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

Optimal. Leaf size=238 \[ \frac{9 b^7 (11 b B-16 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{13/2}}-\frac{9 b^5 (b+2 c x) \sqrt{b x+c x^2} (11 b B-16 A c)}{16384 c^6}+\frac{3 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-16 A c)}{2048 c^5}-\frac{3 b^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{640 c^4}+\frac{3 b x \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{448 c^3}-\frac{x^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{5/2}}{8 c} \]

[Out]

(-9*b^5*(11*b*B - 16*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^6) + (3*b^3*(1
1*b*B - 16*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(2048*c^5) - (3*b^2*(11*b*B - 1
6*A*c)*(b*x + c*x^2)^(5/2))/(640*c^4) + (3*b*(11*b*B - 16*A*c)*x*(b*x + c*x^2)^(
5/2))/(448*c^3) - ((11*b*B - 16*A*c)*x^2*(b*x + c*x^2)^(5/2))/(112*c^2) + (B*x^3
*(b*x + c*x^2)^(5/2))/(8*c) + (9*b^7*(11*b*B - 16*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[
b*x + c*x^2]])/(16384*c^(13/2))

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Rubi [A]  time = 0.535965, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{9 b^7 (11 b B-16 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{13/2}}-\frac{9 b^5 (b+2 c x) \sqrt{b x+c x^2} (11 b B-16 A c)}{16384 c^6}+\frac{3 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-16 A c)}{2048 c^5}-\frac{3 b^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{640 c^4}+\frac{3 b x \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{448 c^3}-\frac{x^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{5/2}}{8 c} \]

Antiderivative was successfully verified.

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

[Out]

(-9*b^5*(11*b*B - 16*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^6) + (3*b^3*(1
1*b*B - 16*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(2048*c^5) - (3*b^2*(11*b*B - 1
6*A*c)*(b*x + c*x^2)^(5/2))/(640*c^4) + (3*b*(11*b*B - 16*A*c)*x*(b*x + c*x^2)^(
5/2))/(448*c^3) - ((11*b*B - 16*A*c)*x^2*(b*x + c*x^2)^(5/2))/(112*c^2) + (B*x^3
*(b*x + c*x^2)^(5/2))/(8*c) + (9*b^7*(11*b*B - 16*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[
b*x + c*x^2]])/(16384*c^(13/2))

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Rubi in Sympy [A]  time = 34.7411, size = 235, normalized size = 0.99 \[ \frac{B x^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{8 c} - \frac{9 b^{7} \left (16 A c - 11 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16384 c^{\frac{13}{2}}} + \frac{9 b^{5} \left (b + 2 c x\right ) \left (16 A c - 11 B b\right ) \sqrt{b x + c x^{2}}}{16384 c^{6}} - \frac{3 b^{3} \left (b + 2 c x\right ) \left (16 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{2048 c^{5}} + \frac{3 b^{2} \left (16 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{640 c^{4}} - \frac{3 b x \left (16 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{448 c^{3}} + \frac{x^{2} \left (16 A c - 11 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{112 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**3*(b*x + c*x**2)**(5/2)/(8*c) - 9*b**7*(16*A*c - 11*B*b)*atanh(sqrt(c)*x/sq
rt(b*x + c*x**2))/(16384*c**(13/2)) + 9*b**5*(b + 2*c*x)*(16*A*c - 11*B*b)*sqrt(
b*x + c*x**2)/(16384*c**6) - 3*b**3*(b + 2*c*x)*(16*A*c - 11*B*b)*(b*x + c*x**2)
**(3/2)/(2048*c**5) + 3*b**2*(16*A*c - 11*B*b)*(b*x + c*x**2)**(5/2)/(640*c**4)
- 3*b*x*(16*A*c - 11*B*b)*(b*x + c*x**2)**(5/2)/(448*c**3) + x**2*(16*A*c - 11*B
*b)*(b*x + c*x**2)**(5/2)/(112*c**2)

