3.135 \(\int \frac{A+B x}{x \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*A)/(5*b*x*(b*x + c*x^2)^(3/2)) - (2*(5*b*B - 8*A*c)*(b + 2*c*x))/(15*b^3*(b*
x + c*x^2)^(3/2)) + (16*c*(5*b*B - 8*A*c)*(b + 2*c*x))/(15*b^5*Sqrt[b*x + c*x^2]
)

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Rubi [A]  time = 0.169657, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{16 c (b+2 c x) (5 b B-8 A c)}{15 b^5 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (5 b B-8 A c)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{5 b x \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(5*b*x*(b*x + c*x^2)^(3/2)) - (2*(5*b*B - 8*A*c)*(b + 2*c*x))/(15*b^3*(b*
x + c*x^2)^(3/2)) + (16*c*(5*b*B - 8*A*c)*(b + 2*c*x))/(15*b^5*Sqrt[b*x + c*x^2]
)

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Rubi in Sympy [A]  time = 9.91328, size = 94, normalized size = 0.98 \[ - \frac{2 A}{5 b x \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (b + 2 c x\right ) \left (8 A c - 5 B b\right )}{15 b^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{8 c \left (2 b + 4 c x\right ) \left (8 A c - 5 B b\right )}{15 b^{5} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x/(c*x**2+b*x)**(5/2),x)

[Out]

-2*A/(5*b*x*(b*x + c*x**2)**(3/2)) + 2*(b + 2*c*x)*(8*A*c - 5*B*b)/(15*b**3*(b*x
+ c*x**2)**(3/2)) - 8*c*(2*b + 4*c*x)*(8*A*c - 5*B*b)/(15*b**5*sqrt(b*x + c*x**
2))

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Mathematica [A]  time = 0.132455, size = 98, normalized size = 1.02 \[ -\frac{2 \left (A \left (3 b^4-8 b^3 c x+48 b^2 c^2 x^2+192 b c^3 x^3+128 c^4 x^4\right )+5 b B x \left (b^3-6 b^2 c x-24 b c^2 x^2-16 c^3 x^3\right )\right )}{15 b^5 x (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5*b*B*x*(b^3 - 6*b^2*c*x - 24*b*c^2*x^2 - 16*c^3*x^3) + A*(3*b^4 - 8*b^3*c*
x + 48*b^2*c^2*x^2 + 192*b*c^3*x^3 + 128*c^4*x^4)))/(15*b^5*x*(x*(b + c*x))^(3/2
))

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Maple [A]  time = 0.008, size = 107, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,A{c}^{4}{x}^{4}-80\,Bb{c}^{3}{x}^{4}+192\,Ab{c}^{3}{x}^{3}-120\,B{b}^{2}{c}^{2}{x}^{3}+48\,A{b}^{2}{c}^{2}{x}^{2}-30\,B{b}^{3}c{x}^{2}-8\,A{b}^{3}cx+5\,{b}^{4}Bx+3\,A{b}^{4} \right ) }{15\,{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*x+b)*(128*A*c^4*x^4-80*B*b*c^3*x^4+192*A*b*c^3*x^3-120*B*b^2*c^2*x^3+48
*A*b^2*c^2*x^2-30*B*b^3*c*x^2-8*A*b^3*c*x+5*B*b^4*x+3*A*b^4)/b^5/(c*x^2+b*x)^(5/
2)

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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((B*x + A)/((c*x^2 + b*x)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298233, size = 158, normalized size = 1.65 \[ -\frac{2 \,{\left (3 \, A b^{4} - 16 \,{\left (5 \, B b c^{3} - 8 \, A c^{4}\right )} x^{4} - 24 \,{\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{3} - 6 \,{\left (5 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{2} +{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} x\right )}}{15 \,{\left (b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3*A*b^4 - 16*(5*B*b*c^3 - 8*A*c^4)*x^4 - 24*(5*B*b^2*c^2 - 8*A*b*c^3)*x^3
- 6*(5*B*b^3*c - 8*A*b^2*c^2)*x^2 + (5*B*b^4 - 8*A*b^3*c)*x)/((b^5*c*x^3 + b^6*
x^2)*sqrt(c*x^2 + b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((A + B*x)/(x*(x*(b + c*x))**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x), x)