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

Optimal. Leaf size=271 \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac{e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac{3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

[Out]

(-3*b^2*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x +
c*x^2])/(1024*c^5) + ((2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*
c*x)*(b*x + c*x^2)^(3/2))/(128*c^4) + (e*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c)
+ (e*(128*c^2*d^2 - 98*b*c*d*e + 21*b^2*e^2 + 30*c*e*(2*c*d - b*e)*x)*(b*x + c*x
^2)^(5/2))/(280*c^3) + (3*b^4*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi [A]  time = 0.786367, antiderivative size = 271, 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 \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac{e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac{3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b^2*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x +
c*x^2])/(1024*c^5) + ((2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*
c*x)*(b*x + c*x^2)^(3/2))/(128*c^4) + (e*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c)
+ (e*(128*c^2*d^2 - 98*b*c*d*e + 21*b^2*e^2 + 30*c*e*(2*c*d - b*e)*x)*(b*x + c*x
^2)^(5/2))/(280*c^3) + (3*b^4*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(11/2))

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Rubi in Sympy [A]  time = 50.0585, size = 274, normalized size = 1.01 \[ - \frac{3 b^{4} \left (b e - 2 c d\right ) \left (3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{1024 c^{\frac{11}{2}}} + \frac{3 b^{2} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{1024 c^{5}} + \frac{e \left (d + e x\right )^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} + \frac{e \left (b x + c x^{2}\right )^{\frac{5}{2}} \left (\frac{63 b^{2} e^{2}}{4} - \frac{147 b c d e}{2} + 96 c^{2} d^{2} - \frac{45 c e x \left (b e - 2 c d\right )}{2}\right )}{210 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{128 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b**4*(b*e - 2*c*d)*(3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atanh(sqrt(c)*x/sq
rt(b*x + c*x**2))/(1024*c**(11/2)) + 3*b**2*(b + 2*c*x)*(b*e - 2*c*d)*sqrt(b*x +
c*x**2)*(3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/(1024*c**5) + e*(d + e*x)**2*(b
*x + c*x**2)**(5/2)/(7*c) + e*(b*x + c*x**2)**(5/2)*(63*b**2*e**2/4 - 147*b*c*d*
e/2 + 96*c**2*d**2 - 45*c*e*x*(b*e - 2*c*d)/2)/(210*c**3) - (b + 2*c*x)*(b*e - 2
*c*d)*(b*x + c*x**2)**(3/2)*(3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/(128*c**4)

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Mathematica [A]  time = 0.585208, size = 314, normalized size = 1.16 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )-16 b^3 c^3 \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+32 b^2 c^4 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )+128 b c^5 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )+256 c^6 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )-\frac{105 b^4 \left (3 b^3 e^3-14 b^2 c d e^2+24 b c^2 d^2 e-16 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{35840 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(315*b^6*e^3 - 210*b^5*c*e^2*(7*d + e*x) + 28*b^4*c^
2*e*(90*d^2 + 35*d*e*x + 6*e^2*x^2) + 32*b^2*c^4*x*(35*d^3 + 42*d^2*e*x + 21*d*e
^2*x^2 + 4*e^3*x^3) - 16*b^3*c^3*(105*d^3 + 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x
^3) + 256*c^6*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 128*b*c^5*
x^2*(105*d^3 + 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) - (105*b^4*(-16*c^3*d^
3 + 24*b*c^2*d^2*e - 14*b^2*c*d*e^2 + 3*b^3*e^3)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b
+ c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(35840*c^(11/2))

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Maple [B]  time = 0.016, size = 629, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*e^2*x*(c*x^2+b*x)^(5/2)/c+1/7*e^3*x^2*(c*x^2+b*x)^(5/2)/c+3/40*e^3*b^2/c^3
*(c*x^2+b*x)^(5/2)-3/128*e^3*b^4/c^4*(c*x^2+b*x)^(3/2)+9/1024*e^3*b^6/c^5*(c*x^2
+b*x)^(1/2)-9/2048*e^3*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/
8*d^3/c*(c*x^2+b*x)^(3/2)*b-3/64*d^3*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*d^3*b^4/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/5*d^2*e*(c*x^2+b*x)^(5/2)/c-3/8
*d^2*e*b/c*(c*x^2+b*x)^(3/2)*x+9/64*d^2*e*b^3/c^2*(c*x^2+b*x)^(1/2)*x+7/32*d*e^2
*b^2/c^2*(c*x^2+b*x)^(3/2)*x-21/256*d*e^2*b^4/c^3*(c*x^2+b*x)^(1/2)*x+7/64*d*e^2
*b^3/c^3*(c*x^2+b*x)^(3/2)-7/20*d*e^2*b/c^2*(c*x^2+b*x)^(5/2)-21/512*d*e^2*b^5/c
^4*(c*x^2+b*x)^(1/2)+21/1024*d*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
)^(1/2))-3/64*e^3*b^3/c^3*(c*x^2+b*x)^(3/2)*x+9/512*e^3*b^5/c^4*(c*x^2+b*x)^(1/2
)*x-3/16*d^2*e*b^2/c^2*(c*x^2+b*x)^(3/2)+9/128*d^2*e*b^4/c^3*(c*x^2+b*x)^(1/2)-9
/256*d^2*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-3/28*e^3*b/c^2*
x*(c*x^2+b*x)^(5/2)-3/32*d^3*b^2/c*(c*x^2+b*x)^(1/2)*x+1/4*d^3*(c*x^2+b*x)^(3/2)
*x

