3.16 \(\int \frac{1}{\left (b x+c x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{256 c^2 (b+2 c x)}{15 b^6 \sqrt{b x+c x^2}}+\frac{32 c (b+2 c x)}{15 b^4 \left (b x+c x^2\right )^{3/2}}-\frac{2 (b+2 c x)}{5 b^2 \left (b x+c x^2\right )^{5/2}} \]

[Out]

(-2*(b + 2*c*x))/(5*b^2*(b*x + c*x^2)^(5/2)) + (32*c*(b + 2*c*x))/(15*b^4*(b*x +
c*x^2)^(3/2)) - (256*c^2*(b + 2*c*x))/(15*b^6*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0519505, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{256 c^2 (b+2 c x)}{15 b^6 \sqrt{b x+c x^2}}+\frac{32 c (b+2 c x)}{15 b^4 \left (b x+c x^2\right )^{3/2}}-\frac{2 (b+2 c x)}{5 b^2 \left (b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-7/2),x]

[Out]

(-2*(b + 2*c*x))/(5*b^2*(b*x + c*x^2)^(5/2)) + (32*c*(b + 2*c*x))/(15*b^4*(b*x +
c*x^2)^(3/2)) - (256*c^2*(b + 2*c*x))/(15*b^6*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 4.94692, size = 82, normalized size = 0.99 \[ - \frac{2 \left (b + 2 c x\right )}{5 b^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}} + \frac{32 c \left (b + 2 c x\right )}{15 b^{4} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{128 c^{2} \left (2 b + 4 c x\right )}{15 b^{6} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b + 2*c*x)/(5*b**2*(b*x + c*x**2)**(5/2)) + 32*c*(b + 2*c*x)/(15*b**4*(b*x +
c*x**2)**(3/2)) - 128*c**2*(2*b + 4*c*x)/(15*b**6*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.06572, size = 70, normalized size = 0.84 \[ -\frac{2 \left (3 b^5-10 b^4 c x+80 b^3 c^2 x^2+480 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right )}{15 b^6 (x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3*b^5 - 10*b^4*c*x + 80*b^3*c^2*x^2 + 480*b^2*c^3*x^3 + 640*b*c^4*x^4 + 256
*c^5*x^5))/(15*b^6*(x*(b + c*x))^(5/2))

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Maple [A]  time = 0.006, size = 75, normalized size = 0.9 \[ -{\frac{2\,x \left ( cx+b \right ) \left ( 256\,{c}^{5}{x}^{5}+640\,{c}^{4}{x}^{4}b+480\,{c}^{3}{x}^{3}{b}^{2}+80\,{c}^{2}{x}^{2}{b}^{3}-10\,cx{b}^{4}+3\,{b}^{5} \right ) }{15\,{b}^{6}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(7/2),x)

[Out]

-2/15*x*(c*x+b)*(256*c^5*x^5+640*b*c^4*x^4+480*b^2*c^3*x^3+80*b^3*c^2*x^2-10*b^4
*c*x+3*b^5)/b^6/(c*x^2+b*x)^(7/2)

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Maxima [A]  time = 0.738817, size = 150, normalized size = 1.81 \[ -\frac{4 \, c x}{5 \,{\left (c x^{2} + b x\right )}^{\frac{5}{2}} b^{2}} + \frac{64 \, c^{2} x}{15 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{4}} - \frac{512 \, c^{3} x}{15 \, \sqrt{c x^{2} + b x} b^{6}} - \frac{2}{5 \,{\left (c x^{2} + b x\right )}^{\frac{5}{2}} b} + \frac{32 \, c}{15 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{3}} - \frac{256 \, c^{2}}{15 \, \sqrt{c x^{2} + b x} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/5*c*x/((c*x^2 + b*x)^(5/2)*b^2) + 64/15*c^2*x/((c*x^2 + b*x)^(3/2)*b^4) - 512
/15*c^3*x/(sqrt(c*x^2 + b*x)*b^6) - 2/5/((c*x^2 + b*x)^(5/2)*b) + 32/15*c/((c*x^
2 + b*x)^(3/2)*b^3) - 256/15*c^2/(sqrt(c*x^2 + b*x)*b^5)

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Fricas [A]  time = 0.216603, size = 127, normalized size = 1.53 \[ -\frac{2 \,{\left (256 \, c^{5} x^{5} + 640 \, b c^{4} x^{4} + 480 \, b^{2} c^{3} x^{3} + 80 \, b^{3} c^{2} x^{2} - 10 \, b^{4} c x + 3 \, b^{5}\right )}}{15 \,{\left (b^{6} c^{2} x^{4} + 2 \, b^{7} c x^{3} + b^{8} x^{2}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(256*c^5*x^5 + 640*b*c^4*x^4 + 480*b^2*c^3*x^3 + 80*b^3*c^2*x^2 - 10*b^4*c
*x + 3*b^5)/((b^6*c^2*x^4 + 2*b^7*c*x^3 + b^8*x^2)*sqrt(c*x^2 + b*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b*x + c*x**2)**(-7/2), x)

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GIAC/XCAS [A]  time = 0.222061, size = 100, normalized size = 1.2 \[ -\frac{2 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \, x{\left (\frac{2 \, c^{5} x}{b^{6}} + \frac{5 \, c^{4}}{b^{5}}\right )} + \frac{15 \, c^{3}}{b^{4}}\right )} x + \frac{5 \, c^{2}}{b^{3}}\right )} x - \frac{5 \, c}{b^{2}}\right )} x + \frac{3}{b}\right )}}{15 \,{\left (c x^{2} + b x\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(2*(8*(2*(4*x*(2*c^5*x/b^6 + 5*c^4/b^5) + 15*c^3/b^4)*x + 5*c^2/b^3)*x - 5
*c/b^2)*x + 3/b)/(c*x^2 + b*x)^(5/2)