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

Optimal. Leaf size=151 \[ \frac{b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (7 b B-12 A c)}{512 c^4}+\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (7 b B-12 A c)}{192 c^3}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]

[Out]

-(b^3*(7*b*B - 12*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + (b*(7*b*B - 12
*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) - ((7*b*B - 12*A*c - 10*B*c*x)*
(b*x + c*x^2)^(5/2))/(60*c^2) + (b^5*(7*b*B - 12*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b
*x + c*x^2]])/(512*c^(9/2))

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Rubi [A]  time = 0.174524, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}-\frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (7 b B-12 A c)}{512 c^4}+\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (7 b B-12 A c)}{192 c^3}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b^3*(7*b*B - 12*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + (b*(7*b*B - 12
*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) - ((7*b*B - 12*A*c - 10*B*c*x)*
(b*x + c*x^2)^(5/2))/(60*c^2) + (b^5*(7*b*B - 12*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b
*x + c*x^2]])/(512*c^(9/2))

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Rubi in Sympy [A]  time = 20.7259, size = 158, normalized size = 1.05 \[ \frac{B x \left (b x + c x^{2}\right )^{\frac{5}{2}}}{6 c} - \frac{b^{5} \left (12 A c - 7 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} + \frac{b^{3} \left (b + 2 c x\right ) \left (12 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{512 c^{4}} - \frac{b \left (b + 2 c x\right ) \left (12 A c - 7 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{3}} + \frac{\left (12 A c - 7 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{60 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x*(b*x + c*x**2)**(5/2)/(6*c) - b**5*(12*A*c - 7*B*b)*atanh(sqrt(c)*x/sqrt(b*x
+ c*x**2))/(512*c**(9/2)) + b**3*(b + 2*c*x)*(12*A*c - 7*B*b)*sqrt(b*x + c*x**2
)/(512*c**4) - b*(b + 2*c*x)*(12*A*c - 7*B*b)*(b*x + c*x**2)**(3/2)/(192*c**3) +
(12*A*c - 7*B*b)*(b*x + c*x**2)**(5/2)/(60*c**2)

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Mathematica [A]  time = 0.30448, size = 168, normalized size = 1.11 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^5 (7 b B-12 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (10 b^4 c (18 A+7 B x)-8 b^3 c^2 x (15 A+7 B x)+48 b^2 c^3 x^2 (2 A+B x)+64 b c^4 x^3 (33 A+26 B x)+256 c^5 x^4 (6 A+5 B x)-105 b^5 B\right )\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^5*B + 48*b^2*c^3*x^2*(2*A + B*x) + 256*c^5*x
^4*(6*A + 5*B*x) - 8*b^3*c^2*x*(15*A + 7*B*x) + 10*b^4*c*(18*A + 7*B*x) + 64*b*c
^4*x^3*(33*A + 26*B*x)) + (15*b^5*(7*b*B - 12*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[
b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(7680*c^(9/2))

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Maple [B]  time = 0.01, size = 283, normalized size = 1.9 \[{\frac{A}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{Abx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}A}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{b}^{3}x}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{4}}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{Bx}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,Bb}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}Bx}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,B{b}^{3}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{4}Bx}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,B{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,B{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*A*(c*x^2+b*x)^(5/2)/c-1/8*A*b/c*(c*x^2+b*x)^(3/2)*x-1/16*A*b^2/c^2*(c*x^2+b*
x)^(3/2)+3/64*A*b^3/c^2*(c*x^2+b*x)^(1/2)*x+3/128*A*b^4/c^3*(c*x^2+b*x)^(1/2)-3/
256*A*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/6*B*x*(c*x^2+b*x)^
(5/2)/c-7/60*B*b/c^2*(c*x^2+b*x)^(5/2)+7/96*B*b^2/c^2*(c*x^2+b*x)^(3/2)*x+7/192*
B*b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*B*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*B*b^5/c^4*
(c*x^2+b*x)^(1/2)+7/1024*B*b^6/c^(9/2)*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 = 0.283116, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (1280 \, B c^{5} x^{5} - 105 \, B b^{5} + 180 \, A b^{4} c + 128 \,{\left (13 \, B b c^{4} + 12 \, A c^{5}\right )} x^{4} + 48 \,{\left (B b^{2} c^{3} + 44 \, A b c^{4}\right )} x^{3} - 8 \,{\left (7 \, B b^{3} c^{2} - 12 \, A b^{2} c^{3}\right )} x^{2} + 10 \,{\left (7 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{9}{2}}}, \frac{{\left (1280 \, B c^{5} x^{5} - 105 \, B b^{5} + 180 \, A b^{4} c + 128 \,{\left (13 \, B b c^{4} + 12 \, A c^{5}\right )} x^{4} + 48 \,{\left (B b^{2} c^{3} + 44 \, A b c^{4}\right )} x^{3} - 8 \,{\left (7 \, B b^{3} c^{2} - 12 \, A b^{2} c^{3}\right )} x^{2} + 10 \,{\left (7 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 15 \,{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{4}}\right ] \]

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]

[1/15360*(2*(1280*B*c^5*x^5 - 105*B*b^5 + 180*A*b^4*c + 128*(13*B*b*c^4 + 12*A*c
^5)*x^4 + 48*(B*b^2*c^3 + 44*A*b*c^4)*x^3 - 8*(7*B*b^3*c^2 - 12*A*b^2*c^3)*x^2 +
10*(7*B*b^4*c - 12*A*b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(7*B*b^6 - 12*A
*b^5*c)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/c^(9/2), 1/7680*((1280
*B*c^5*x^5 - 105*B*b^5 + 180*A*b^4*c + 128*(13*B*b*c^4 + 12*A*c^5)*x^4 + 48*(B*b
^2*c^3 + 44*A*b*c^4)*x^3 - 8*(7*B*b^3*c^2 - 12*A*b^2*c^3)*x^2 + 10*(7*B*b^4*c -
12*A*b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 15*(7*B*b^6 - 12*A*b^5*c)*arctan(s
qrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282601, size = 262, normalized size = 1.74 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c x + \frac{13 \, B b c^{5} + 12 \, A c^{6}}{c^{5}}\right )} x + \frac{3 \,{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )}}{c^{5}}\right )} x - \frac{7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}}{c^{5}}\right )} x + \frac{5 \,{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )}}{c^{5}}\right )} x - \frac{15 \,{\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )}}{c^{5}}\right )} - \frac{{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]

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]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*c*x + (13*B*b*c^5 + 12*A*c^6)/c^5)*x
+ 3*(B*b^2*c^4 + 44*A*b*c^5)/c^5)*x - (7*B*b^3*c^3 - 12*A*b^2*c^4)/c^5)*x + 5*(7
*B*b^4*c^2 - 12*A*b^3*c^3)/c^5)*x - 15*(7*B*b^5*c - 12*A*b^4*c^2)/c^5) - 1/1024*
(7*B*b^6 - 12*A*b^5*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c
^(9/2)