3.188 \(\int \frac{x^{7/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{3/2} c^{5/2}}-\frac{\sqrt{x} (A c+3 b B)}{4 b c^2 (b+c x)}-\frac{x^{3/2} (b B-A c)}{2 b c (b+c x)^2} \]

[Out]

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

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Rubi [A]  time = 0.120171, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{3/2} c^{5/2}}-\frac{\sqrt{x} (A c+3 b B)}{4 b c^2 (b+c x)}-\frac{x^{3/2} (b B-A c)}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.4236, size = 85, normalized size = 0.85 \[ \frac{x^{\frac{3}{2}} \left (A c - B b\right )}{2 b c \left (b + c x\right )^{2}} - \frac{\sqrt{x} \left (A c + 3 B b\right )}{4 b c^{2} \left (b + c x\right )} + \frac{\left (A c + 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.167852, size = 85, normalized size = 0.85 \[ \frac{\frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\sqrt{c} \sqrt{x} \left (-b c (A+5 B x)+A c^2 x-3 b^2 B\right )}{b (b+c x)^2}}{4 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.019, size = 94, normalized size = 0.9 \[ 2\,{\frac{1}{ \left ( cx+b \right ) ^{2}} \left ( 1/8\,{\frac{ \left ( Ac-5\,Bb \right ){x}^{3/2}}{bc}}-1/8\,{\frac{ \left ( Ac+3\,Bb \right ) \sqrt{x}}{{c}^{2}}} \right ) }+{\frac{A}{4\,bc}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{4\,{c}^{2}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(c*x^2+b*x)^3,x)

[Out]

2*(1/8*(A*c-5*B*b)/b/c*x^(3/2)-1/8*(A*c+3*B*b)/c^2*x^(1/2))/(c*x+b)^2+1/4/c/b/(b
*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+3/4/c^2/(b*c)^(1/2)*arctan(c*x^(1/2)/(
b*c)^(1/2))*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^(7/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292169, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B b^{2} + A b c +{\left (5 \, B b c - A c^{2}\right )} x\right )} \sqrt{-b c} \sqrt{x} -{\left (3 \, B b^{3} + A b^{2} c +{\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x\right )} \log \left (\frac{2 \, b c \sqrt{x} + \sqrt{-b c}{\left (c x - b\right )}}{c x + b}\right )}{8 \,{\left (b c^{4} x^{2} + 2 \, b^{2} c^{3} x + b^{3} c^{2}\right )} \sqrt{-b c}}, -\frac{{\left (3 \, B b^{2} + A b c +{\left (5 \, B b c - A c^{2}\right )} x\right )} \sqrt{b c} \sqrt{x} +{\left (3 \, B b^{3} + A b^{2} c +{\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x\right )} \arctan \left (\frac{b}{\sqrt{b c} \sqrt{x}}\right )}{4 \,{\left (b c^{4} x^{2} + 2 \, b^{2} c^{3} x + b^{3} c^{2}\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(2*(3*B*b^2 + A*b*c + (5*B*b*c - A*c^2)*x)*sqrt(-b*c)*sqrt(x) - (3*B*b^3 +
A*b^2*c + (3*B*b*c^2 + A*c^3)*x^2 + 2*(3*B*b^2*c + A*b*c^2)*x)*log((2*b*c*sqrt(
x) + sqrt(-b*c)*(c*x - b))/(c*x + b)))/((b*c^4*x^2 + 2*b^2*c^3*x + b^3*c^2)*sqrt
(-b*c)), -1/4*((3*B*b^2 + A*b*c + (5*B*b*c - A*c^2)*x)*sqrt(b*c)*sqrt(x) + (3*B*
b^3 + A*b^2*c + (3*B*b*c^2 + A*c^3)*x^2 + 2*(3*B*b^2*c + A*b*c^2)*x)*arctan(b/(s
qrt(b*c)*sqrt(x))))/((b*c^4*x^2 + 2*b^2*c^3*x + b^3*c^2)*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**(7/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270987, size = 111, normalized size = 1.11 \[ \frac{{\left (3 \, B b + A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b c^{2}} - \frac{5 \, B b c x^{\frac{3}{2}} - A c^{2} x^{\frac{3}{2}} + 3 \, B b^{2} \sqrt{x} + A b c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

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