3.1544 \(\int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=200 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)} \]

[Out]

-((b*d - a*e)^3*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)) + (
b*(b*d - a*e)^2*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)) - (
3*b^2*(b*d - a*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)) +
(b^3*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^4*(a + b*x))

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Rubi [A]  time = 0.415269, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-((b*d - a*e)^3*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)) + (
b*(b*d - a*e)^2*(d + e*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)) - (
3*b^2*(b*d - a*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)) +
(b^3*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^4*(a + b*x))

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Rubi in Sympy [A]  time = 25.562, size = 163, normalized size = 0.82 \[ \frac{\left (d + e x\right )^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 e} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56 e^{2}} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56 e^{3}} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 e^{4} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

(d + e*x)**5*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*e) + (3*a + 3*b*x)*(d + e*x)
**5*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(56*e**2) + (d + e*x)**5*(a*e -
b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(56*e**3) + (d + e*x)**5*(a*e - b*d)**
3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*e**4*(a + b*x))

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Mathematica [A]  time = 0.147611, size = 215, normalized size = 1.08 \[ \frac{x \sqrt{(a+b x)^2} \left (56 a^3 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+28 a^2 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+8 a b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )}{280 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(56*a^3*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3
+ e^4*x^4) + 28*a^2*b*x*(15*d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5
*e^4*x^4) + 8*a*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 +
15*e^4*x^4) + b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 +
35*e^4*x^4)))/(280*(a + b*x))

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Maple [A]  time = 0.011, size = 264, normalized size = 1.3 \[{\frac{x \left ( 35\,{b}^{3}{e}^{4}{x}^{7}+120\,{x}^{6}a{b}^{2}{e}^{4}+160\,{x}^{6}{b}^{3}d{e}^{3}+140\,{x}^{5}{a}^{2}b{e}^{4}+560\,{x}^{5}a{b}^{2}d{e}^{3}+280\,{x}^{5}{b}^{3}{d}^{2}{e}^{2}+56\,{x}^{4}{a}^{3}{e}^{4}+672\,{x}^{4}{a}^{2}bd{e}^{3}+1008\,{x}^{4}a{b}^{2}{d}^{2}{e}^{2}+224\,{x}^{4}{b}^{3}{d}^{3}e+280\,{x}^{3}{a}^{3}d{e}^{3}+1260\,{x}^{3}{a}^{2}b{d}^{2}{e}^{2}+840\,{x}^{3}a{b}^{2}{d}^{3}e+70\,{x}^{3}{b}^{3}{d}^{4}+560\,{a}^{3}{d}^{2}{e}^{2}{x}^{2}+1120\,{a}^{2}b{d}^{3}e{x}^{2}+280\,a{b}^{2}{d}^{4}{x}^{2}+560\,x{a}^{3}{d}^{3}e+420\,x{a}^{2}b{d}^{4}+280\,{a}^{3}{d}^{4} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/280*x*(35*b^3*e^4*x^7+120*a*b^2*e^4*x^6+160*b^3*d*e^3*x^6+140*a^2*b*e^4*x^5+56
0*a*b^2*d*e^3*x^5+280*b^3*d^2*e^2*x^5+56*a^3*e^4*x^4+672*a^2*b*d*e^3*x^4+1008*a*
b^2*d^2*e^2*x^4+224*b^3*d^3*e*x^4+280*a^3*d*e^3*x^3+1260*a^2*b*d^2*e^2*x^3+840*a
*b^2*d^3*e*x^3+70*b^3*d^4*x^3+560*a^3*d^2*e^2*x^2+1120*a^2*b*d^3*e*x^2+280*a*b^2
*d^4*x^2+560*a^3*d^3*e*x+420*a^2*b*d^4*x+280*a^3*d^4)*((b*x+a)^2)^(3/2)/(b*x+a)^
3

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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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206402, size = 304, normalized size = 1.52 \[ \frac{1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} +{\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^4,x, algorithm="fricas")

[Out]

1/8*b^3*e^4*x^8 + a^3*d^4*x + 1/7*(4*b^3*d*e^3 + 3*a*b^2*e^4)*x^7 + 1/2*(2*b^3*d
^2*e^2 + 4*a*b^2*d*e^3 + a^2*b*e^4)*x^6 + 1/5*(4*b^3*d^3*e + 18*a*b^2*d^2*e^2 +
12*a^2*b*d*e^3 + a^3*e^4)*x^5 + 1/4*(b^3*d^4 + 12*a*b^2*d^3*e + 18*a^2*b*d^2*e^2
+ 4*a^3*d*e^3)*x^4 + (a*b^2*d^4 + 4*a^2*b*d^3*e + 2*a^3*d^2*e^2)*x^3 + 1/2*(3*a
^2*b*d^4 + 4*a^3*d^3*e)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((d + e*x)**4*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.21543, size = 482, normalized size = 2.41 \[ \frac{1}{8} \, b^{3} x^{8} e^{4}{\rm sign}\left (b x + a\right ) + \frac{4}{7} \, b^{3} d x^{7} e^{3}{\rm sign}\left (b x + a\right ) + b^{3} d^{2} x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{4}{5} \, b^{3} d^{3} x^{5} e{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d^{4} x^{4}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, a b^{2} x^{7} e^{4}{\rm sign}\left (b x + a\right ) + 2 \, a b^{2} d x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{18}{5} \, a b^{2} d^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b^{2} d^{3} x^{4} e{\rm sign}\left (b x + a\right ) + a b^{2} d^{4} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{2} b x^{6} e^{4}{\rm sign}\left (b x + a\right ) + \frac{12}{5} \, a^{2} b d x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{9}{2} \, a^{2} b d^{2} x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a^{2} b d^{3} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d^{4} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a^{3} x^{5} e^{4}{\rm sign}\left (b x + a\right ) + a^{3} d x^{4} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a^{3} d^{2} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a^{3} d^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a^{3} d^{4} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^4,x, algorithm="giac")

[Out]

1/8*b^3*x^8*e^4*sign(b*x + a) + 4/7*b^3*d*x^7*e^3*sign(b*x + a) + b^3*d^2*x^6*e^
2*sign(b*x + a) + 4/5*b^3*d^3*x^5*e*sign(b*x + a) + 1/4*b^3*d^4*x^4*sign(b*x + a
) + 3/7*a*b^2*x^7*e^4*sign(b*x + a) + 2*a*b^2*d*x^6*e^3*sign(b*x + a) + 18/5*a*b
^2*d^2*x^5*e^2*sign(b*x + a) + 3*a*b^2*d^3*x^4*e*sign(b*x + a) + a*b^2*d^4*x^3*s
ign(b*x + a) + 1/2*a^2*b*x^6*e^4*sign(b*x + a) + 12/5*a^2*b*d*x^5*e^3*sign(b*x +
a) + 9/2*a^2*b*d^2*x^4*e^2*sign(b*x + a) + 4*a^2*b*d^3*x^3*e*sign(b*x + a) + 3/
2*a^2*b*d^4*x^2*sign(b*x + a) + 1/5*a^3*x^5*e^4*sign(b*x + a) + a^3*d*x^4*e^3*si
gn(b*x + a) + 2*a^3*d^2*x^3*e^2*sign(b*x + a) + 2*a^3*d^3*x^2*e*sign(b*x + a) +
a^3*d^4*x*sign(b*x + a)