3.95 \(\int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=189 \[ -\frac{5 b^7 (9 b B-16 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 b^5 (b+2 c x) \sqrt{b x+c x^2} (9 b B-16 A c)}{16384 c^5}-\frac{5 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-16 A c)}{6144 c^4}+\frac{b (b+2 c x) \left (b x+c x^2\right )^{5/2} (9 b B-16 A c)}{384 c^3}-\frac{\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]

[Out]

(5*b^5*(9*b*B - 16*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^3*(9*b
*B - 16*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + (b*(9*b*B - 16*A*c)*(
b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B - 16*A*c - 14*B*c*x)*(b*x +
c*x^2)^(7/2))/(112*c^2) - (5*b^7*(9*b*B - 16*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(16384*c^(11/2))

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Rubi [A]  time = 0.223903, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 b^7 (9 b B-16 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 b^5 (b+2 c x) \sqrt{b x+c x^2} (9 b B-16 A c)}{16384 c^5}-\frac{5 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-16 A c)}{6144 c^4}+\frac{b (b+2 c x) \left (b x+c x^2\right )^{5/2} (9 b B-16 A c)}{384 c^3}-\frac{\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^5*(9*b*B - 16*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^3*(9*b
*B - 16*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + (b*(9*b*B - 16*A*c)*(
b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B - 16*A*c - 14*B*c*x)*(b*x +
c*x^2)^(7/2))/(112*c^2) - (5*b^7*(9*b*B - 16*A*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(16384*c^(11/2))

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Rubi in Sympy [A]  time = 27.1864, size = 201, normalized size = 1.06 \[ \frac{B x \left (b x + c x^{2}\right )^{\frac{7}{2}}}{8 c} + \frac{5 b^{7} \left (16 A c - 9 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16384 c^{\frac{11}{2}}} - \frac{5 b^{5} \left (b + 2 c x\right ) \left (16 A c - 9 B b\right ) \sqrt{b x + c x^{2}}}{16384 c^{5}} + \frac{5 b^{3} \left (b + 2 c x\right ) \left (16 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{6144 c^{4}} - \frac{b \left (b + 2 c x\right ) \left (16 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{384 c^{3}} + \frac{\left (16 A c - 9 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{112 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

B*x*(b*x + c*x**2)**(7/2)/(8*c) + 5*b**7*(16*A*c - 9*B*b)*atanh(sqrt(c)*x/sqrt(b
*x + c*x**2))/(16384*c**(11/2)) - 5*b**5*(b + 2*c*x)*(16*A*c - 9*B*b)*sqrt(b*x +
c*x**2)/(16384*c**5) + 5*b**3*(b + 2*c*x)*(16*A*c - 9*B*b)*(b*x + c*x**2)**(3/2
)/(6144*c**4) - b*(b + 2*c*x)*(16*A*c - 9*B*b)*(b*x + c*x**2)**(5/2)/(384*c**3)
+ (16*A*c - 9*B*b)*(b*x + c*x**2)**(7/2)/(112*c**2)

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Mathematica [A]  time = 0.414835, size = 206, normalized size = 1.09 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-210 b^6 c (8 A+3 B x)+56 b^5 c^2 x (20 A+9 B x)-16 b^4 c^3 x^2 (56 A+27 B x)+384 b^3 c^4 x^3 (2 A+B x)+256 b^2 c^5 x^4 (296 A+243 B x)+1024 b c^6 x^5 (116 A+99 B x)+6144 c^7 x^6 (8 A+7 B x)+945 b^7 B\right )-\frac{105 b^7 (9 b B-16 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{344064 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^7*B + 384*b^3*c^4*x^3*(2*A + B*x) - 210*b^6*c
*(8*A + 3*B*x) + 6144*c^7*x^6*(8*A + 7*B*x) + 56*b^5*c^2*x*(20*A + 9*B*x) - 16*b
^4*c^3*x^2*(56*A + 27*B*x) + 1024*b*c^6*x^5*(116*A + 99*B*x) + 256*b^2*c^5*x^4*(
296*A + 243*B*x)) - (105*b^7*(9*b*B - 16*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c
*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(344064*c^(11/2))

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Maple [B]  time = 0.01, size = 365, normalized size = 1.9 \[{\frac{A}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{Abx}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}A}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{b}^{3}x}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{4}}{384\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{b}^{5}x}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,A{b}^{6}}{1024\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{7}}{2048}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{Bx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{9\,Bb}{112\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{b}^{2}Bx}{64\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,B{b}^{3}}{128\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{b}^{4}Bx}{1024\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{15\,B{b}^{5}}{2048\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{b}^{6}Bx}{8192\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{45\,B{b}^{7}}{16384\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{45\,B{b}^{8}}{32768}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(c*x^2+b*x)^(5/2),x)

[Out]

