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

Optimal. Leaf size=203 \[ -\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{b^4 (b+2 c x) \sqrt{b x+c x^2} (9 b B-14 A c)}{1024 c^5}-\frac{b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-14 A c)}{384 c^4}+\frac{b \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{120 c^3}-\frac{x \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c} \]

[Out]

(b^4*(9*b*B - 14*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) - (b^2*(9*b*B -
14*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^4) + (b*(9*b*B - 14*A*c)*(b*x +
c*x^2)^(5/2))/(120*c^3) - ((9*b*B - 14*A*c)*x*(b*x + c*x^2)^(5/2))/(84*c^2) + (B
*x^2*(b*x + c*x^2)^(5/2))/(7*c) - (b^6*(9*b*B - 14*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt
[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi [A]  time = 0.440589, antiderivative size = 203, 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 \[ -\frac{b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{b^4 (b+2 c x) \sqrt{b x+c x^2} (9 b B-14 A c)}{1024 c^5}-\frac{b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-14 A c)}{384 c^4}+\frac{b \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{120 c^3}-\frac{x \left (b x+c x^2\right )^{5/2} (9 b B-14 A c)}{84 c^2}+\frac{B x^2 \left (b x+c x^2\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(b^4*(9*b*B - 14*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) - (b^2*(9*b*B -
14*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^4) + (b*(9*b*B - 14*A*c)*(b*x +
c*x^2)^(5/2))/(120*c^3) - ((9*b*B - 14*A*c)*x*(b*x + c*x^2)^(5/2))/(84*c^2) + (B
*x^2*(b*x + c*x^2)^(5/2))/(7*c) - (b^6*(9*b*B - 14*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt
[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi in Sympy [A]  time = 27.4985, size = 192, normalized size = 0.95 \[ \frac{B x^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} + \frac{b^{6} \left (14 A c - 9 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{1024 c^{\frac{11}{2}}} - \frac{b^{4} \left (b + 2 c x\right ) \left (14 A c - 9 B b\right ) \sqrt{b x + c x^{2}}}{1024 c^{5}} + \frac{b^{2} \left (b + 2 c x\right ) \left (14 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{4}} - \frac{b \left (14 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{120 c^{3}} + \frac{x \left (14 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{84 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(b*x + c*x**2)**(5/2)/(7*c) + b**6*(14*A*c - 9*B*b)*atanh(sqrt(c)*x/sqrt(
b*x + c*x**2))/(1024*c**(11/2)) - b**4*(b + 2*c*x)*(14*A*c - 9*B*b)*sqrt(b*x + c
*x**2)/(1024*c**5) + b**2*(b + 2*c*x)*(14*A*c - 9*B*b)*(b*x + c*x**2)**(3/2)/(38
4*c**4) - b*(14*A*c - 9*B*b)*(b*x + c*x**2)**(5/2)/(120*c**3) + x*(14*A*c - 9*B*
b)*(b*x + c*x**2)**(5/2)/(84*c**2)

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Mathematica [A]  time = 0.371823, size = 188, normalized size = 0.93 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-210 b^5 c (7 A+3 B x)+28 b^4 c^2 x (35 A+18 B x)-16 b^3 c^3 x^2 (49 A+27 B x)+96 b^2 c^4 x^3 (7 A+4 B x)+256 b c^5 x^4 (91 A+75 B x)+2560 c^6 x^5 (7 A+6 B x)+945 b^6 B\right )-\frac{105 b^6 (9 b B-14 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{107520 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^6*B - 210*b^5*c*(7*A + 3*B*x) + 96*b^2*c^4*x^
3*(7*A + 4*B*x) + 2560*c^6*x^5*(7*A + 6*B*x) + 28*b^4*c^2*x*(35*A + 18*B*x) - 16
*b^3*c^3*x^2*(49*A + 27*B*x) + 256*b*c^5*x^4*(91*A + 75*B*x)) - (105*b^6*(9*b*B
- 14*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(107
520*c^(11/2))

