3.67 \(\int x^3 (A+B x) \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{11/2}}+\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2} (3 b B-4 A c)}{512 c^5}-\frac{7 b^2 \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{192 c^4}+\frac{7 b x \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{160 c^3}-\frac{x^2 \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{3/2}}{6 c} \]

[Out]

(7*b^3*(3*b*B - 4*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^5) - (7*b^2*(3*b*B
- 4*A*c)*(b*x + c*x^2)^(3/2))/(192*c^4) + (7*b*(3*b*B - 4*A*c)*x*(b*x + c*x^2)^(
3/2))/(160*c^3) - ((3*b*B - 4*A*c)*x^2*(b*x + c*x^2)^(3/2))/(20*c^2) + (B*x^3*(b
*x + c*x^2)^(3/2))/(6*c) - (7*b^5*(3*b*B - 4*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(512*c^(11/2))

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Rubi [A]  time = 0.440358, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{11/2}}+\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2} (3 b B-4 A c)}{512 c^5}-\frac{7 b^2 \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{192 c^4}+\frac{7 b x \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{160 c^3}-\frac{x^2 \left (b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(7*b^3*(3*b*B - 4*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^5) - (7*b^2*(3*b*B
- 4*A*c)*(b*x + c*x^2)^(3/2))/(192*c^4) + (7*b*(3*b*B - 4*A*c)*x*(b*x + c*x^2)^(
3/2))/(160*c^3) - ((3*b*B - 4*A*c)*x^2*(b*x + c*x^2)^(3/2))/(20*c^2) + (B*x^3*(b
*x + c*x^2)^(3/2))/(6*c) - (7*b^5*(3*b*B - 4*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(512*c^(11/2))

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Rubi in Sympy [A]  time = 28.3072, size = 196, normalized size = 0.98 \[ \frac{B x^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{6 c} + \frac{7 b^{5} \left (4 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{11}{2}}} - \frac{7 b^{3} \left (b + 2 c x\right ) \left (4 A c - 3 B b\right ) \sqrt{b x + c x^{2}}}{512 c^{5}} + \frac{7 b^{2} \left (4 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{4}} - \frac{7 b x \left (4 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{160 c^{3}} + \frac{x^{2} \left (4 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{20 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(c*x**2+b*x)**(1/2),x)

[Out]

B*x**3*(b*x + c*x**2)**(3/2)/(6*c) + 7*b**5*(4*A*c - 3*B*b)*atanh(sqrt(c)*x/sqrt
(b*x + c*x**2))/(512*c**(11/2)) - 7*b**3*(b + 2*c*x)*(4*A*c - 3*B*b)*sqrt(b*x +
c*x**2)/(512*c**5) + 7*b**2*(4*A*c - 3*B*b)*(b*x + c*x**2)**(3/2)/(192*c**4) - 7
*b*x*(4*A*c - 3*B*b)*(b*x + c*x**2)**(3/2)/(160*c**3) + x**2*(4*A*c - 3*B*b)*(b*
x + c*x**2)**(3/2)/(20*c**2)

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Mathematica [A]  time = 0.311663, size = 168, normalized size = 0.84 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-210 b^4 c (2 A+B x)+56 b^3 c^2 x (5 A+3 B x)-16 b^2 c^3 x^2 (14 A+9 B x)+64 b c^4 x^3 (3 A+2 B x)+256 c^5 x^4 (6 A+5 B x)+315 b^5 B\right )-\frac{105 b^5 (3 b B-4 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{7680 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(315*b^5*B - 210*b^4*c*(2*A + B*x) + 64*b*c^4*x^3*(3
*A + 2*B*x) + 56*b^3*c^2*x*(5*A + 3*B*x) + 256*c^5*x^4*(6*A + 5*B*x) - 16*b^2*c^
3*x^2*(14*A + 9*B*x)) - (105*b^5*(3*b*B - 4*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b
+ c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(7680*c^(11/2))

