3.93 \(\int x^3 (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=276 \[ -\frac{11 b^9 (13 b B-20 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{131072 c^{15/2}}+\frac{11 b^7 (b+2 c x) \sqrt{b x+c x^2} (13 b B-20 A c)}{131072 c^7}-\frac{11 b^5 (b+2 c x) \left (b x+c x^2\right )^{3/2} (13 b B-20 A c)}{49152 c^6}+\frac{11 b^3 (b+2 c x) \left (b x+c x^2\right )^{5/2} (13 b B-20 A c)}{15360 c^5}-\frac{11 b^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{4480 c^4}+\frac{11 b x \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{2880 c^3}-\frac{x^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c} \]

[Out]

(11*b^7*(13*b*B - 20*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(131072*c^7) - (11*b^5*
(13*b*B - 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(49152*c^6) + (11*b^3*(13*b*B
- 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(15360*c^5) - (11*b^2*(13*b*B - 20*A
*c)*(b*x + c*x^2)^(7/2))/(4480*c^4) + (11*b*(13*b*B - 20*A*c)*x*(b*x + c*x^2)^(7
/2))/(2880*c^3) - ((13*b*B - 20*A*c)*x^2*(b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3
*(b*x + c*x^2)^(7/2))/(10*c) - (11*b^9*(13*b*B - 20*A*c)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(131072*c^(15/2))

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Rubi [A]  time = 0.631634, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{11 b^9 (13 b B-20 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{131072 c^{15/2}}+\frac{11 b^7 (b+2 c x) \sqrt{b x+c x^2} (13 b B-20 A c)}{131072 c^7}-\frac{11 b^5 (b+2 c x) \left (b x+c x^2\right )^{3/2} (13 b B-20 A c)}{49152 c^6}+\frac{11 b^3 (b+2 c x) \left (b x+c x^2\right )^{5/2} (13 b B-20 A c)}{15360 c^5}-\frac{11 b^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{4480 c^4}+\frac{11 b x \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{2880 c^3}-\frac{x^2 \left (b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (b x+c x^2\right )^{7/2}}{10 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(11*b^7*(13*b*B - 20*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(131072*c^7) - (11*b^5*
(13*b*B - 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(49152*c^6) + (11*b^3*(13*b*B
- 20*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(15360*c^5) - (11*b^2*(13*b*B - 20*A
*c)*(b*x + c*x^2)^(7/2))/(4480*c^4) + (11*b*(13*b*B - 20*A*c)*x*(b*x + c*x^2)^(7
/2))/(2880*c^3) - ((13*b*B - 20*A*c)*x^2*(b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3
*(b*x + c*x^2)^(7/2))/(10*c) - (11*b^9*(13*b*B - 20*A*c)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(131072*c^(15/2))

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Rubi in Sympy [A]  time = 42.751, size = 274, normalized size = 0.99 \[ \frac{B x^{3} \left (b x + c x^{2}\right )^{\frac{7}{2}}}{10 c} + \frac{11 b^{9} \left (20 A c - 13 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{131072 c^{\frac{15}{2}}} - \frac{11 b^{7} \left (b + 2 c x\right ) \left (20 A c - 13 B b\right ) \sqrt{b x + c x^{2}}}{131072 c^{7}} + \frac{11 b^{5} \left (b + 2 c x\right ) \left (20 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{49152 c^{6}} - \frac{11 b^{3} \left (b + 2 c x\right ) \left (20 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15360 c^{5}} + \frac{11 b^{2} \left (20 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{4480 c^{4}} - \frac{11 b x \left (20 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{2880 c^{3}} + \frac{x^{2} \left (20 A c - 13 B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{180 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

