3.99 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=155 \[ -\frac{5 b^3 (b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (b B-8 A c)}{4 b x}+\frac{5}{24} \left (b x+c x^2\right )^{3/2} (b B-8 A c)+\frac{5 b (b+2 c x) \sqrt{b x+c x^2} (b B-8 A c)}{64 c}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3} \]

[Out]

(5*b*(b*B - 8*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c) + (5*(b*B - 8*A*c)*(b*x
+ c*x^2)^(3/2))/24 + ((b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(4*b*x) + (2*A*(b*x +
c*x^2)^(7/2))/(b*x^3) - (5*b^3*(b*B - 8*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(64*c^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.368312, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 b^3 (b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (b B-8 A c)}{4 b x}+\frac{5}{24} \left (b x+c x^2\right )^{3/2} (b B-8 A c)+\frac{5 b (b+2 c x) \sqrt{b x+c x^2} (b B-8 A c)}{64 c}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^3,x]

[Out]

(5*b*(b*B - 8*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c) + (5*(b*B - 8*A*c)*(b*x
+ c*x^2)^(3/2))/24 + ((b*B - 8*A*c)*(b*x + c*x^2)^(5/2))/(4*b*x) + (2*A*(b*x +
c*x^2)^(7/2))/(b*x^3) - (5*b^3*(b*B - 8*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(64*c^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.6492, size = 144, normalized size = 0.93 \[ \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{b x^{3}} + \frac{5 b^{3} \left (8 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{3}{2}}} - \frac{5 b \left (b + 2 c x\right ) \left (8 A c - B b\right ) \sqrt{b x + c x^{2}}}{64 c} - \left (\frac{5 A c}{3} - \frac{5 B b}{24}\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} - \frac{\left (8 A c - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{4 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**3,x)

[Out]

2*A*(b*x + c*x**2)**(7/2)/(b*x**3) + 5*b**3*(8*A*c - B*b)*atanh(sqrt(c)*x/sqrt(b
*x + c*x**2))/(64*c**(3/2)) - 5*b*(b + 2*c*x)*(8*A*c - B*b)*sqrt(b*x + c*x**2)/(
64*c) - (5*A*c/3 - 5*B*b/24)*(b*x + c*x**2)**(3/2) - (8*A*c - B*b)*(b*x + c*x**2
)**(5/2)/(4*b*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.237266, size = 130, normalized size = 0.84 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (2 b^2 c (132 A+59 B x)+8 b c^2 x (26 A+17 B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )-\frac{15 b^3 (b B-8 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^3,x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*B + 16*c^3*x^2*(4*A + 3*B*x) + 8*b*c^2*x*(26
*A + 17*B*x) + 2*b^2*c*(132*A + 59*B*x)) - (15*b^3*(b*B - 8*A*c)*Log[c*Sqrt[x] +
Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 306, normalized size = 2. \[ 2\,{\frac{A \left ( c{x}^{2}+bx \right ) ^{7/2}}{b{x}^{3}}}-{\frac{16\,Ac}{3\,{b}^{2}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{16\,A{c}^{2}}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{10\,Ax{c}^{2}}{3\,b} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ac}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Abcx}{4}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{2}A}{8}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{2\,B}{3\,b{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Bc}{3\,b} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bcx}{12} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bb}{24} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{b}^{3}}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^3,x)

[Out]

2*A*(c*x^2+b*x)^(7/2)/b/x^3-16/3*A*c/b^2/x^2*(c*x^2+b*x)^(7/2)+16/3*A*c^2/b^2*(c
*x^2+b*x)^(5/2)+10/3*A*c^2/b*(c*x^2+b*x)^(3/2)*x+5/3*A*c*(c*x^2+b*x)^(3/2)-5/4*A
*c*b*(c*x^2+b*x)^(1/2)*x-5/8*A*b^2*(c*x^2+b*x)^(1/2)+5/16*A/c^(1/2)*b^3*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/3*B/b/x^2*(c*x^2+b*x)^(7/2)-2/3*B*c/b*(c*x^2
+b*x)^(5/2)-5/12*B*c*(c*x^2+b*x)^(3/2)*x-5/24*B*b*(c*x^2+b*x)^(3/2)+5/32*B*b^2*(
c*x^2+b*x)^(1/2)*x+5/64*B/c*b^3*(c*x^2+b*x)^(1/2)-5/128*B/c^(3/2)*b^4*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)^(5/2)*(B*x + A)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.294773, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 264 \, A b^{2} c + 8 \,{\left (17 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (59 \, B b^{2} c + 104 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{3}{2}}}, \frac{{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 264 \, A b^{2} c + 8 \,{\left (17 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (59 \, B b^{2} c + 104 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 15 \,{\left (B b^{4} - 8 \, A b^{3} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

[1/384*(2*(48*B*c^3*x^3 + 15*B*b^3 + 264*A*b^2*c + 8*(17*B*b*c^2 + 8*A*c^3)*x^2
+ 2*(59*B*b^2*c + 104*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(B*b^4 - 8*A*b^
3*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(3/2), 1/192*((48*B*c^3
*x^3 + 15*B*b^3 + 264*A*b^2*c + 8*(17*B*b*c^2 + 8*A*c^3)*x^2 + 2*(59*B*b^2*c + 1
04*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(B*b^4 - 8*A*b^3*c)*arctan(sqrt(c
*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**3,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**3, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284069, size = 190, normalized size = 1.23 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B c^{2} x + \frac{17 \, B b c^{4} + 8 \, A c^{5}}{c^{3}}\right )} x + \frac{59 \, B b^{2} c^{3} + 104 \, A b c^{4}}{c^{3}}\right )} x + \frac{3 \,{\left (5 \, B b^{3} c^{2} + 88 \, A b^{2} c^{3}\right )}}{c^{3}}\right )} + \frac{5 \,{\left (B b^{4} - 8 \, A b^{3} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*c^2*x + (17*B*b*c^4 + 8*A*c^5)/c^3)*x + (59*B
*b^2*c^3 + 104*A*b*c^4)/c^3)*x + 3*(5*B*b^3*c^2 + 88*A*b^2*c^3)/c^3) + 5/128*(B*
b^4 - 8*A*b^3*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(3/2)