3.187 \(\int \frac{x^{9/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}}+\frac{3 \sqrt{x} (5 b B-A c)}{4 b c^3}-\frac{x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}-\frac{x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \]

[Out]

(3*(5*b*B - A*c)*Sqrt[x])/(4*b*c^3) - ((b*B - A*c)*x^(5/2))/(2*b*c*(b + c*x)^2)
- ((5*b*B - A*c)*x^(3/2))/(4*b*c^2*(b + c*x)) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]
*Sqrt[x])/Sqrt[b]])/(4*Sqrt[b]*c^(7/2))

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Rubi [A]  time = 0.149619, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}}+\frac{3 \sqrt{x} (5 b B-A c)}{4 b c^3}-\frac{x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}-\frac{x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

(3*(5*b*B - A*c)*Sqrt[x])/(4*b*c^3) - ((b*B - A*c)*x^(5/2))/(2*b*c*(b + c*x)^2)
- ((5*b*B - A*c)*x^(3/2))/(4*b*c^2*(b + c*x)) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]
*Sqrt[x])/Sqrt[b]])/(4*Sqrt[b]*c^(7/2))

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Rubi in Sympy [A]  time = 18.481, size = 109, normalized size = 0.87 \[ \frac{x^{\frac{5}{2}} \left (A c - B b\right )}{2 b c \left (b + c x\right )^{2}} + \frac{x^{\frac{3}{2}} \left (A c - 5 B b\right )}{4 b c^{2} \left (b + c x\right )} - \frac{3 \sqrt{x} \left (A c - 5 B b\right )}{4 b c^{3}} + \frac{3 \left (A c - 5 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 \sqrt{b} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

x**(5/2)*(A*c - B*b)/(2*b*c*(b + c*x)**2) + x**(3/2)*(A*c - 5*B*b)/(4*b*c**2*(b
+ c*x)) - 3*sqrt(x)*(A*c - 5*B*b)/(4*b*c**3) + 3*(A*c - 5*B*b)*atan(sqrt(c)*sqrt
(x)/sqrt(b))/(4*sqrt(b)*c**(7/2))

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Mathematica [A]  time = 0.151945, size = 92, normalized size = 0.73 \[ \frac{\sqrt{x} \left (b (25 B c x-3 A c)+c^2 x (8 B x-5 A)+15 b^2 B\right )}{4 c^3 (b+c x)^2}-\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

(Sqrt[x]*(15*b^2*B + c^2*x*(-5*A + 8*B*x) + b*(-3*A*c + 25*B*c*x)))/(4*c^3*(b +
c*x)^2) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*Sqrt[b]*c^(7/2)
)

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Maple [A]  time = 0.022, size = 125, normalized size = 1. \[ 2\,{\frac{B\sqrt{x}}{{c}^{3}}}-{\frac{5\,A}{4\,c \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,Bb}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{3\,Ab}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{7\,{b}^{2}B}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{3\,A}{4\,{c}^{2}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{15\,Bb}{4\,{c}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^3,x)

[Out]

2*B*x^(1/2)/c^3-5/4/c/(c*x+b)^2*x^(3/2)*A+9/4/c^2/(c*x+b)^2*x^(3/2)*B*b-3/4/c^2/
(c*x+b)^2*A*x^(1/2)*b+7/4/c^3/(c*x+b)^2*B*x^(1/2)*b^2+3/4/c^2/(b*c)^(1/2)*arctan
(c*x^(1/2)/(b*c)^(1/2))*A-15/4/c^3/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*B*b

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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)*x^(9/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.3044, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B c^{2} x^{2} + 15 \, B b^{2} - 3 \, A b c + 5 \,{\left (5 \, B b c - A c^{2}\right )} x\right )} \sqrt{-b c} \sqrt{x} - 3 \,{\left (5 \, B b^{3} - A b^{2} c +{\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \log \left (\frac{2 \, b c \sqrt{x} + \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right )}{8 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )} \sqrt{-b c}}, \frac{{\left (8 \, B c^{2} x^{2} + 15 \, B b^{2} - 3 \, A b c + 5 \,{\left (5 \, B b c - A c^{2}\right )} x\right )} \sqrt{b c} \sqrt{x} + 3 \,{\left (5 \, B b^{3} - A b^{2} c +{\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right )}{4 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(2*(8*B*c^2*x^2 + 15*B*b^2 - 3*A*b*c + 5*(5*B*b*c - A*c^2)*x)*sqrt(-b*c)*sq
rt(x) - 3*(5*B*b^3 - A*b^2*c + (5*B*b*c^2 - A*c^3)*x^2 + 2*(5*B*b^2*c - A*b*c^2)
*x)*log((2*b*c*sqrt(x) + sqrt(-b*c)*(c*x - b))/(c*x + b)))/((c^5*x^2 + 2*b*c^4*x
+ b^2*c^3)*sqrt(-b*c)), 1/4*((8*B*c^2*x^2 + 15*B*b^2 - 3*A*b*c + 5*(5*B*b*c - A
*c^2)*x)*sqrt(b*c)*sqrt(x) + 3*(5*B*b^3 - A*b^2*c + (5*B*b*c^2 - A*c^3)*x^2 + 2*
(5*B*b^2*c - A*b*c^2)*x)*arctan(b/(sqrt(b*c)*sqrt(x))))/((c^5*x^2 + 2*b*c^4*x +
b^2*c^3)*sqrt(b*c))]

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270609, size = 117, normalized size = 0.93 \[ \frac{2 \, B \sqrt{x}}{c^{3}} - \frac{3 \,{\left (5 \, B b - A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} c^{3}} + \frac{9 \, B b c x^{\frac{3}{2}} - 5 \, A c^{2} x^{\frac{3}{2}} + 7 \, B b^{2} \sqrt{x} - 3 \, A b c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

2*B*sqrt(x)/c^3 - 3/4*(5*B*b - A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c^3)
+ 1/4*(9*B*b*c*x^(3/2) - 5*A*c^2*x^(3/2) + 7*B*b^2*sqrt(x) - 3*A*b*c*sqrt(x))/((
c*x + b)^2*c^3)