B*x**3*(b*x + c*x**2)**(7/2)/(10*c) + 11*b**9*(20*A*c - 13*B*b)*atanh(sqrt(c)*x/
sqrt(b*x + c*x**2))/(131072*c**(15/2)) - 11*b**7*(b + 2*c*x)*(20*A*c - 13*B*b)*s
qrt(b*x + c*x**2)/(131072*c**7) + 11*b**5*(b + 2*c*x)*(20*A*c - 13*B*b)*(b*x + c
*x**2)**(3/2)/(49152*c**6) - 11*b**3*(b + 2*c*x)*(20*A*c - 13*B*b)*(b*x + c*x**2
)**(5/2)/(15360*c**5) + 11*b**2*(20*A*c - 13*B*b)*(b*x + c*x**2)**(7/2)/(4480*c*
*4) - 11*b*x*(20*A*c - 13*B*b)*(b*x + c*x**2)**(7/2)/(2880*c**3) + x**2*(20*A*c
- 13*B*b)*(b*x + c*x**2)**(7/2)/(180*c**2)

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Mathematica [A]  time = 0.449743, size = 259, normalized size = 0.94 \[ \frac{\sqrt{x} \sqrt{b+c x} \left (\sqrt{c} \sqrt{x} \sqrt{b+c x} \left (-2310 b^8 c (30 A+13 B x)+1848 b^7 c^2 x (25 A+13 B x)-528 b^6 c^3 x^2 (70 A+39 B x)+704 b^5 c^4 x^3 (45 A+26 B x)-1280 b^4 c^5 x^4 (22 A+13 B x)+5120 b^3 c^6 x^5 (5 A+3 B x)+2048 b^2 c^7 x^6 (3090 A+2681 B x)+57344 b c^8 x^7 (185 A+164 B x)+458752 c^9 x^8 (10 A+9 B x)+45045 b^9 B\right )-3465 b^9 (13 b B-20 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{41287680 c^{15/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(A + B*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(45045*b^9*B + 5120*b^3*c^
6*x^5*(5*A + 3*B*x) + 458752*c^9*x^8*(10*A + 9*B*x) - 1280*b^4*c^5*x^4*(22*A + 1
3*B*x) + 1848*b^7*c^2*x*(25*A + 13*B*x) - 2310*b^8*c*(30*A + 13*B*x) + 704*b^5*c
^4*x^3*(45*A + 26*B*x) - 528*b^6*c^3*x^2*(70*A + 39*B*x) + 57344*b*c^8*x^7*(185*
A + 164*B*x) + 2048*b^2*c^7*x^6*(3090*A + 2681*B*x)) - 3465*b^9*(13*b*B - 20*A*c
)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]]))/(41287680*c^(15/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.015, size = 455, normalized size = 1.7 \[{\frac{A{x}^{2}}{9\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{11\,Abx}{144\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{b}^{2}A}{224\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{11\,A{b}^{3}x}{384\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{11\,A{b}^{4}}{768\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{55\,A{b}^{5}x}{6144\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{55\,A{b}^{6}}{12288\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{55\,A{b}^{7}x}{16384\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{55\,A{b}^{8}}{32768\,{c}^{6}}\sqrt{c{x}^{2}+bx}}+{\frac{55\,A{b}^{9}}{65536}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{13}{2}}}}+{\frac{B{x}^{3}}{10\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{13\,Bb{x}^{2}}{180\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{143\,{b}^{2}Bx}{2880\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{143\,B{b}^{3}}{4480\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{143\,{b}^{4}Bx}{7680\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{143\,B{b}^{5}}{15360\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{143\,{b}^{6}Bx}{24576\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{143\,B{b}^{7}}{49152\,{c}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{143\,B{b}^{8}x}{65536\,{c}^{6}}\sqrt{c{x}^{2}+bx}}+{\frac{143\,B{b}^{9}}{131072\,{c}^{7}}\sqrt{c{x}^{2}+bx}}-{\frac{143\,B{b}^{10}}{262144}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+b*x)^(5/2),x)

[Out]

