3.137 \(\int \frac{A+B x}{x^3 \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{256 c^3 (b+2 c x) (3 b B-4 A c)}{63 b^7 \sqrt{b x+c x^2}}-\frac{32 c^2 (b+2 c x) (3 b B-4 A c)}{63 b^5 \left (b x+c x^2\right )^{3/2}}+\frac{4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac{2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*A)/(9*b*x^3*(b*x + c*x^2)^(3/2)) - (2*(3*b*B - 4*A*c))/(21*b^2*x^2*(b*x + c*
x^2)^(3/2)) + (4*c*(3*b*B - 4*A*c))/(21*b^3*x*(b*x + c*x^2)^(3/2)) - (32*c^2*(3*
b*B - 4*A*c)*(b + 2*c*x))/(63*b^5*(b*x + c*x^2)^(3/2)) + (256*c^3*(3*b*B - 4*A*c
)*(b + 2*c*x))/(63*b^7*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.346902, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{256 c^3 (b+2 c x) (3 b B-4 A c)}{63 b^7 \sqrt{b x+c x^2}}-\frac{32 c^2 (b+2 c x) (3 b B-4 A c)}{63 b^5 \left (b x+c x^2\right )^{3/2}}+\frac{4 c (3 b B-4 A c)}{21 b^3 x \left (b x+c x^2\right )^{3/2}}-\frac{2 (3 b B-4 A c)}{21 b^2 x^2 \left (b x+c x^2\right )^{3/2}}-\frac{2 A}{9 b x^3 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^3*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*A)/(9*b*x^3*(b*x + c*x^2)^(3/2)) - (2*(3*b*B - 4*A*c))/(21*b^2*x^2*(b*x + c*
x^2)^(3/2)) + (4*c*(3*b*B - 4*A*c))/(21*b^3*x*(b*x + c*x^2)^(3/2)) - (32*c^2*(3*
b*B - 4*A*c)*(b + 2*c*x))/(63*b^5*(b*x + c*x^2)^(3/2)) + (256*c^3*(3*b*B - 4*A*c
)*(b + 2*c*x))/(63*b^7*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 20.5959, size = 163, normalized size = 0.98 \[ - \frac{2 A}{9 b x^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (4 A c - 3 B b\right )}{21 b^{2} x^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 c \left (4 A c - 3 B b\right )}{21 b^{3} x \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{32 c^{2} \left (b + 2 c x\right ) \left (4 A c - 3 B b\right )}{63 b^{5} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{128 c^{3} \left (2 b + 4 c x\right ) \left (4 A c - 3 B b\right )}{63 b^{7} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**3/(c*x**2+b*x)**(5/2),x)

[Out]

-2*A/(9*b*x**3*(b*x + c*x**2)**(3/2)) + 2*(4*A*c - 3*B*b)/(21*b**2*x**2*(b*x + c
*x**2)**(3/2)) - 4*c*(4*A*c - 3*B*b)/(21*b**3*x*(b*x + c*x**2)**(3/2)) + 32*c**2
*(b + 2*c*x)*(4*A*c - 3*B*b)/(63*b**5*(b*x + c*x**2)**(3/2)) - 128*c**3*(2*b + 4
*c*x)*(4*A*c - 3*B*b)/(63*b**7*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.208849, size = 145, normalized size = 0.87 \[ \frac{6 b B x \left (-3 b^5+6 b^4 c x-16 b^3 c^2 x^2+96 b^2 c^3 x^3+384 b c^4 x^4+256 c^5 x^5\right )-2 A \left (7 b^6-12 b^5 c x+24 b^4 c^2 x^2-64 b^3 c^3 x^3+384 b^2 c^4 x^4+1536 b c^5 x^5+1024 c^6 x^6\right )}{63 b^7 x^3 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^3*(b*x + c*x^2)^(5/2)),x]

[Out]

(6*b*B*x*(-3*b^5 + 6*b^4*c*x - 16*b^3*c^2*x^2 + 96*b^2*c^3*x^3 + 384*b*c^4*x^4 +
256*c^5*x^5) - 2*A*(7*b^6 - 12*b^5*c*x + 24*b^4*c^2*x^2 - 64*b^3*c^3*x^3 + 384*
b^2*c^4*x^4 + 1536*b*c^5*x^5 + 1024*c^6*x^6))/(63*b^7*x^3*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.01, size = 158, normalized size = 1. \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 1024\,A{c}^{6}{x}^{6}-768\,Bb{c}^{5}{x}^{6}+1536\,Ab{c}^{5}{x}^{5}-1152\,B{b}^{2}{c}^{4}{x}^{5}+384\,A{b}^{2}{c}^{4}{x}^{4}-288\,B{b}^{3}{c}^{3}{x}^{4}-64\,A{b}^{3}{c}^{3}{x}^{3}+48\,B{b}^{4}{c}^{2}{x}^{3}+24\,A{b}^{4}{c}^{2}{x}^{2}-18\,B{b}^{5}c{x}^{2}-12\,A{b}^{5}cx+9\,{b}^{6}Bx+7\,A{b}^{6} \right ) }{63\,{x}^{2}{b}^{7}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/(c*x^2+b*x)^(5/2),x)

[Out]

-2/63*(c*x+b)*(1024*A*c^6*x^6-768*B*b*c^5*x^6+1536*A*b*c^5*x^5-1152*B*b^2*c^4*x^
5+384*A*b^2*c^4*x^4-288*B*b^3*c^3*x^4-64*A*b^3*c^3*x^3+48*B*b^4*c^2*x^3+24*A*b^4
*c^2*x^2-18*B*b^5*c*x^2-12*A*b^5*c*x+9*B*b^6*x+7*A*b^6)/x^2/b^7/(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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298893, size = 224, normalized size = 1.35 \[ -\frac{2 \,{\left (7 \, A b^{6} - 256 \,{\left (3 \, B b c^{5} - 4 \, A c^{6}\right )} x^{6} - 384 \,{\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{5} - 96 \,{\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} + 16 \,{\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{3} - 6 \,{\left (3 \, B b^{5} c - 4 \, A b^{4} c^{2}\right )} x^{2} + 3 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} x\right )}}{63 \,{\left (b^{7} c x^{5} + b^{8} x^{4}\right )} \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^3),x, algorithm="fricas")

[Out]

-2/63*(7*A*b^6 - 256*(3*B*b*c^5 - 4*A*c^6)*x^6 - 384*(3*B*b^2*c^4 - 4*A*b*c^5)*x
^5 - 96*(3*B*b^3*c^3 - 4*A*b^2*c^4)*x^4 + 16*(3*B*b^4*c^2 - 4*A*b^3*c^3)*x^3 - 6
*(3*B*b^5*c - 4*A*b^4*c^2)*x^2 + 3*(3*B*b^6 - 4*A*b^5*c)*x)/((b^7*c*x^5 + b^8*x^
4)*sqrt(c*x^2 + b*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{3} \left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**3/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((A + B*x)/(x**3*(x*(b + c*x))**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^3),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^3), x)