + 51480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b^{2} c^{2} + 38115 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{3} c^{\frac{3}{2}} + 15785 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{4} c + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{5} \sqrt{c} + 315 \, b^{6}\right )}}{3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(11088*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*c^3 + 36960*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^5*b*c^(5/2) + 51480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^2*c^2 + 3811
5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^3*c^(3/2) + 15785*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*b^4*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^5*sqrt(c) + 315*b^6)/(s
qrt(c)*x - sqrt(c*x^2 + b*x))^11