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

Optimal. Leaf size=118 \[ \frac{\left (b x+c x^2\right )^{3/2} (4 A c+b B)}{2 b x}+\frac{3}{4} \sqrt{b x+c x^2} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{b x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.269753, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (b x+c x^2\right )^{3/2} (4 A c+b B)}{2 b x}+\frac{3}{4} \sqrt{b x+c x^2} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{b x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.7891, size = 107, normalized size = 0.91 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{b x^{3}} + \frac{3 b \left (4 A c + B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 \sqrt{c}} + \left (3 A c + \frac{3 B b}{4}\right ) \sqrt{b x + c x^{2}} + \frac{\left (4 A c + B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.1715, size = 94, normalized size = 0.8 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b \sqrt{x} (4 A c+b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{b+c x}}+A (4 c x-8 b)+B x (5 b+2 c x)\right )}{4 x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(B*x*(5*b + 2*c*x) + A*(-8*b + 4*c*x) + (3*b*(b*B + 4*A*c)*Sq
rt[x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[c]*Sqrt[b + c*x])))/(4*x)

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 232, normalized size = 2. \[ -2\,{\frac{A \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{3}}}+8\,{\frac{Ac \left ( c{x}^{2}+bx \right ) ^{5/2}}{{b}^{2}{x}^{2}}}-8\,{\frac{A{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}{{b}^{2}}}-6\,{\frac{A{c}^{2}\sqrt{c{x}^{2}+bx}x}{b}}-3\,Ac\sqrt{c{x}^{2}+bx}+{\frac{3\,Ab}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) }+2\,{\frac{B \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{2}}}-2\,{\frac{Bc \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{3\,Bcx}{2}\sqrt{c{x}^{2}+bx}}-{\frac{3\,Bb}{4}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(c*x^2+b*x)^(5/2)/b/x^3+8*A*c/b^2/x^2*(c*x^2+b*x)^(5/2)-8*A*c^2/b^2*(c*x^2+
b*x)^(3/2)-6*A*c^2/b*(c*x^2+b*x)^(1/2)*x-3*A*c*(c*x^2+b*x)^(1/2)+3/2*A*c^(1/2)*b
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2*B/b/x^2*(c*x^2+b*x)^(5/2)-2*B*c/b*(
c*x^2+b*x)^(3/2)-3/2*B*c*(c*x^2+b*x)^(1/2)*x-3/4*B*b*(c*x^2+b*x)^(1/2)+3/8*B/c^(
1/2)*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.284797, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} x \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (2 \, B c x^{2} - 8 \, A b +{\left (5 \, B b + 4 \, A c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{8 \, \sqrt{c} x}, \frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (2 \, B c x^{2} - 8 \, A b +{\left (5 \, B b + 4 \, A c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{4 \, \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287873, size = 147, normalized size = 1.25 \[ \frac{2 \, A b^{2}}{\sqrt{c} x - \sqrt{c x^{2} + b x}} + \frac{1}{4} \,{\left (2 \, B c x + \frac{5 \, B b c + 4 \, A c^{2}}{c}\right )} \sqrt{c x^{2} + b x} - \frac{3 \,{\left (B b^{2} + 4 \, A b c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b^2/(sqrt(c)*x - sqrt(c*x^2 + b*x)) + 1/4*(2*B*c*x + (5*B*b*c + 4*A*c^2)/c)*
sqrt(c*x^2 + b*x) - 3/8*(B*b^2 + 4*A*b*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*
x))*sqrt(c) - b))/sqrt(c)