1/7*A*(c*x^2+b*x)^(7/2)/c-1/12*A*b/c*(c*x^2+b*x)^(5/2)*x-1/24*A*b^2/c^2*(c*x^2+b
*x)^(5/2)+5/192*A*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/384*A*b^4/c^3*(c*x^2+b*x)^(3/2)-
5/512*A*b^5/c^3*(c*x^2+b*x)^(1/2)*x-5/1024*A*b^6/c^4*(c*x^2+b*x)^(1/2)+5/2048*A*
b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*B*x*(c*x^2+b*x)^(7/2)/
c-9/112*B*b/c^2*(c*x^2+b*x)^(7/2)+3/64*B*b^2/c^2*(c*x^2+b*x)^(5/2)*x+3/128*B*b^3
/c^3*(c*x^2+b*x)^(5/2)-15/1024*B*b^4/c^3*(c*x^2+b*x)^(3/2)*x-15/2048*B*b^5/c^4*(
c*x^2+b*x)^(3/2)+45/8192*B*b^6/c^4*(c*x^2+b*x)^(1/2)*x+45/16384*B*b^7/c^5*(c*x^2
+b*x)^(1/2)-45/32768*B*b^8/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)^(5/2)*(B*x + A)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306139, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (43008 \, B c^{7} x^{7} + 945 \, B b^{7} - 1680 \, A b^{6} c + 3072 \,{\left (33 \, B b c^{6} + 16 \, A c^{7}\right )} x^{6} + 256 \,{\left (243 \, B b^{2} c^{5} + 464 \, A b c^{6}\right )} x^{5} + 128 \,{\left (3 \, B b^{3} c^{4} + 592 \, A b^{2} c^{5}\right )} x^{4} - 48 \,{\left (9 \, B b^{4} c^{3} - 16 \, A b^{3} c^{4}\right )} x^{3} + 56 \,{\left (9 \, B b^{5} c^{2} - 16 \, A b^{4} c^{3}\right )} x^{2} - 70 \,{\left (9 \, B b^{6} c - 16 \, A b^{5} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{688128 \, c^{\frac{11}{2}}}, \frac{{\left (43008 \, B c^{7} x^{7} + 945 \, B b^{7} - 1680 \, A b^{6} c + 3072 \,{\left (33 \, B b c^{6} + 16 \, A c^{7}\right )} x^{6} + 256 \,{\left (243 \, B b^{2} c^{5} + 464 \, A b c^{6}\right )} x^{5} + 128 \,{\left (3 \, B b^{3} c^{4} + 592 \, A b^{2} c^{5}\right )} x^{4} - 48 \,{\left (9 \, B b^{4} c^{3} - 16 \, A b^{3} c^{4}\right )} x^{3} + 56 \,{\left (9 \, B b^{5} c^{2} - 16 \, A b^{4} c^{3}\right )} x^{2} - 70 \,{\left (9 \, B b^{6} c - 16 \, A b^{5} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 105 \,{\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{344064 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x,x, algorithm="fricas")

[Out]

[1/688128*(2*(43008*B*c^7*x^7 + 945*B*b^7 - 1680*A*b^6*c + 3072*(33*B*b*c^6 + 16
*A*c^7)*x^6 + 256*(243*B*b^2*c^5 + 464*A*b*c^6)*x^5 + 128*(3*B*b^3*c^4 + 592*A*b
^2*c^5)*x^4 - 48*(9*B*b^4*c^3 - 16*A*b^3*c^4)*x^3 + 56*(9*B*b^5*c^2 - 16*A*b^4*c
^3)*x^2 - 70*(9*B*b^6*c - 16*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 105*(9*B*
b^8 - 16*A*b^7*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(11/2), 1/
344064*((43008*B*c^7*x^7 + 945*B*b^7 - 1680*A*b^6*c + 3072*(33*B*b*c^6 + 16*A*c^
7)*x^6 + 256*(243*B*b^2*c^5 + 464*A*b*c^6)*x^5 + 128*(3*B*b^3*c^4 + 592*A*b^2*c^
5)*x^4 - 48*(9*B*b^4*c^3 - 16*A*b^3*c^4)*x^3 + 56*(9*B*b^5*c^2 - 16*A*b^4*c^3)*x
^2 - 70*(9*B*b^6*c - 16*A*b^5*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(9*B*b^8
- 16*A*b^7*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. \[ \int x \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x*(x*(b + c*x))**(5/2)*(A + B*x), x)

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GIAC/XCAS [A]  time = 0.286641, size = 342, normalized size = 1.81 \[ \frac{1}{344064} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, B c^{2} x + \frac{33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac{243 \, B b^{2} c^{7} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac{3 \, B b^{3} c^{6} + 592 \, A b^{2} c^{7}}{c^{7}}\right )} x - \frac{3 \,{\left (9 \, B b^{4} c^{5} - 16 \, A b^{3} c^{6}\right )}}{c^{7}}\right )} x + \frac{7 \,{\left (9 \, B b^{5} c^{4} - 16 \, A b^{4} c^{5}\right )}}{c^{7}}\right )} x - \frac{35 \,{\left (9 \, B b^{6} c^{3} - 16 \, A b^{5} c^{4}\right )}}{c^{7}}\right )} x + \frac{105 \,{\left (9 \, B b^{7} c^{2} - 16 \, A b^{6} c^{3}\right )}}{c^{7}}\right )} + \frac{5 \,{\left (9 \, B b^{8} - 16 \, A b^{7} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x,x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x + (33*B*b*c^8 + 16*A*c
^9)/c^7)*x + (243*B*b^2*c^7 + 464*A*b*c^8)/c^7)*x + (3*B*b^3*c^6 + 592*A*b^2*c^7
)/c^7)*x - 3*(9*B*b^4*c^5 - 16*A*b^3*c^6)/c^7)*x + 7*(9*B*b^5*c^4 - 16*A*b^4*c^5
)/c^7)*x - 35*(9*B*b^6*c^3 - 16*A*b^5*c^4)/c^7)*x + 105*(9*B*b^7*c^2 - 16*A*b^6*
c^3)/c^7) + 5/32768*(9*B*b^8 - 16*A*b^7*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b
*x))*sqrt(c) - b))/c^(11/2)