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

Optimal. Leaf size=169 \[ \frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}+\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2} \]

[Out]

(7*(5*b*B - 9*A*c))/(20*b^3*c*x^(5/2)) - (7*(5*b*B - 9*A*c))/(12*b^4*x^(3/2)) +
(7*c*(5*b*B - 9*A*c))/(4*b^5*Sqrt[x]) - (b*B - A*c)/(2*b*c*x^(5/2)*(b + c*x)^2)
- (5*b*B - 9*A*c)/(4*b^2*c*x^(5/2)*(b + c*x)) + (7*c^(3/2)*(5*b*B - 9*A*c)*ArcTa
n[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.205458, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}+\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(7*(5*b*B - 9*A*c))/(20*b^3*c*x^(5/2)) - (7*(5*b*B - 9*A*c))/(12*b^4*x^(3/2)) +
(7*c*(5*b*B - 9*A*c))/(4*b^5*Sqrt[x]) - (b*B - A*c)/(2*b*c*x^(5/2)*(b + c*x)^2)
- (5*b*B - 9*A*c)/(4*b^2*c*x^(5/2)*(b + c*x)) + (7*c^(3/2)*(5*b*B - 9*A*c)*ArcTa
n[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.1966, size = 156, normalized size = 0.92 \[ \frac{A c - B b}{2 b c x^{\frac{5}{2}} \left (b + c x\right )^{2}} + \frac{9 A c - 5 B b}{4 b^{2} c x^{\frac{5}{2}} \left (b + c x\right )} - \frac{7 \left (9 A c - 5 B b\right )}{20 b^{3} c x^{\frac{5}{2}}} + \frac{7 \left (9 A c - 5 B b\right )}{12 b^{4} x^{\frac{3}{2}}} - \frac{7 c \left (9 A c - 5 B b\right )}{4 b^{5} \sqrt{x}} - \frac{7 c^{\frac{3}{2}} \left (9 A c - 5 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(c*x**2+b*x)**3/x**(1/2),x)

[Out]

(A*c - B*b)/(2*b*c*x**(5/2)*(b + c*x)**2) + (9*A*c - 5*B*b)/(4*b**2*c*x**(5/2)*(
b + c*x)) - 7*(9*A*c - 5*B*b)/(20*b**3*c*x**(5/2)) + 7*(9*A*c - 5*B*b)/(12*b**4*
x**(3/2)) - 7*c*(9*A*c - 5*B*b)/(4*b**5*sqrt(x)) - 7*c**(3/2)*(9*A*c - 5*B*b)*at
an(sqrt(c)*sqrt(x)/sqrt(b))/(4*b**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.226095, size = 140, normalized size = 0.83 \[ \frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}+\frac{5 b B x \left (-8 b^3+56 b^2 c x+175 b c^2 x^2+105 c^3 x^3\right )-3 A \left (8 b^4-24 b^3 c x+168 b^2 c^2 x^2+525 b c^3 x^3+315 c^4 x^4\right )}{60 b^5 x^{5/2} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*b*B*x*(-8*b^3 + 56*b^2*c*x + 175*b*c^2*x^2 + 105*c^3*x^3) - 3*A*(8*b^4 - 24*b
^3*c*x + 168*b^2*c^2*x^2 + 525*b*c^3*x^3 + 315*c^4*x^4))/(60*b^5*x^(5/2)*(b + c*
x)^2) + (7*c^(3/2)*(5*b*B - 9*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2
))

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 178, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+2\,{\frac{Ac}{{x}^{3/2}{b}^{4}}}-{\frac{2\,B}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-12\,{\frac{A{c}^{2}}{{b}^{5}\sqrt{x}}}+6\,{\frac{Bc}{{b}^{4}\sqrt{x}}}-{\frac{15\,{c}^{4}A}{4\,{b}^{5} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,B{c}^{3}}{4\,{b}^{4} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,A{c}^{3}}{4\,{b}^{4} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{13\,B{c}^{2}}{4\,{b}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,A{c}^{3}}{4\,{b}^{5}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{35\,B{c}^{2}}{4\,{b}^{4}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x)^3/x^(1/2),x)

