3.243 \(\int \frac{x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{16 b \sqrt{x} (2 b B-A c)}{3 c^4 \sqrt{b x+c x^2}}-\frac{8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt{b x+c x^2}}+\frac{2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(3*b*c*(b*x + c*x^2)^(3/2)) - (16*b*(2*b*B - A*c)*Sqrt[
x])/(3*c^4*Sqrt[b*x + c*x^2]) - (8*(2*b*B - A*c)*x^(3/2))/(3*c^3*Sqrt[b*x + c*x^
2]) + (2*(2*b*B - A*c)*x^(5/2))/(3*b*c^2*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.27842, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{16 b \sqrt{x} (2 b B-A c)}{3 c^4 \sqrt{b x+c x^2}}-\frac{8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt{b x+c x^2}}+\frac{2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(3*b*c*(b*x + c*x^2)^(3/2)) - (16*b*(2*b*B - A*c)*Sqrt[
x])/(3*c^4*Sqrt[b*x + c*x^2]) - (8*(2*b*B - A*c)*x^(3/2))/(3*c^3*Sqrt[b*x + c*x^
2]) + (2*(2*b*B - A*c)*x^(5/2))/(3*b*c^2*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 18.3841, size = 129, normalized size = 0.9 \[ \frac{16 b \sqrt{x} \left (A c - 2 B b\right )}{3 c^{4} \sqrt{b x + c x^{2}}} + \frac{16 x^{\frac{3}{2}} \left (\frac{A c}{2} - B b\right )}{3 c^{3} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{9}{2}} \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 x^{\frac{5}{2}} \left (\frac{A c}{2} - B b\right )}{3 b c^{2} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

16*b*sqrt(x)*(A*c - 2*B*b)/(3*c**4*sqrt(b*x + c*x**2)) + 16*x**(3/2)*(A*c/2 - B*
b)/(3*c**3*sqrt(b*x + c*x**2)) + 2*x**(9/2)*(A*c - B*b)/(3*b*c*(b*x + c*x**2)**(
3/2)) - 4*x**(5/2)*(A*c/2 - B*b)/(3*b*c**2*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0869109, size = 70, normalized size = 0.49 \[ \frac{2 x^{3/2} \left (8 b^2 c (A-3 B x)-6 b c^2 x (B x-2 A)+c^3 x^2 (3 A+B x)-16 b^3 B\right )}{3 c^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 82, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( B{c}^{3}{x}^{3}+3\,A{c}^{3}{x}^{2}-6\,Bb{c}^{2}{x}^{2}+12\,Ab{c}^{2}x-24\,B{b}^{2}cx+8\,A{b}^{2}c-16\,B{b}^{3} \right ) }{3\,{c}^{4}}{x}^{{\frac{5}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x)

[Out]

2/3*(c*x+b)*(B*c^3*x^3+3*A*c^3*x^2-6*B*b*c^2*x^2+12*A*b*c^2*x-24*B*b^2*c*x+8*A*b
^2*c-16*B*b^3)*x^(5/2)/c^4/(c*x^2+b*x)^(5/2)

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Maxima [A]  time = 0.795003, size = 239, normalized size = 1.67 \[ \frac{2}{3} \,{\left (\frac{3 \, \sqrt{c x + b} x^{2}}{c^{3} x^{2} + 2 \, b c^{2} x + b^{2} c} + \frac{4 \,{\left (3 \, c x + 2 \, b\right )} b}{{\left (c x + b\right )}^{\frac{3}{2}} c^{3}}\right )} A + \frac{2}{3} \, B{\left (\frac{{\left (c^{3} x^{2} - b c^{2} x - 2 \, b^{2} c\right )} x^{3} - 4 \,{\left (b c^{2} x^{2} + 2 \, b^{2} c x + b^{3}\right )} x^{2}}{{\left (c^{5} x^{3} + 2 \, b c^{4} x^{2} + b^{2} c^{3} x\right )} \sqrt{c x + b}} - \frac{4 \,{\left (5 \,{\left (c x + b\right )} b^{2} - b^{3}\right )}}{{\left (c x + b\right )}^{\frac{3}{2}} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.294119, size = 127, normalized size = 0.89 \[ \frac{2 \,{\left (B c^{3} x^{4} - 3 \,{\left (2 \, B b c^{2} - A c^{3}\right )} x^{3} - 12 \,{\left (2 \, B b^{2} c - A b c^{2}\right )} x^{2} - 8 \,{\left (2 \, B b^{3} - A b^{2} c\right )} x\right )}}{3 \,{\left (c^{5} x + b c^{4}\right )} \sqrt{c x^{2} + b x} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*c^3*x^4 - 3*(2*B*b*c^2 - A*c^3)*x^3 - 12*(2*B*b^2*c - A*b*c^2)*x^2 - 8*(2
*B*b^3 - A*b^2*c)*x)/((c^5*x + b*c^4)*sqrt(c*x^2 + b*x)*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277444, size = 136, normalized size = 0.95 \[ \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B - 9 \, \sqrt{c x + b} B b + 3 \, \sqrt{c x + b} A c - \frac{9 \,{\left (c x + b\right )} B b^{2} - B b^{3} - 6 \,{\left (c x + b\right )} A b c + A b^{2} c}{{\left (c x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, c^{4}} + \frac{16 \,{\left (2 \, B b^{2} - A b c\right )}}{3 \, \sqrt{b} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")

[Out]

2/3*((c*x + b)^(3/2)*B - 9*sqrt(c*x + b)*B*b + 3*sqrt(c*x + b)*A*c - (9*(c*x + b
)*B*b^2 - B*b^3 - 6*(c*x + b)*A*b*c + A*b^2*c)/(c*x + b)^(3/2))/c^4 + 16/3*(2*B*
b^2 - A*b*c)/(sqrt(b)*c^4)