1/9*A*x^2*(c*x^2+b*x)^(7/2)/c-11/144*A*b/c^2*x*(c*x^2+b*x)^(7/2)+11/224*A*b^2/c^
3*(c*x^2+b*x)^(7/2)-11/384*A*b^3/c^3*(c*x^2+b*x)^(5/2)*x-11/768*A*b^4/c^4*(c*x^2
+b*x)^(5/2)+55/6144*A*b^5/c^4*(c*x^2+b*x)^(3/2)*x+55/12288*A*b^6/c^5*(c*x^2+b*x)
^(3/2)-55/16384*A*b^7/c^5*(c*x^2+b*x)^(1/2)*x-55/32768*A*b^8/c^6*(c*x^2+b*x)^(1/
2)+55/65536*A*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/10*B*x^3*
(c*x^2+b*x)^(7/2)/c-13/180*B*b/c^2*x^2*(c*x^2+b*x)^(7/2)+143/2880*B*b^2/c^3*x*(c
*x^2+b*x)^(7/2)-143/4480*B*b^3/c^4*(c*x^2+b*x)^(7/2)+143/7680*B*b^4/c^4*(c*x^2+b
*x)^(5/2)*x+143/15360*B*b^5/c^5*(c*x^2+b*x)^(5/2)-143/24576*B*b^6/c^5*(c*x^2+b*x
)^(3/2)*x-143/49152*B*b^7/c^6*(c*x^2+b*x)^(3/2)+143/65536*B*b^8/c^6*(c*x^2+b*x)^
(1/2)*x+143/131072*B*b^9/c^7*(c*x^2+b*x)^(1/2)-143/262144*B*b^10/c^(15/2)*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30991, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (4128768 \, B c^{9} x^{9} + 45045 \, B b^{9} - 69300 \, A b^{8} c + 229376 \,{\left (41 \, B b c^{8} + 20 \, A c^{9}\right )} x^{8} + 14336 \,{\left (383 \, B b^{2} c^{7} + 740 \, A b c^{8}\right )} x^{7} + 15360 \,{\left (B b^{3} c^{6} + 412 \, A b^{2} c^{7}\right )} x^{6} - 1280 \,{\left (13 \, B b^{4} c^{5} - 20 \, A b^{3} c^{6}\right )} x^{5} + 1408 \,{\left (13 \, B b^{5} c^{4} - 20 \, A b^{4} c^{5}\right )} x^{4} - 1584 \,{\left (13 \, B b^{6} c^{3} - 20 \, A b^{5} c^{4}\right )} x^{3} + 1848 \,{\left (13 \, B b^{7} c^{2} - 20 \, A b^{6} c^{3}\right )} x^{2} - 2310 \,{\left (13 \, B b^{8} c - 20 \, A b^{7} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3465 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{82575360 \, c^{\frac{15}{2}}}, \frac{{\left (4128768 \, B c^{9} x^{9} + 45045 \, B b^{9} - 69300 \, A b^{8} c + 229376 \,{\left (41 \, B b c^{8} + 20 \, A c^{9}\right )} x^{8} + 14336 \,{\left (383 \, B b^{2} c^{7} + 740 \, A b c^{8}\right )} x^{7} + 15360 \,{\left (B b^{3} c^{6} + 412 \, A b^{2} c^{7}\right )} x^{6} - 1280 \,{\left (13 \, B b^{4} c^{5} - 20 \, A b^{3} c^{6}\right )} x^{5} + 1408 \,{\left (13 \, B b^{5} c^{4} - 20 \, A b^{4} c^{5}\right )} x^{4} - 1584 \,{\left (13 \, B b^{6} c^{3} - 20 \, A b^{5} c^{4}\right )} x^{3} + 1848 \,{\left (13 \, B b^{7} c^{2} - 20 \, A b^{6} c^{3}\right )} x^{2} - 2310 \,{\left (13 \, B b^{8} c - 20 \, A b^{7} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3465 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{41287680 \, \sqrt{-c} c^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x^3,x, algorithm="fricas")

[Out]

