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

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

[Out]

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

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Rubi [A]  time = 0.170136, 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 \[ -\frac{5 \sqrt{c} (3 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}-\frac{5 (3 b B-7 A c)}{4 b^4 \sqrt{x}}+\frac{5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac{3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac{b B-A c}{2 b c x^{3/2} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.2396, size = 133, normalized size = 0.9 \[ \frac{A c - B b}{2 b c x^{\frac{3}{2}} \left (b + c x\right )^{2}} + \frac{7 A c - 3 B b}{4 b^{2} c x^{\frac{3}{2}} \left (b + c x\right )} - \frac{5 \left (7 A c - 3 B b\right )}{12 b^{3} c x^{\frac{3}{2}}} + \frac{5 \left (7 A c - 3 B b\right )}{4 b^{4} \sqrt{x}} + \frac{5 \sqrt{c} \left (7 A c - 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.212004, size = 117, normalized size = 0.8 \[ \frac{5 \sqrt{c} (7 A c-3 b B) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{9/2}}+\frac{A \left (-8 b^3+56 b^2 c x+175 b c^2 x^2+105 c^3 x^3\right )-3 b B x \left (8 b^2+25 b c x+15 c^2 x^2\right )}{12 b^4 x^{3/2} (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*B*x*(8*b^2 + 25*b*c*x + 15*c^2*x^2) + A*(-8*b^3 + 56*b^2*c*x + 175*b*c^2*x
^2 + 105*c^3*x^3))/(12*b^4*x^(3/2)*(b + c*x)^2) + (5*Sqrt[c]*(-3*b*B + 7*A*c)*Ar
cTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(9/2))

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Maple [A]  time = 0.03, size = 152, normalized size = 1. \[ -{\frac{2\,A}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{Ac}{\sqrt{x}{b}^{4}}}-2\,{\frac{B}{\sqrt{x}{b}^{3}}}+{\frac{11\,A{c}^{3}}{4\,{b}^{4} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{7\,B{c}^{2}}{4\,{b}^{3} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,A{c}^{2}}{4\,{b}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{9\,Bc}{4\,{b}^{2} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{35\,A{c}^{2}}{4\,{b}^{4}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{15\,Bc}{4\,{b}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306238, size = 1, normalized size = 0.01 \[ \left [-\frac{16 \, A b^{3} + 30 \,{\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 50 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 15 \,{\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 2 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} +{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt{x} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 16 \,{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x}{24 \,{\left (b^{4} c^{2} x^{3} + 2 \, b^{5} c x^{2} + b^{6} x\right )} \sqrt{x}}, -\frac{8 \, A b^{3} + 15 \,{\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} - 15 \,{\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 2 \,{\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} +{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt{x} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) + 8 \,{\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x}{12 \,{\left (b^{4} c^{2} x^{3} + 2 \, b^{5} c x^{2} + b^{6} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(16*A*b^3 + 30*(3*B*b*c^2 - 7*A*c^3)*x^3 + 50*(3*B*b^2*c - 7*A*b*c^2)*x^2
+ 15*((3*B*b*c^2 - 7*A*c^3)*x^3 + 2*(3*B*b^2*c - 7*A*b*c^2)*x^2 + (3*B*b^3 - 7*
A*b^2*c)*x)*sqrt(x)*sqrt(-c/b)*log((c*x + 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b))
+ 16*(3*B*b^3 - 7*A*b^2*c)*x)/((b^4*c^2*x^3 + 2*b^5*c*x^2 + b^6*x)*sqrt(x)), -1
/12*(8*A*b^3 + 15*(3*B*b*c^2 - 7*A*c^3)*x^3 + 25*(3*B*b^2*c - 7*A*b*c^2)*x^2 - 1
5*((3*B*b*c^2 - 7*A*c^3)*x^3 + 2*(3*B*b^2*c - 7*A*b*c^2)*x^2 + (3*B*b^3 - 7*A*b^
2*c)*x)*sqrt(x)*sqrt(c/b)*arctan(b*sqrt(c/b)/(c*sqrt(x))) + 8*(3*B*b^3 - 7*A*b^2
*c)*x)/((b^4*c^2*x^3 + 2*b^5*c*x^2 + b^6*x)*sqrt(x))]

