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

Optimal. Leaf size=132 \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]

[Out]

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

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Rubi [A]  time = 0.240908, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.2428, size = 119, normalized size = 0.9 \[ \frac{B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{4 c x} - \frac{b^{3} \left (8 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{5}{2}}} + \frac{b \left (b + 2 c x\right ) \left (8 A c - 3 B b\right ) \sqrt{b x + c x^{2}}}{64 c^{2}} + \frac{\left (8 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.228852, size = 130, normalized size = 0.98 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b^3 (3 b B-8 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (6 b^2 c (4 A+B x)+8 b c^2 x (14 A+9 B x)+16 c^3 x^2 (4 A+3 B x)-9 b^3 B\right )\right )}{192 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-9*b^3*B + 6*b^2*c*(4*A + B*x) + 16*c^3*x^2*(4*A +
3*B*x) + 8*b*c^2*x*(14*A + 9*B*x)) + (3*b^3*(3*b*B - 8*A*c)*Log[c*Sqrt[x] + Sqrt
[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^(5/2))

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Maple [A]  time = 0.01, size = 192, normalized size = 1.5 \[{\frac{Bx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}Bx}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,B{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Abx}{4}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{2}A}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284719, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} x^{3} - 9 \, B b^{3} + 24 \, A b^{2} c + 8 \,{\left (9 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + 56 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{5}{2}}}, \frac{{\left (48 \, B c^{3} x^{3} - 9 \, B b^{3} + 24 \, A b^{2} c + 8 \,{\left (9 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + 56 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*B*c^3*x^3 - 9*B*b^3 + 24*A*b^2*c + 8*(9*B*b*c^2 + 8*A*c^3)*x^2 + 2
*(3*B*b^2*c + 56*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 3*(3*B*b^4 - 8*A*b^3*c)
*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/c^(5/2), 1/192*((48*B*c^3*x^3
 - 9*B*b^3 + 24*A*b^2*c + 8*(9*B*b*c^2 + 8*A*c^3)*x^2 + 2*(3*B*b^2*c + 56*A*b*c^
2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 3*(3*B*b^4 - 8*A*b^3*c)*arctan(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. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285228, size = 186, normalized size = 1.41 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B c x + \frac{9 \, B b c^{3} + 8 \, A c^{4}}{c^{3}}\right )} x + \frac{3 \, B b^{2} c^{2} + 56 \, A b c^{3}}{c^{3}}\right )} x - \frac{3 \,{\left (3 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )}}{c^{3}}\right )} - \frac{{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*c*x + (9*B*b*c^3 + 8*A*c^4)/c^3)*x + (3*B*b^2
*c^2 + 56*A*b*c^3)/c^3)*x - 3*(3*B*b^3*c - 8*A*b^2*c^2)/c^3) - 1/128*(3*B*b^4 -
8*A*b^3*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)