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Mathematica [A]  time = 0.409369, size = 207, normalized size = 0.87 \[ \frac{\sqrt{x (b+c x)} \left (\frac{315 b^7 (11 b B-16 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (210 b^6 c (24 A+11 B x)-168 b^5 c^2 x (20 A+11 B x)+48 b^4 c^3 x^2 (56 A+33 B x)-128 b^3 c^4 x^3 (18 A+11 B x)+256 b^2 c^5 x^4 (8 A+5 B x)+5120 b c^6 x^5 (20 A+17 B x)+10240 c^7 x^6 (8 A+7 B x)-3465 b^7 B\right )\right )}{573440 c^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3465*b^7*B + 256*b^2*c^5*x^4*(8*A + 5*B*x) + 10240
*c^7*x^6*(8*A + 7*B*x) - 128*b^3*c^4*x^3*(18*A + 11*B*x) - 168*b^5*c^2*x*(20*A +
11*B*x) + 210*b^6*c*(24*A + 11*B*x) + 5120*b*c^6*x^5*(20*A + 17*B*x) + 48*b^4*c
^3*x^2*(56*A + 33*B*x)) + (315*b^7*(11*b*B - 16*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqr
t[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(573440*c^(13/2))

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Maple [A]  time = 0.015, size = 373, normalized size = 1.6 \[{\frac{A{x}^{2}}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,Abx}{28\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{2}A}{40\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{3\,A{b}^{3}x}{64\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{4}}{128\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{9\,A{b}^{5}x}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{9\,A{b}^{6}}{1024\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{9\,A{b}^{7}}{2048}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}}+{\frac{B{x}^{3}}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{11\,Bb{x}^{2}}{112\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{33\,{b}^{2}Bx}{448\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{33\,B{b}^{3}}{640\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{33\,{b}^{4}Bx}{1024\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{33\,B{b}^{5}}{2048\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{99\,{b}^{6}Bx}{8192\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{99\,B{b}^{7}}{16384\,{c}^{6}}\sqrt{c{x}^{2}+bx}}+{\frac{99\,B{b}^{8}}{32768}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*A*x^2*(c*x^2+b*x)^(5/2)/c-3/28*A*b/c^2*x*(c*x^2+b*x)^(5/2)+3/40*A*b^2/c^3*(c
*x^2+b*x)^(5/2)-3/64*A*b^3/c^3*(c*x^2+b*x)^(3/2)*x-3/128*A*b^4/c^4*(c*x^2+b*x)^(
3/2)+9/512*A*b^5/c^4*(c*x^2+b*x)^(1/2)*x+9/1024*A*b^6/c^5*(c*x^2+b*x)^(1/2)-9/20
48*A*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*B*x^3*(c*x^2+b*x
)^(5/2)/c-11/112*B*b/c^2*x^2*(c*x^2+b*x)^(5/2)+33/448*B*b^2/c^3*x*(c*x^2+b*x)^(5
/2)-33/640*B*b^3/c^4*(c*x^2+b*x)^(5/2)+33/1024*B*b^4/c^4*(c*x^2+b*x)^(3/2)*x+33/
2048*B*b^5/c^5*(c*x^2+b*x)^(3/2)-99/8192*B*b^6/c^5*(c*x^2+b*x)^(1/2)*x-99/16384*
