3.331 \(\int \frac{(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.110874, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(b*x + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.3552, size = 76, normalized size = 0.97 \[ - \frac{2 \left (b + 2 c x\right ) \left (d + e x\right )^{2}}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \left (2 b d - x \left (2 b e - 4 c d\right )\right ) \left (b e - 2 c d\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(c*x**2+b*x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.122241, size = 95, normalized size = 1.22 \[ \frac{-2 b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )+16 b c^2 d x^2 (3 d-2 e x)+32 c^3 d^2 x^3}{3 b^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(b*x + c*x^2)^(5/2),x]

[Out]

(32*c^3*d^2*x^3 + 16*b*c^2*d*x^2*(3*d - 2*e*x) - 2*b^3*(d^2 + 6*d*e*x - 3*e^2*x^
2) + 4*b^2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2))/(3*b^4*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.009, size = 117, normalized size = 1.5 \[ -{\frac{2\,x \left ( cx+b \right ) \left ( -2\,{b}^{2}c{e}^{2}{x}^{3}+16\,b{c}^{2}de{x}^{3}-16\,{c}^{3}{d}^{2}{x}^{3}-3\,{b}^{3}{e}^{2}{x}^{2}+24\,{b}^{2}cde{x}^{2}-24\,b{c}^{2}{d}^{2}{x}^{2}+6\,{b}^{3}dex-6\,{b}^{2}c{d}^{2}x+{d}^{2}{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3*x*(c*x+b)*(-2*b^2*c*e^2*x^3+16*b*c^2*d*e*x^3-16*c^3*d^2*x^3-3*b^3*e^2*x^2+2
4*b^2*c*d*e*x^2-24*b*c^2*d^2*x^2+6*b^3*d*e*x-6*b^2*c*d^2*x+b^3*d^2)/b^4/(c*x^2+b
*x)^(5/2)

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Maxima [A]  time = 0.695403, size = 274, normalized size = 3.51 \[ -\frac{4 \, c d^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} d^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} + \frac{4 \, d e x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{32 \, c d e x}{3 \, \sqrt{c x^{2} + b x} b^{3}} + \frac{4 \, e^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, e^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{2 \, d^{2}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c d^{2}}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{16 \, d e}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, e^{2}}{3 \, \sqrt{c x^{2} + b x} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

-4/3*c*d^2*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d^2*x/(sqrt(c*x^2 + b*x)*b^4)
+ 4/3*d*e*x/((c*x^2 + b*x)^(3/2)*b) - 32/3*c*d*e*x/(sqrt(c*x^2 + b*x)*b^3) + 4/3
*e^2*x/(sqrt(c*x^2 + b*x)*b^2) - 2/3*e^2*x/((c*x^2 + b*x)^(3/2)*c) - 2/3*d^2/((c
*x^2 + b*x)^(3/2)*b) + 16/3*c*d^2/(sqrt(c*x^2 + b*x)*b^3) - 16/3*d*e/(sqrt(c*x^2
+ b*x)*b^2) + 2/3*e^2/(sqrt(c*x^2 + b*x)*b*c)

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Fricas [A]  time = 0.217537, size = 157, normalized size = 2.01 \[ -\frac{2 \,{\left (b^{3} d^{2} - 2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 6 \,{\left (b^{2} c d^{2} - b^{3} d e\right )} x\right )}}{3 \,{\left (b^{4} c x^{2} + b^{5} x\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((d + e*x)**2/(x*(b + c*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.218134, size = 166, normalized size = 2.13 \[ \frac{{\left (x{\left (\frac{2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{6 \,{\left (b^{2} c d^{2} - b^{3} d e\right )}}{b^{4} c^{2}}\right )} x - \frac{d^{2}}{b c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*x^2 + b*x)^(5/2),x, algorithm="giac")

[Out]

1/3*((x*(2*(8*c^3*d^2 - 8*b*c^2*d*e + b^2*c*e^2)*x/(b^4*c^2) + 3*(8*b*c^2*d^2 -
8*b^2*c*d*e + b^3*e^2)/(b^4*c^2)) + 6*(b^2*c*d^2 - b^3*d*e)/(b^4*c^2))*x - d^2/(
b*c^2))/(c*x^2 + b*x)^(3/2)