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Sympy [A]  time = 57.178, size = 780, normalized size = 5.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*A*c**3*x**(3/2)/(8*b**6 + 16*b**5*c*x + 8*b**4*c**2*x**2) + 10*A*c**2*sqrt(x)/
(8*b**5 + 16*b**4*c*x + 8*b**3*c**2*x**2) + 4*A*c**2*sqrt(x)/(2*b**5 + 2*b**4*c*
x) - 3*A*c**2*sqrt(-1/(b**5*c))*log(-b**3*sqrt(-1/(b**5*c)) + sqrt(x))/(8*b**2)
+ 3*A*c**2*sqrt(-1/(b**5*c))*log(b**3*sqrt(-1/(b**5*c)) + sqrt(x))/(8*b**2) - A*
c**2*sqrt(-1/(b**3*c))*log(-b**2*sqrt(-1/(b**3*c)) + sqrt(x))/b**3 + A*c**2*sqrt
(-1/(b**3*c))*log(b**2*sqrt(-1/(b**3*c)) + sqrt(x))/b**3 - 2*A/(3*b**3*x**(3/2))
+ 6*A*c**2*Piecewise((atan(sqrt(x)/sqrt(b/c))/(c*sqrt(b/c)), b/c > 0), (-acoth(
sqrt(x)/sqrt(-b/c))/(c*sqrt(-b/c)), (b/c < 0) & (x > -b/c)), (-atanh(sqrt(x)/sqr
t(-b/c))/(c*sqrt(-b/c)), (b/c < 0) & (x < -b/c)))/b**4 + 6*A*c/(b**4*sqrt(x)) -
6*B*c**2*x**(3/2)/(8*b**5 + 16*b**4*c*x + 8*b**3*c**2*x**2) - 10*B*c*sqrt(x)/(8*
b**4 + 16*b**3*c*x + 8*b**2*c**2*x**2) - 2*B*c*sqrt(x)/(2*b**4 + 2*b**3*c*x) + 3
*B*c*sqrt(-1/(b**5*c))*log(-b**3*sqrt(-1/(b**5*c)) + sqrt(x))/(8*b) - 3*B*c*sqrt
(-1/(b**5*c))*log(b**3*sqrt(-1/(b**5*c)) + sqrt(x))/(8*b) + B*c*sqrt(-1/(b**3*c)
)*log(-b**2*sqrt(-1/(b**3*c)) + sqrt(x))/(2*b**2) - B*c*sqrt(-1/(b**3*c))*log(b*
*2*sqrt(-1/(b**3*c)) + sqrt(x))/(2*b**2) - 2*B*c*Piecewise((atan(sqrt(x)/sqrt(b/
c))/(c*sqrt(b/c)), b/c > 0), (-acoth(sqrt(x)/sqrt(-b/c))/(c*sqrt(-b/c)), (b/c <
0) & (x > -b/c)), (-atanh(sqrt(x)/sqrt(-b/c))/(c*sqrt(-b/c)), (b/c < 0) & (x < -
b/c)))/b**3 - 2*B/(b**3*sqrt(x))

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GIAC/XCAS [A]  time = 0.271042, size = 146, normalized size = 0.99 \[ -\frac{5 \,{\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b^{4}} - \frac{2 \,{\left (3 \, B b x - 9 \, A c x + A b\right )}}{3 \, b^{4} x^{\frac{3}{2}}} - \frac{7 \, B b c^{2} x^{\frac{3}{2}} - 11 \, A c^{3} x^{\frac{3}{2}} + 9 \, B b^{2} c \sqrt{x} - 13 \, A b c^{2} \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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