3.1972 \(\int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=172 \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \]

[Out]

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

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Rubi [A]  time = 0.424911, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.2438, size = 143, normalized size = 0.83 \[ \frac{\left (d + e x\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{8 b} - \frac{3 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{56 b^{2}} + \frac{\left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{56 b^{3}} - \frac{\left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{280 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.140192, size = 212, normalized size = 1.23 \[ \frac{x \sqrt{(a+b x)^2} \left (70 a^4 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+56 a^3 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+28 a^2 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+8 a b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )}{280 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(70*a^4*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 56*a^
3*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 28*a^2*b^2*x^2*(20*d^3
+ 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 8*a*b^3*x^3*(35*d^3 + 84*d^2*e*x + 7
0*d*e^2*x^2 + 20*e^3*x^3) + b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e
^3*x^3)))/(280*(a + b*x))

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Maple [B]  time = 0.012, size = 264, normalized size = 1.5 \[{\frac{x \left ( 35\,{b}^{4}{e}^{3}{x}^{7}+160\,{x}^{6}{e}^{3}a{b}^{3}+120\,{x}^{6}d{e}^{2}{b}^{4}+280\,{x}^{5}{e}^{3}{a}^{2}{b}^{2}+560\,{x}^{5}d{e}^{2}a{b}^{3}+140\,{x}^{5}{d}^{2}e{b}^{4}+224\,{x}^{4}{a}^{3}b{e}^{3}+1008\,{x}^{4}{a}^{2}{b}^{2}d{e}^{2}+672\,{x}^{4}a{b}^{3}{d}^{2}e+56\,{x}^{4}{d}^{3}{b}^{4}+70\,{x}^{3}{e}^{3}{a}^{4}+840\,{x}^{3}d{e}^{2}{a}^{3}b+1260\,{x}^{3}{d}^{2}e{a}^{2}{b}^{2}+280\,{x}^{3}{d}^{3}a{b}^{3}+280\,{a}^{4}d{e}^{2}{x}^{2}+1120\,{a}^{3}b{d}^{2}e{x}^{2}+560\,{a}^{2}{b}^{2}{d}^{3}{x}^{2}+420\,x{d}^{2}e{a}^{4}+560\,x{d}^{3}{a}^{3}b+280\,{a}^{4}{d}^{3} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281861, size = 304, normalized size = 1.77 \[ \frac{1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285177, size = 486, normalized size = 2.83 \[ \frac{1}{8} \, b^{4} x^{8} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, b^{4} d x^{7} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b^{4} d^{2} x^{6} e{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, b^{4} d^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{4}{7} \, a b^{3} x^{7} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a b^{3} d x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{12}{5} \, a b^{3} d^{2} x^{5} e{\rm sign}\left (b x + a\right ) + a b^{3} d^{3} x^{4}{\rm sign}\left (b x + a\right ) + a^{2} b^{2} x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{18}{5} \, a^{2} b^{2} d x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{9}{2} \, a^{2} b^{2} d^{2} x^{4} e{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{4}{5} \, a^{3} b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{3} b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a^{4} x^{4} e^{3}{\rm sign}\left (b x + a\right ) + a^{4} d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{4} d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a^{4} d^{3} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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