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Maple [A]  time = 0.013, size = 291, normalized size = 1.5 \[{\frac{A{x}^{2}}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,Abx}{40\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}A}{48\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{3}x}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,A{b}^{4}}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,A{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{B{x}^{3}}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Bb{x}^{2}}{20\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{21\,{b}^{2}Bx}{160\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,B{b}^{3}}{64\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{21\,{b}^{4}Bx}{256\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{21\,B{b}^{5}}{512\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{21\,B{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+b*x)^(1/2),x)

[Out]

1/5*A*x^2*(c*x^2+b*x)^(3/2)/c-7/40*A*b/c^2*x*(c*x^2+b*x)^(3/2)+7/48*A*b^2/c^3*(c
*x^2+b*x)^(3/2)-7/64*A*b^3/c^3*(c*x^2+b*x)^(1/2)*x-7/128*A*b^4/c^4*(c*x^2+b*x)^(
1/2)+7/256*A*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/6*B*x^3*(c*
x^2+b*x)^(3/2)/c-3/20*B*b/c^2*x^2*(c*x^2+b*x)^(3/2)+21/160*B*b^2/c^3*x*(c*x^2+b*
x)^(3/2)-7/64*B*b^3/c^4*(c*x^2+b*x)^(3/2)+21/256*B*b^4/c^4*(c*x^2+b*x)^(1/2)*x+2
1/512*B*b^5/c^5*(c*x^2+b*x)^(1/2)-21/1024*B*b^6/c^(11/2)*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292339, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B c^{5} x^{5} + 315 \, B b^{5} - 420 \, A b^{4} c + 128 \,{\left (B b c^{4} + 12 \, A c^{5}\right )} x^{4} - 48 \,{\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} x^{3} + 56 \,{\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} x^{2} - 70 \,{\left (3 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{11}{2}}}, \frac{{\left (1280 \, B c^{5} x^{5} + 315 \, B b^{5} - 420 \, A b^{4} c + 128 \,{\left (B b c^{4} + 12 \, A c^{5}\right )} x^{4} - 48 \,{\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} x^{3} + 56 \,{\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} x^{2} - 70 \,{\left (3 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 105 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(2*(1280*B*c^5*x^5 + 315*B*b^5 - 420*A*b^4*c + 128*(B*b*c^4 + 12*A*c^5)
*x^4 - 48*(3*B*b^2*c^3 - 4*A*b*c^4)*x^3 + 56*(3*B*b^3*c^2 - 4*A*b^2*c^3)*x^2 - 7
0*(3*B*b^4*c - 4*A*b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 105*(3*B*b^6 - 4*A*b^
5*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(11/2), 1/7680*((1280*B
*c^5*x^5 + 315*B*b^5 - 420*A*b^4*c + 128*(B*b*c^4 + 12*A*c^5)*x^4 - 48*(3*B*b^2*
c^3 - 4*A*b*c^4)*x^3 + 56*(3*B*b^3*c^2 - 4*A*b^2*c^3)*x^2 - 70*(3*B*b^4*c - 4*A*
b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(3*B*b^6 - 4*A*b^5*c)*arctan(sqrt(c
*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(c*x**2+b*x)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.282328, size = 254, normalized size = 1.27 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B x + \frac{B b c^{4} + 12 \, A c^{5}}{c^{5}}\right )} x - \frac{3 \,{\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )}}{c^{5}}\right )} x + \frac{7 \,{\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )}}{c^{5}}\right )} x - \frac{35 \,{\left (3 \, B b^{4} c - 4 \, A b^{3} c^{2}\right )}}{c^{5}}\right )} x + \frac{105 \,{\left (3 \, B b^{5} - 4 \, A b^{4} c\right )}}{c^{5}}\right )} + \frac{7 \,{\left (3 \, B b^{6} - 4 \, A b^{5} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*x + (B*b*c^4 + 12*A*c^5)/c^5)*x - 3*(
3*B*b^2*c^3 - 4*A*b*c^4)/c^5)*x + 7*(3*B*b^3*c^2 - 4*A*b^2*c^3)/c^5)*x - 35*(3*B
*b^4*c - 4*A*b^3*c^2)/c^5)*x + 105*(3*B*b^5 - 4*A*b^4*c)/c^5) + 7/1024*(3*B*b^6
- 4*A*b^5*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)