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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)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237555, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (5120 \, c^{6} e^{3} x^{6} - 1680 \, b^{3} c^{3} d^{3} + 2520 \, b^{4} c^{2} d^{2} e - 1470 \, b^{5} c d e^{2} + 315 \, b^{6} e^{3} + 1280 \,{\left (14 \, c^{6} d e^{2} + 5 \, b c^{5} e^{3}\right )} x^{5} + 128 \,{\left (168 \, c^{6} d^{2} e + 182 \, b c^{5} d e^{2} + b^{2} c^{4} e^{3}\right )} x^{4} + 16 \,{\left (560 \, c^{6} d^{3} + 1848 \, b c^{5} d^{2} e + 42 \, b^{2} c^{4} d e^{2} - 9 \, b^{3} c^{3} e^{3}\right )} x^{3} + 56 \,{\left (240 \, b c^{5} d^{3} + 24 \, b^{2} c^{4} d^{2} e - 14 \, b^{3} c^{3} d e^{2} + 3 \, b^{4} c^{2} e^{3}\right )} x^{2} + 70 \,{\left (16 \, b^{2} c^{4} d^{3} - 24 \, b^{3} c^{3} d^{2} e + 14 \, b^{4} c^{2} d e^{2} - 3 \, b^{5} c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{71680 \, c^{\frac{11}{2}}}, \frac{{\left (5120 \, c^{6} e^{3} x^{6} - 1680 \, b^{3} c^{3} d^{3} + 2520 \, b^{4} c^{2} d^{2} e - 1470 \, b^{5} c d e^{2} + 315 \, b^{6} e^{3} + 1280 \,{\left (14 \, c^{6} d e^{2} + 5 \, b c^{5} e^{3}\right )} x^{5} + 128 \,{\left (168 \, c^{6} d^{2} e + 182 \, b c^{5} d e^{2} + b^{2} c^{4} e^{3}\right )} x^{4} + 16 \,{\left (560 \, c^{6} d^{3} + 1848 \, b c^{5} d^{2} e + 42 \, b^{2} c^{4} d e^{2} - 9 \, b^{3} c^{3} e^{3}\right )} x^{3} + 56 \,{\left (240 \, b c^{5} d^{3} + 24 \, b^{2} c^{4} d^{2} e - 14 \, b^{3} c^{3} d e^{2} + 3 \, b^{4} c^{2} e^{3}\right )} x^{2} + 70 \,{\left (16 \, b^{2} c^{4} d^{3} - 24 \, b^{3} c^{3} d^{2} e + 14 \, b^{4} c^{2} d e^{2} - 3 \, b^{5} c e^{3}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 105 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{35840 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/71680*(2*(5120*c^6*e^3*x^6 - 1680*b^3*c^3*d^3 + 2520*b^4*c^2*d^2*e - 1470*b^5
*c*d*e^2 + 315*b^6*e^3 + 1280*(14*c^6*d*e^2 + 5*b*c^5*e^3)*x^5 + 128*(168*c^6*d^
2*e + 182*b*c^5*d*e^2 + b^2*c^4*e^3)*x^4 + 16*(560*c^6*d^3 + 1848*b*c^5*d^2*e +
42*b^2*c^4*d*e^2 - 9*b^3*c^3*e^3)*x^3 + 56*(240*b*c^5*d^3 + 24*b^2*c^4*d^2*e - 1
4*b^3*c^3*d*e^2 + 3*b^4*c^2*e^3)*x^2 + 70*(16*b^2*c^4*d^3 - 24*b^3*c^3*d^2*e + 1
4*b^4*c^2*d*e^2 - 3*b^5*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 105*(16*b^4*c^3*d^
3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*log((2*c*x + b)*sqrt(c) - 2*s
qrt(c*x^2 + b*x)*c))/c^(11/2), 1/35840*((5120*c^6*e^3*x^6 - 1680*b^3*c^3*d^3 + 2
520*b^4*c^2*d^2*e - 1470*b^5*c*d*e^2 + 315*b^6*e^3 + 1280*(14*c^6*d*e^2 + 5*b*c^
5*e^3)*x^5 + 128*(168*c^6*d^2*e + 182*b*c^5*d*e^2 + b^2*c^4*e^3)*x^4 + 16*(560*c
^6*d^3 + 1848*b*c^5*d^2*e + 42*b^2*c^4*d*e^2 - 9*b^3*c^3*e^3)*x^3 + 56*(240*b*c^
5*d^3 + 24*b^2*c^4*d^2*e - 14*b^3*c^3*d*e^2 + 3*b^4*c^2*e^3)*x^2 + 70*(16*b^2*c^
4*d^3 - 24*b^3*c^3*d^2*e + 14*b^4*c^2*d*e^2 - 3*b^5*c*e^3)*x)*sqrt(c*x^2 + b*x)*
sqrt(-c) + 105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*
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. \[ \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223506, size = 493, normalized size = 1.82 \[ \frac{1}{35840} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c x e^{3} + \frac{14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac{168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac{560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac{7 \,{\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac{105 \,{\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac{3 \,{\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35840*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(4*c*x*e^3 + (14*c^7*d*e^2 + 5*b*c^6*e
^3)/c^6)*x + (168*c^7*d^2*e + 182*b*c^6*d*e^2 + b^2*c^5*e^3)/c^6)*x + (560*c^7*d
^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)/c^6)*x + 7*(240*b*c^6*
d^3 + 24*b^2*c^5*d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)/c^6)*x + 35*(16*b^2*c
^5*d^3 - 24*b^3*c^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)/c^6)*x - 105*(16*b
^3*c^4*d^3 - 24*b^4*c^3*d^2*e + 14*b^5*c^2*d*e^2 - 3*b^6*c*e^3)/c^6) - 3/2048*(1
6*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)