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Maple [A]  time = 0.011, size = 327, normalized size = 1.6 \[{\frac{Ax}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,Ab}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,Ax{b}^{2}}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{3}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,A{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,A{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{B{x}^{2}}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,xBb}{28\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{2}B}{40\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,Bx{b}^{3}}{64\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{4}B}{128\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{9\,B{b}^{5}x}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{9\,B{b}^{6}}{1024\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{9\,B{b}^{7}}{2048}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*A*x*(c*x^2+b*x)^(5/2)/c-7/60*A*b/c^2*(c*x^2+b*x)^(5/2)+7/96*A*b^2/c^2*(c*x^2
+b*x)^(3/2)*x+7/192*A*b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*A*b^4/c^3*(c*x^2+b*x)^(1/2
)*x-7/512*A*b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*A*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x)^(1/2))+1/7*B*x^2*(c*x^2+b*x)^(5/2)/c-3/28*B*b/c^2*x*(c*x^2+b*x)^(5
/2)+3/40*B*b^2/c^3*(c*x^2+b*x)^(5/2)-3/64*B*b^3/c^3*(c*x^2+b*x)^(3/2)*x-3/128*B*
b^4/c^4*(c*x^2+b*x)^(3/2)+9/512*B*b^5/c^4*(c*x^2+b*x)^(1/2)*x+9/1024*B*b^6/c^5*(
c*x^2+b*x)^(1/2)-9/2048*B*b^7/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((c*x^2 + b*x)^(3/2)*(B*x + A)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291487, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (15360 \, B c^{6} x^{6} + 945 \, B b^{6} - 1470 \, A b^{5} c + 1280 \,{\left (15 \, B b c^{5} + 14 \, A c^{6}\right )} x^{5} + 128 \,{\left (3 \, B b^{2} c^{4} + 182 \, A b c^{5}\right )} x^{4} - 48 \,{\left (9 \, B b^{3} c^{3} - 14 \, A b^{2} c^{4}\right )} x^{3} + 56 \,{\left (9 \, B b^{4} c^{2} - 14 \, A b^{3} c^{3}\right )} x^{2} - 70 \,{\left (9 \, B b^{5} c - 14 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{215040 \, c^{\frac{11}{2}}}, \frac{{\left (15360 \, B c^{6} x^{6} + 945 \, B b^{6} - 1470 \, A b^{5} c + 1280 \,{\left (15 \, B b c^{5} + 14 \, A c^{6}\right )} x^{5} + 128 \,{\left (3 \, B b^{2} c^{4} + 182 \, A b c^{5}\right )} x^{4} - 48 \,{\left (9 \, B b^{3} c^{3} - 14 \, A b^{2} c^{4}\right )} x^{3} + 56 \,{\left (9 \, B b^{4} c^{2} - 14 \, A b^{3} c^{3}\right )} x^{2} - 70 \,{\left (9 \, B b^{5} c - 14 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 105 \,{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{107520 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/215040*(2*(15360*B*c^6*x^6 + 945*B*b^6 - 1470*A*b^5*c + 1280*(15*B*b*c^5 + 14
*A*c^6)*x^5 + 128*(3*B*b^2*c^4 + 182*A*b*c^5)*x^4 - 48*(9*B*b^3*c^3 - 14*A*b^2*c
^4)*x^3 + 56*(9*B*b^4*c^2 - 14*A*b^3*c^3)*x^2 - 70*(9*B*b^5*c - 14*A*b^4*c^2)*x)
*sqrt(c*x^2 + b*x)*sqrt(c) - 105*(9*B*b^7 - 14*A*b^6*c)*log((2*c*x + b)*sqrt(c)
+ 2*sqrt(c*x^2 + b*x)*c))/c^(11/2), 1/107520*((15360*B*c^6*x^6 + 945*B*b^6 - 147
0*A*b^5*c + 1280*(15*B*b*c^5 + 14*A*c^6)*x^5 + 128*(3*B*b^2*c^4 + 182*A*b*c^5)*x
^4 - 48*(9*B*b^3*c^3 - 14*A*b^2*c^4)*x^3 + 56*(9*B*b^4*c^2 - 14*A*b^3*c^3)*x^2 -
70*(9*B*b^5*c - 14*A*b^4*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(9*B*b^7 - 14
*A*b^6*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. \[ \int x^{2} \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285887, size = 300, normalized size = 1.48 \[ \frac{1}{107520} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x + \frac{15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{2} c^{5} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac{3 \,{\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )}}{c^{6}}\right )} x - \frac{35 \,{\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (9 \, B b^{6} c - 14 \, A b^{5} c^{2}\right )}}{c^{6}}\right )} + \frac{{\left (9 \, B b^{7} - 14 \, A b^{6} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*x + (15*B*b*c^6 + 14*A*c^7)/c
^6)*x + (3*B*b^2*c^5 + 182*A*b*c^6)/c^6)*x - 3*(9*B*b^3*c^4 - 14*A*b^2*c^5)/c^6)
*x + 7*(9*B*b^4*c^3 - 14*A*b^3*c^4)/c^6)*x - 35*(9*B*b^5*c^2 - 14*A*b^4*c^3)/c^6
)*x + 105*(9*B*b^6*c - 14*A*b^5*c^2)/c^6) + 1/2048*(9*B*b^7 - 14*A*b^6*c)*ln(abs
(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)