3.69 \(\int x (A+B x) \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2} (5 b B-8 A c)}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]

[Out]

(b*(5*b*B - 8*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) - ((5*b*B - 8*A*c - 6
*B*c*x)*(b*x + c*x^2)^(3/2))/(24*c^2) - (b^3*(5*b*B - 8*A*c)*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(64*c^(7/2))

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Rubi [A]  time = 0.137167, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2} (5 b B-8 A c)}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(b*(5*b*B - 8*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64*c^3) - ((5*b*B - 8*A*c - 6
*B*c*x)*(b*x + c*x^2)^(3/2))/(24*c^2) - (b^3*(5*b*B - 8*A*c)*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(64*c^(7/2))

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Rubi in Sympy [A]  time = 15.6031, size = 121, normalized size = 1.07 \[ \frac{B x \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 c} + \frac{b^{3} \left (8 A c - 5 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} - \frac{b \left (b + 2 c x\right ) \left (8 A c - 5 B b\right ) \sqrt{b x + c x^{2}}}{64 c^{3}} + \frac{\left (8 A c - 5 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.221047, size = 130, normalized size = 1.15 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-2 b^2 c (12 A+5 B x)+8 b c^2 x (2 A+B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )-\frac{3 b^3 (5 b B-8 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*B + 8*b*c^2*x*(2*A + B*x) + 16*c^3*x^2*(4*A
+ 3*B*x) - 2*b^2*c*(12*A + 5*B*x)) - (3*b^3*(5*b*B - 8*A*c)*Log[c*Sqrt[x] + Sqrt
[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^(7/2))

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Maple [B]  time = 0.01, size = 201, normalized size = 1.8 \[{\frac{A}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{Abx}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}A}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Bx}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bb}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281557, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} - 24 \, A b^{2} c + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{7}{2}}}, \frac{{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} - 24 \, A b^{2} c + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*B*c^3*x^3 + 15*B*b^3 - 24*A*b^2*c + 8*(B*b*c^2 + 8*A*c^3)*x^2 - 2*
(5*B*b^2*c - 8*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 3*(5*B*b^4 - 8*A*b^3*c)*l
og((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(7/2), 1/192*((48*B*c^3*x^3 +
15*B*b^3 - 24*A*b^2*c + 8*(B*b*c^2 + 8*A*c^3)*x^2 - 2*(5*B*b^2*c - 8*A*b*c^2)*x
)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3*(5*B*b^4 - 8*A*b^3*c)*arctan(sqrt(c*x^2 + b*x)*
sqrt(-c)/(c*x)))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(x*(b + c*x))*(A + B*x), x)

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GIAC/XCAS [A]  time = 0.281923, size = 178, normalized size = 1.58 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B x + \frac{B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac{5 \, B b^{2} c - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac{3 \,{\left (5 \, B b^{3} - 8 \, A b^{2} c\right )}}{c^{3}}\right )} + \frac{{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*x + (B*b*c^2 + 8*A*c^3)/c^3)*x - (5*B*b^2*c -
8*A*b*c^2)/c^3)*x + 3*(5*B*b^3 - 8*A*b^2*c)/c^3) + 1/128*(5*B*b^4 - 8*A*b^3*c)*
ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)