3.296 \(\int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

[Out]

-(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*
c^4) + ((24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(
192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(b*x
+ c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt
[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.408209, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*
c^4) + ((24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(
192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(b*x
+ c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt
[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.2419, size = 207, normalized size = 0.97 \[ \frac{b^{4} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} - \frac{b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{512 c^{4}} + \frac{e \left (d + e x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{6 c} - \frac{7 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{60 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{192 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(c*x**2+b*x)**(3/2),x)

[Out]

b**4*(7*b**2*e**2 - 24*b*c*d*e + 24*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2
))/(512*c**(9/2)) - b**2*(b + 2*c*x)*sqrt(b*x + c*x**2)*(7*b**2*e**2 - 24*b*c*d*
e + 24*c**2*d**2)/(512*c**4) + e*(d + e*x)*(b*x + c*x**2)**(5/2)/(6*c) - 7*e*(b*
e - 2*c*d)*(b*x + c*x**2)**(5/2)/(60*c**2) + (b + 2*c*x)*(b*x + c*x**2)**(3/2)*(
7*b**2*e**2 - 24*b*c*d*e + 24*c**2*d**2)/(192*c**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.382251, size = 226, normalized size = 1.06 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)-8 b^3 c^2 \left (45 d^2+30 d e x+7 e^2 x^2\right )+48 b^2 c^3 x \left (5 d^2+4 d e x+e^2 x^2\right )+64 b c^4 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )+128 c^5 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^5*e^2 + 10*b^4*c*e*(36*d + 7*e*x) + 48*b^2*c
^3*x*(5*d^2 + 4*d*e*x + e^2*x^2) - 8*b^3*c^2*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + 1
28*c^5*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 64*b*c^4*x^2*(45*d^2 + 66*d*e*x +
26*e^2*x^2)) + (15*b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*Log[c*Sqrt[x] + Sqr
t[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(7680*c^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 420, normalized size = 2. \[{\frac{{d}^{2}x}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{2}b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}{d}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{d}^{2}{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{d}^{2}{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{e}^{2}x}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{2}b}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{e}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}{e}^{2}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{2}{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{e}^{2}{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{e}^{2}{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{2\,de}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bdex}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}de}{8\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,de{b}^{3}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,de{b}^{4}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,de{b}^{5}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d^2*(c*x^2+b*x)^(3/2)*x+1/8*d^2/c*(c*x^2+b*x)^(3/2)*b-3/32*d^2*b^2/c*(c*x^2+
b*x)^(1/2)*x-3/64*d^2*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*d^2*b^4/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/6*e^2*x*(c*x^2+b*x)^(5/2)/c-7/60*e^2*b/c^2*(c*
x^2+b*x)^(5/2)+7/96*e^2*b^2/c^2*(c*x^2+b*x)^(3/2)*x+7/192*e^2*b^3/c^3*(c*x^2+b*x
)^(3/2)-7/256*e^2*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*e^2*b^5/c^4*(c*x^2+b*x)^(1/2
)+7/1024*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/5*d*e*(c*x^
2+b*x)^(5/2)/c-1/4*d*e*b/c*(c*x^2+b*x)^(3/2)*x-1/8*d*e*b^2/c^2*(c*x^2+b*x)^(3/2)
+3/32*d*e*b^3/c^2*(c*x^2+b*x)^(1/2)*x+3/64*d*e*b^4/c^3*(c*x^2+b*x)^(1/2)-3/128*d
*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

_______________________________________________________________________________________

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)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231582, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, c^{5} e^{2} x^{5} - 360 \, b^{3} c^{2} d^{2} + 360 \, b^{4} c d e - 105 \, b^{5} e^{2} + 128 \,{\left (24 \, c^{5} d e + 13 \, b c^{4} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{5} d^{2} + 88 \, b c^{4} d e + b^{2} c^{3} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{4} d^{2} + 24 \, b^{2} c^{3} d e - 7 \, b^{3} c^{2} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{3} d^{2} - 24 \, b^{3} c^{2} d e + 7 \, b^{4} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{9}{2}}}, \frac{{\left (1280 \, c^{5} e^{2} x^{5} - 360 \, b^{3} c^{2} d^{2} + 360 \, b^{4} c d e - 105 \, b^{5} e^{2} + 128 \,{\left (24 \, c^{5} d e + 13 \, b c^{4} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{5} d^{2} + 88 \, b c^{4} d e + b^{2} c^{3} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{4} d^{2} + 24 \, b^{2} c^{3} d e - 7 \, b^{3} c^{2} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{3} d^{2} - 24 \, b^{3} c^{2} d e + 7 \, b^{4} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(2*(1280*c^5*e^2*x^5 - 360*b^3*c^2*d^2 + 360*b^4*c*d*e - 105*b^5*e^2 +
128*(24*c^5*d*e + 13*b*c^4*e^2)*x^4 + 48*(40*c^5*d^2 + 88*b*c^4*d*e + b^2*c^3*e^
2)*x^3 + 8*(360*b*c^4*d^2 + 24*b^2*c^3*d*e - 7*b^3*c^2*e^2)*x^2 + 10*(24*b^2*c^3
*d^2 - 24*b^3*c^2*d*e + 7*b^4*c*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 15*(24*b^4*c
^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)
*c))/c^(9/2), 1/7680*((1280*c^5*e^2*x^5 - 360*b^3*c^2*d^2 + 360*b^4*c*d*e - 105*
b^5*e^2 + 128*(24*c^5*d*e + 13*b*c^4*e^2)*x^4 + 48*(40*c^5*d^2 + 88*b*c^4*d*e +
b^2*c^3*e^2)*x^3 + 8*(360*b*c^4*d^2 + 24*b^2*c^3*d*e - 7*b^3*c^2*e^2)*x^2 + 10*(
24*b^2*c^3*d^2 - 24*b^3*c^2*d*e + 7*b^4*c*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 1
5*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/
(c*x)))/(sqrt(-c)*c^4)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(c*x**2+b*x)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(d + e*x)**2, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221905, size = 354, normalized size = 1.65 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{3 \,{\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac{5 \,{\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac{15 \,{\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^2)/c
^5)*x + 3*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)/c^5)*x + (360*b*c^5*d^2 + 24
*b^2*c^4*d*e - 7*b^3*c^3*e^2)/c^5)*x + 5*(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^
4*c^2*e^2)/c^5)*x - 15*(24*b^3*c^3*d^2 - 24*b^4*c^2*d*e + 7*b^5*c*e^2)/c^5) - 1/
1024*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x
^2 + b*x))*sqrt(c) - b))/c^(9/2)