3.186 \(\int \frac{x^{11/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}}-\frac{5 \sqrt{x} (7 b B-3 A c)}{4 c^4}+\frac{5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac{x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}-\frac{x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \]

[Out]

(-5*(7*b*B - 3*A*c)*Sqrt[x])/(4*c^4) + (5*(7*b*B - 3*A*c)*x^(3/2))/(12*b*c^3) -
((b*B - A*c)*x^(7/2))/(2*b*c*(b + c*x)^2) - ((7*b*B - 3*A*c)*x^(5/2))/(4*b*c^2*(
b + c*x)) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*c^(
9/2))

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Rubi [A]  time = 0.179086, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}}-\frac{5 \sqrt{x} (7 b B-3 A c)}{4 c^4}+\frac{5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac{x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}-\frac{x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(11/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

(-5*(7*b*B - 3*A*c)*Sqrt[x])/(4*c^4) + (5*(7*b*B - 3*A*c)*x^(3/2))/(12*b*c^3) -
((b*B - A*c)*x^(7/2))/(2*b*c*(b + c*x)^2) - ((7*b*B - 3*A*c)*x^(5/2))/(4*b*c^2*(
b + c*x)) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*c^(
9/2))

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Rubi in Sympy [A]  time = 22.5292, size = 136, normalized size = 0.93 \[ - \frac{5 \sqrt{b} \left (3 A c - 7 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 c^{\frac{9}{2}}} + \frac{5 \sqrt{x} \left (3 A c - 7 B b\right )}{4 c^{4}} + \frac{x^{\frac{7}{2}} \left (A c - B b\right )}{2 b c \left (b + c x\right )^{2}} + \frac{x^{\frac{5}{2}} \left (3 A c - 7 B b\right )}{4 b c^{2} \left (b + c x\right )} - \frac{5 x^{\frac{3}{2}} \left (3 A c - 7 B b\right )}{12 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(11/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

-5*sqrt(b)*(3*A*c - 7*B*b)*atan(sqrt(c)*sqrt(x)/sqrt(b))/(4*c**(9/2)) + 5*sqrt(x
)*(3*A*c - 7*B*b)/(4*c**4) + x**(7/2)*(A*c - B*b)/(2*b*c*(b + c*x)**2) + x**(5/2
)*(3*A*c - 7*B*b)/(4*b*c**2*(b + c*x)) - 5*x**(3/2)*(3*A*c - 7*B*b)/(12*b*c**3)

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Mathematica [A]  time = 0.208108, size = 110, normalized size = 0.75 \[ \frac{\sqrt{x} \left (5 b^2 c (9 A-35 B x)+b c^2 x (75 A-56 B x)+8 c^3 x^2 (3 A+B x)-105 b^3 B\right )}{12 c^4 (b+c x)^2}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(11/2)*(A + B*x))/(b*x + c*x^2)^3,x]

[Out]

(Sqrt[x]*(-105*b^3*B + b*c^2*x*(75*A - 56*B*x) + 5*b^2*c*(9*A - 35*B*x) + 8*c^3*
x^2*(3*A + B*x)))/(12*c^4*(b + c*x)^2) + (5*Sqrt[b]*(7*b*B - 3*A*c)*ArcTan[(Sqrt
[c]*Sqrt[x])/Sqrt[b]])/(4*c^(9/2))

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Maple [A]  time = 0.023, size = 152, normalized size = 1. \[{\frac{2\,B}{3\,{c}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{3}}}-6\,{\frac{\sqrt{x}Bb}{{c}^{4}}}+{\frac{9\,Ab}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{13\,{b}^{2}B}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}A}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{11\,B{b}^{3}}{4\,{c}^{4} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,Ab}{4\,{c}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{35\,{b}^{2}B}{4\,{c}^{4}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)*(B*x+A)/(c*x^2+b*x)^3,x)

[Out]

2/3/c^3*B*x^(3/2)+2/c^3*A*x^(1/2)-6/c^4*x^(1/2)*B*b+9/4*b/c^2/(c*x+b)^2*x^(3/2)*
A-13/4*b^2/c^3/(c*x+b)^2*x^(3/2)*B+7/4*b^2/c^3/(c*x+b)^2*A*x^(1/2)-11/4*b^3/c^4/
(c*x+b)^2*B*x^(1/2)-15/4*b/c^3/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+35/4*
b^2/c^4/(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^(11/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295607, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c +{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x - 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{x}}{24 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}, \frac{15 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c +{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{b}{c}}}\right ) +{\left (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{x}}{12 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(11/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[-1/24*(15*(7*B*b^3 - 3*A*b^2*c + (7*B*b*c^2 - 3*A*c^3)*x^2 + 2*(7*B*b^2*c - 3*A
*b*c^2)*x)*sqrt(-b/c)*log((c*x - 2*c*sqrt(x)*sqrt(-b/c) - b)/(c*x + b)) - 2*(8*B
*c^3*x^3 - 105*B*b^3 + 45*A*b^2*c - 8*(7*B*b*c^2 - 3*A*c^3)*x^2 - 25*(7*B*b^2*c
- 3*A*b*c^2)*x)*sqrt(x))/(c^6*x^2 + 2*b*c^5*x + b^2*c^4), 1/12*(15*(7*B*b^3 - 3*
A*b^2*c + (7*B*b*c^2 - 3*A*c^3)*x^2 + 2*(7*B*b^2*c - 3*A*b*c^2)*x)*sqrt(b/c)*arc
tan(sqrt(x)/sqrt(b/c)) + (8*B*c^3*x^3 - 105*B*b^3 + 45*A*b^2*c - 8*(7*B*b*c^2 -
3*A*c^3)*x^2 - 25*(7*B*b^2*c - 3*A*b*c^2)*x)*sqrt(x))/(c^6*x^2 + 2*b*c^5*x + b^2
*c^4)]

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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**(11/2)*(B*x+A)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269193, size = 161, normalized size = 1.1 \[ \frac{5 \,{\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} c^{4}} - \frac{13 \, B b^{2} c x^{\frac{3}{2}} - 9 \, A b c^{2} x^{\frac{3}{2}} + 11 \, B b^{3} \sqrt{x} - 7 \, A b^{2} c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} c^{4}} + \frac{2 \,{\left (B c^{6} x^{\frac{3}{2}} - 9 \, B b c^{5} \sqrt{x} + 3 \, A c^{6} \sqrt{x}\right )}}{3 \, c^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(11/2)/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

5/4*(7*B*b^2 - 3*A*b*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*c^4) - 1/4*(13*B*
b^2*c*x^(3/2) - 9*A*b*c^2*x^(3/2) + 11*B*b^3*sqrt(x) - 7*A*b^2*c*sqrt(x))/((c*x
+ b)^2*c^4) + 2/3*(B*c^6*x^(3/2) - 9*B*b*c^5*sqrt(x) + 3*A*c^6*sqrt(x))/c^9