[1/82575360*(2*(4128768*B*c^9*x^9 + 45045*B*b^9 - 69300*A*b^8*c + 229376*(41*B*b
*c^8 + 20*A*c^9)*x^8 + 14336*(383*B*b^2*c^7 + 740*A*b*c^8)*x^7 + 15360*(B*b^3*c^
6 + 412*A*b^2*c^7)*x^6 - 1280*(13*B*b^4*c^5 - 20*A*b^3*c^6)*x^5 + 1408*(13*B*b^5
*c^4 - 20*A*b^4*c^5)*x^4 - 1584*(13*B*b^6*c^3 - 20*A*b^5*c^4)*x^3 + 1848*(13*B*b
^7*c^2 - 20*A*b^6*c^3)*x^2 - 2310*(13*B*b^8*c - 20*A*b^7*c^2)*x)*sqrt(c*x^2 + b*
x)*sqrt(c) - 3465*(13*B*b^10 - 20*A*b^9*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^
2 + b*x)*c))/c^(15/2), 1/41287680*((4128768*B*c^9*x^9 + 45045*B*b^9 - 69300*A*b^
8*c + 229376*(41*B*b*c^8 + 20*A*c^9)*x^8 + 14336*(383*B*b^2*c^7 + 740*A*b*c^8)*x
^7 + 15360*(B*b^3*c^6 + 412*A*b^2*c^7)*x^6 - 1280*(13*B*b^4*c^5 - 20*A*b^3*c^6)*
x^5 + 1408*(13*B*b^5*c^4 - 20*A*b^4*c^5)*x^4 - 1584*(13*B*b^6*c^3 - 20*A*b^5*c^4
)*x^3 + 1848*(13*B*b^7*c^2 - 20*A*b^6*c^3)*x^2 - 2310*(13*B*b^8*c - 20*A*b^7*c^2
)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 3465*(13*B*b^10 - 20*A*b^9*c)*arctan(sqrt(c*x^
2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral(x**3*(x*(b + c*x))**(5/2)*(A + B*x), x)

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GIAC/XCAS [A]  time = 0.290267, size = 417, normalized size = 1.51 \[ \frac{1}{41287680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \,{\left (18 \, B c^{2} x + \frac{41 \, B b c^{10} + 20 \, A c^{11}}{c^{9}}\right )} x + \frac{383 \, B b^{2} c^{9} + 740 \, A b c^{10}}{c^{9}}\right )} x + \frac{15 \,{\left (B b^{3} c^{8} + 412 \, A b^{2} c^{9}\right )}}{c^{9}}\right )} x - \frac{5 \,{\left (13 \, B b^{4} c^{7} - 20 \, A b^{3} c^{8}\right )}}{c^{9}}\right )} x + \frac{11 \,{\left (13 \, B b^{5} c^{6} - 20 \, A b^{4} c^{7}\right )}}{c^{9}}\right )} x - \frac{99 \,{\left (13 \, B b^{6} c^{5} - 20 \, A b^{5} c^{6}\right )}}{c^{9}}\right )} x + \frac{231 \,{\left (13 \, B b^{7} c^{4} - 20 \, A b^{6} c^{5}\right )}}{c^{9}}\right )} x - \frac{1155 \,{\left (13 \, B b^{8} c^{3} - 20 \, A b^{7} c^{4}\right )}}{c^{9}}\right )} x + \frac{3465 \,{\left (13 \, B b^{9} c^{2} - 20 \, A b^{8} c^{3}\right )}}{c^{9}}\right )} + \frac{11 \,{\left (13 \, B b^{10} - 20 \, A b^{9} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{262144 \, c^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*x^3,x, algorithm="giac")

[Out]

1/41287680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + (41*B*b*c^1
0 + 20*A*c^11)/c^9)*x + (383*B*b^2*c^9 + 740*A*b*c^10)/c^9)*x + 15*(B*b^3*c^8 +
412*A*b^2*c^9)/c^9)*x - 5*(13*B*b^4*c^7 - 20*A*b^3*c^8)/c^9)*x + 11*(13*B*b^5*c^
6 - 20*A*b^4*c^7)/c^9)*x - 99*(13*B*b^6*c^5 - 20*A*b^5*c^6)/c^9)*x + 231*(13*B*b
^7*c^4 - 20*A*b^6*c^5)/c^9)*x - 1155*(13*B*b^8*c^3 - 20*A*b^7*c^4)/c^9)*x + 3465
*(13*B*b^9*c^2 - 20*A*b^8*c^3)/c^9) + 11/262144*(13*B*b^10 - 20*A*b^9*c)*ln(abs(
-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(15/2)