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

Optimal. Leaf size=169 \[ \frac{b^4 (3 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2}}-\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} (3 b B-10 A c)}{128 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (3 b B-10 A c)}{48 c}+\frac{\left (b x+c x^2\right )^{5/2} (3 b B-10 A c)}{15 b}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^2} \]

[Out]

-(b^2*(3*b*B - 10*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^2) + ((3*b*B - 10*A
*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(48*c) + ((3*b*B - 10*A*c)*(b*x + c*x^2)^(5
/2))/(15*b) + (2*A*(b*x + c*x^2)^(7/2))/(3*b*x^2) + (b^4*(3*b*B - 10*A*c)*ArcTan
h[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2))

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Rubi [A]  time = 0.370795, antiderivative size = 169, 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 \[ \frac{b^4 (3 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2}}-\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} (3 b B-10 A c)}{128 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (3 b B-10 A c)}{48 c}+\frac{\left (b x+c x^2\right )^{5/2} (3 b B-10 A c)}{15 b}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2*(3*b*B - 10*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(128*c^2) + ((3*b*B - 10*A
*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(48*c) + ((3*b*B - 10*A*c)*(b*x + c*x^2)^(5
/2))/(15*b) + (2*A*(b*x + c*x^2)^(7/2))/(3*b*x^2) + (b^4*(3*b*B - 10*A*c)*ArcTan
h[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 20.6837, size = 156, normalized size = 0.92 \[ \frac{2 A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{3 b x^{2}} - \frac{b^{4} \left (10 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} + \frac{b^{2} \left (b + 2 c x\right ) \left (10 A c - 3 B b\right ) \sqrt{b x + c x^{2}}}{128 c^{2}} - \frac{\left (b + 2 c x\right ) \left (10 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{48 c} - \frac{\left (10 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*(b*x + c*x**2)**(7/2)/(3*b*x**2) - b**4*(10*A*c - 3*B*b)*atanh(sqrt(c)*x/sqr
t(b*x + c*x**2))/(128*c**(5/2)) + b**2*(b + 2*c*x)*(10*A*c - 3*B*b)*sqrt(b*x + c
*x**2)/(128*c**2) - (b + 2*c*x)*(10*A*c - 3*B*b)*(b*x + c*x**2)**(3/2)/(48*c) -
(10*A*c - 3*B*b)*(b*x + c*x**2)**(5/2)/(15*b)

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Mathematica [A]  time = 0.291444, size = 149, normalized size = 0.88 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^4 (3 b B-10 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (30 b^3 c (5 A+B x)+4 b^2 c^2 x (295 A+186 B x)+16 b c^3 x^2 (85 A+63 B x)+96 c^4 x^3 (5 A+4 B x)-45 b^4 B\right )\right )}{1920 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-45*b^4*B + 30*b^3*c*(5*A + B*x) + 96*c^4*x^3*(5*A
+ 4*B*x) + 16*b*c^3*x^2*(85*A + 63*B*x) + 4*b^2*c^2*x*(295*A + 186*B*x)) + (15*b
^4*(3*b*B - 10*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*
x])))/(1920*c^(5/2))

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Maple [A]  time = 0.015, size = 266, normalized size = 1.6 \[{\frac{2\,A}{3\,b{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Ac}{3\,b} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Acx}{12} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ab}{24} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ax{b}^{2}}{32}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{3}}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,A{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B}{5} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{xBb}{8} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}B}{16\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Bx{b}^{3}}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{4}B}{128\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,B{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*A*(c*x^2+b*x)^(7/2)/b/x^2-2/3*A*c/b*(c*x^2+b*x)^(5/2)-5/12*A*c*(c*x^2+b*x)^(
3/2)*x-5/24*A*b*(c*x^2+b*x)^(3/2)+5/32*A*b^2*(c*x^2+b*x)^(1/2)*x+5/64*A/c*b^3*(c
*x^2+b*x)^(1/2)-5/128*A/c^(3/2)*b^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/
5*B*(c*x^2+b*x)^(5/2)+1/8*B*b*(c*x^2+b*x)^(3/2)*x+1/16*B/c*(c*x^2+b*x)^(3/2)*b^2
-3/64*B*b^3/c*(c*x^2+b*x)^(1/2)*x-3/128*B*b^4/c^2*(c*x^2+b*x)^(1/2)+3/256*B*b^5/
c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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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((c*x^2 + b*x)^(5/2)*(B*x + A)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301904, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (384 \, B c^{4} x^{4} - 45 \, B b^{4} + 150 \, A b^{3} c + 48 \,{\left (21 \, B b c^{3} + 10 \, A c^{4}\right )} x^{3} + 8 \,{\left (93 \, B b^{2} c^{2} + 170 \, A b c^{3}\right )} x^{2} + 10 \,{\left (3 \, B b^{3} c + 118 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (3 \, B b^{5} - 10 \, A b^{4} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{3840 \, c^{\frac{5}{2}}}, \frac{{\left (384 \, B c^{4} x^{4} - 45 \, B b^{4} + 150 \, A b^{3} c + 48 \,{\left (21 \, B b c^{3} + 10 \, A c^{4}\right )} x^{3} + 8 \,{\left (93 \, B b^{2} c^{2} + 170 \, A b c^{3}\right )} x^{2} + 10 \,{\left (3 \, B b^{3} c + 118 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 15 \,{\left (3 \, B b^{5} - 10 \, A b^{4} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{1920 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(384*B*c^4*x^4 - 45*B*b^4 + 150*A*b^3*c + 48*(21*B*b*c^3 + 10*A*c^4)*
x^3 + 8*(93*B*b^2*c^2 + 170*A*b*c^3)*x^2 + 10*(3*B*b^3*c + 118*A*b^2*c^2)*x)*sqr
t(c*x^2 + b*x)*sqrt(c) - 15*(3*B*b^5 - 10*A*b^4*c)*log((2*c*x + b)*sqrt(c) - 2*s
qrt(c*x^2 + b*x)*c))/c^(5/2), 1/1920*((384*B*c^4*x^4 - 45*B*b^4 + 150*A*b^3*c +
48*(21*B*b*c^3 + 10*A*c^4)*x^3 + 8*(93*B*b^2*c^2 + 170*A*b*c^3)*x^2 + 10*(3*B*b^
3*c + 118*A*b^2*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 15*(3*B*b^5 - 10*A*b^4*c)*a
rctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285737, size = 230, normalized size = 1.36 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B c^{2} x + \frac{21 \, B b c^{5} + 10 \, A c^{6}}{c^{4}}\right )} x + \frac{93 \, B b^{2} c^{4} + 170 \, A b c^{5}}{c^{4}}\right )} x + \frac{5 \,{\left (3 \, B b^{3} c^{3} + 118 \, A b^{2} c^{4}\right )}}{c^{4}}\right )} x - \frac{15 \,{\left (3 \, B b^{4} c^{2} - 10 \, A b^{3} c^{3}\right )}}{c^{4}}\right )} - \frac{{\left (3 \, B b^{5} - 10 \, A b^{4} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x)*(2*(4*(6*(8*B*c^2*x + (21*B*b*c^5 + 10*A*c^6)/c^4)*x +
(93*B*b^2*c^4 + 170*A*b*c^5)/c^4)*x + 5*(3*B*b^3*c^3 + 118*A*b^2*c^4)/c^4)*x - 1
5*(3*B*b^4*c^2 - 10*A*b^3*c^3)/c^4) - 1/256*(3*B*b^5 - 10*A*b^4*c)*ln(abs(-2*(sq
rt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)