[Out]

-2/5*A/b^3/x^(5/2)+2/x^(3/2)/b^4*A*c-2/3/x^(3/2)/b^3*B-12*c^2/b^5/x^(1/2)*A+6*c/
b^4/x^(1/2)*B-15/4/b^5*c^4/(c*x+b)^2*x^(3/2)*A+11/4/b^4*c^3/(c*x+b)^2*x^(3/2)*B-
17/4/b^4*c^3/(c*x+b)^2*A*x^(1/2)+13/4/b^3*c^2/(c*x+b)^2*B*x^(1/2)-63/4/b^5*c^3/(
b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2))*A+35/4/b^4*c^2/(b*c)^(1/2)*arctan(c*x^(
1/2)/(b*c)^(1/2))*B

_______________________________________________________________________________________

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)^3*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.302901, size = 1, normalized size = 0.01 \[ \left [-\frac{48 \, A b^{4} - 210 \,{\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} - 350 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 112 \,{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 105 \,{\left ({\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 2 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} +{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 16 \,{\left (5 \, B b^{4} - 9 \, A b^{3} c\right )} x}{120 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )} \sqrt{x}}, -\frac{24 \, A b^{4} - 105 \,{\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} - 175 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 56 \,{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 105 \,{\left ({\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 2 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} +{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 8 \,{\left (5 \, B b^{4} - 9 \, A b^{3} c\right )} x}{60 \,{\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/120*(48*A*b^4 - 210*(5*B*b*c^3 - 9*A*c^4)*x^4 - 350*(5*B*b^2*c^2 - 9*A*b*c^3
)*x^3 - 112*(5*B*b^3*c - 9*A*b^2*c^2)*x^2 + 105*((5*B*b*c^3 - 9*A*c^4)*x^4 + 2*(
5*B*b^2*c^2 - 9*A*b*c^3)*x^3 + (5*B*b^3*c - 9*A*b^2*c^2)*x^2)*sqrt(x)*sqrt(-c/b)
*log((c*x - 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b)) + 16*(5*B*b^4 - 9*A*b^3*c)*x)
/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)*sqrt(x)), -1/60*(24*A*b^4 - 105*(5*B*b*c
^3 - 9*A*c^4)*x^4 - 175*(5*B*b^2*c^2 - 9*A*b*c^3)*x^3 - 56*(5*B*b^3*c - 9*A*b^2*
c^2)*x^2 + 105*((5*B*b*c^3 - 9*A*c^4)*x^4 + 2*(5*B*b^2*c^2 - 9*A*b*c^3)*x^3 + (5
*B*b^3*c - 9*A*b^2*c^2)*x^2)*sqrt(x)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) +
8*(5*B*b^4 - 9*A*b^3*c)*x)/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)*sqrt(x))]

_______________________________________________________________________________________

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((B*x+A)/(c*x**2+b*x)**3/x**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273287, size = 182, normalized size = 1.08 \[ \frac{7 \,{\left (5 \, B b c^{2} - 9 \, A c^{3}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b^{5}} + \frac{11 \, B b c^{3} x^{\frac{3}{2}} - 15 \, A c^{4} x^{\frac{3}{2}} + 13 \, B b^{2} c^{2} \sqrt{x} - 17 \, A b c^{3} \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b^{5}} + \frac{2 \,{\left (45 \, B b c x^{2} - 90 \, A c^{2} x^{2} - 5 \, B b^{2} x + 15 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{5} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/4*(5*B*b*c^2 - 9*A*c^3)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^5) + 1/4*(11*
B*b*c^3*x^(3/2) - 15*A*c^4*x^(3/2) + 13*B*b^2*c^2*sqrt(x) - 17*A*b*c^3*sqrt(x))/
((c*x + b)^2*b^5) + 2/15*(45*B*b*c*x^2 - 90*A*c^2*x^2 - 5*B*b^2*x + 15*A*b*c*x -
3*A*b^2)/(b^5*x^(5/2))