B*b^7/c^6*(c*x^2+b*x)^(1/2)+99/32768*B*b^8/c^(13/2)*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294276, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (71680 \, B c^{7} x^{7} - 3465 \, B b^{7} + 5040 \, A b^{6} c + 5120 \,{\left (17 \, B b c^{6} + 16 \, A c^{7}\right )} x^{6} + 1280 \,{\left (B b^{2} c^{5} + 80 \, A b c^{6}\right )} x^{5} - 128 \,{\left (11 \, B b^{3} c^{4} - 16 \, A b^{2} c^{5}\right )} x^{4} + 144 \,{\left (11 \, B b^{4} c^{3} - 16 \, A b^{3} c^{4}\right )} x^{3} - 168 \,{\left (11 \, B b^{5} c^{2} - 16 \, A b^{4} c^{3}\right )} x^{2} + 210 \,{\left (11 \, B b^{6} c - 16 \, A b^{5} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 315 \,{\left (11 \, B b^{8} - 16 \, A b^{7} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{1146880 \, c^{\frac{13}{2}}}, \frac{{\left (71680 \, B c^{7} x^{7} - 3465 \, B b^{7} + 5040 \, A b^{6} c + 5120 \,{\left (17 \, B b c^{6} + 16 \, A c^{7}\right )} x^{6} + 1280 \,{\left (B b^{2} c^{5} + 80 \, A b c^{6}\right )} x^{5} - 128 \,{\left (11 \, B b^{3} c^{4} - 16 \, A b^{2} c^{5}\right )} x^{4} + 144 \,{\left (11 \, B b^{4} c^{3} - 16 \, A b^{3} c^{4}\right )} x^{3} - 168 \,{\left (11 \, B b^{5} c^{2} - 16 \, A b^{4} c^{3}\right )} x^{2} + 210 \,{\left (11 \, B b^{6} c - 16 \, A b^{5} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 315 \,{\left (11 \, B b^{8} - 16 \, A b^{7} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{573440 \, \sqrt{-c} c^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1146880*(2*(71680*B*c^7*x^7 - 3465*B*b^7 + 5040*A*b^6*c + 5120*(17*B*b*c^6 +
16*A*c^7)*x^6 + 1280*(B*b^2*c^5 + 80*A*b*c^6)*x^5 - 128*(11*B*b^3*c^4 - 16*A*b^2
*c^5)*x^4 + 144*(11*B*b^4*c^3 - 16*A*b^3*c^4)*x^3 - 168*(11*B*b^5*c^2 - 16*A*b^4
*c^3)*x^2 + 210*(11*B*b^6*c - 16*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 315*(
11*B*b^8 - 16*A*b^7*c)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c))/c^(13/2
), 1/573440*((71680*B*c^7*x^7 - 3465*B*b^7 + 5040*A*b^6*c + 5120*(17*B*b*c^6 + 1
6*A*c^7)*x^6 + 1280*(B*b^2*c^5 + 80*A*b*c^6)*x^5 - 128*(11*B*b^3*c^4 - 16*A*b^2*
c^5)*x^4 + 144*(11*B*b^4*c^3 - 16*A*b^3*c^4)*x^3 - 168*(11*B*b^5*c^2 - 16*A*b^4*
c^3)*x^2 + 210*(11*B*b^6*c - 16*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 315*(
11*B*b^8 - 16*A*b^7*c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283137, size = 336, normalized size = 1.41 \[ \frac{1}{573440} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \,{\left (14 \, B c x + \frac{17 \, B b c^{7} + 16 \, A c^{8}}{c^{7}}\right )} x + \frac{B b^{2} c^{6} + 80 \, A b c^{7}}{c^{7}}\right )} x - \frac{11 \, B b^{3} c^{5} - 16 \, A b^{2} c^{6}}{c^{7}}\right )} x + \frac{9 \,{\left (11 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )}}{c^{7}}\right )} x - \frac{21 \,{\left (11 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )}}{c^{7}}\right )} x + \frac{105 \,{\left (11 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )}}{c^{7}}\right )} x - \frac{315 \,{\left (11 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )}}{c^{7}}\right )} - \frac{9 \,{\left (11 \, B b^{8} - 16 \, A b^{7} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/573440*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(4*(14*B*c*x + (17*B*b*c^7 + 16*A*c^8
)/c^7)*x + (B*b^2*c^6 + 80*A*b*c^7)/c^7)*x - (11*B*b^3*c^5 - 16*A*b^2*c^6)/c^7)*
x + 9*(11*B*b^4*c^4 - 16*A*b^3*c^5)/c^7)*x - 21*(11*B*b^5*c^3 - 16*A*b^4*c^4)/c^
7)*x + 105*(11*B*b^6*c^2 - 16*A*b^5*c^3)/c^7)*x - 315*(11*B*b^7*c - 16*A*b^6*c^2
)/c^7) - 9/32768*(11*B*b^8 - 16*A*b^7*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x
))*sqrt(c) - b))/c^(13/2)