3.1546 \(\int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 e \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{5 b^3}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3} \]

[Out]

((b*d - a*e)^2*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b^3) + (2*e*(b*d -
a*e)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(5*b^3) + (e^2*(a + b*x)*(a^2 + 2*a*b*x +
b^2*x^2)^(5/2))/(6*b^3)

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Rubi [A]  time = 0.142389, antiderivative size = 125, normalized size of antiderivative = 1.1, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)}{5 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^2}{4 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6969, size = 114, normalized size = 1. \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b} - \frac{e \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 b^{3}} + \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{24 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x)*(d + e*x)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*b) - e*(a*e -
b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*b**3) + (2*a + 2*b*x)*(a*e - b*d)*
*2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(24*b**3)

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Mathematica [A]  time = 0.076954, size = 127, normalized size = 1.11 \[ \frac{x \sqrt{(a+b x)^2} \left (20 a^3 \left (3 d^2+3 d e x+e^2 x^2\right )+15 a^2 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+6 a b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )}{60 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(20*a^3*(3*d^2 + 3*d*e*x + e^2*x^2) + 15*a^2*b*x*(6*d^2 + 8
*d*e*x + 3*e^2*x^2) + 6*a*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + b^3*x^3*(15*
d^2 + 24*d*e*x + 10*e^2*x^2)))/(60*(a + b*x))

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Maple [A]  time = 0.01, size = 148, normalized size = 1.3 \[{\frac{x \left ( 10\,{b}^{3}{e}^{2}{x}^{5}+36\,{x}^{4}{e}^{2}a{b}^{2}+24\,{x}^{4}de{b}^{3}+45\,{x}^{3}{e}^{2}{a}^{2}b+90\,{x}^{3}dea{b}^{2}+15\,{x}^{3}{b}^{3}{d}^{2}+20\,{x}^{2}{e}^{2}{a}^{3}+120\,{x}^{2}de{a}^{2}b+60\,{x}^{2}{d}^{2}a{b}^{2}+60\,xde{a}^{3}+90\,x{d}^{2}{a}^{2}b+60\,{a}^{3}{d}^{2} \right ) }{60\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*x*(10*b^3*e^2*x^5+36*a*b^2*e^2*x^4+24*b^3*d*e*x^4+45*a^2*b*e^2*x^3+90*a*b^2
*d*e*x^3+15*b^3*d^2*x^3+20*a^3*e^2*x^2+120*a^2*b*d*e*x^2+60*a*b^2*d^2*x^2+60*a^3
*d*e*x+90*a^2*b*d^2*x+60*a^3*d^2)*((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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204678, size = 167, normalized size = 1.46 \[ \frac{1}{6} \, b^{3} e^{2} x^{6} + a^{3} d^{2} x + \frac{1}{5} \,{\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \]

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)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214309, size = 273, normalized size = 2.39 \[ \frac{1}{6} \, b^{3} x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, b^{3} d x^{5} e{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d^{2} x^{4}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, a b^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a b^{2} d x^{4} e{\rm sign}\left (b x + a\right ) + a b^{2} d^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a^{2} b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b d x^{3} e{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{3} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + a^{3} d x^{2} e{\rm sign}\left (b x + a\right ) + a^{3} d^{2} x{\rm sign}\left (b x + a\right ) \]

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)^2,x, algorithm="giac")

[Out]

1/6*b^3*x^6*e^2*sign(b*x + a) + 2/5*b^3*d*x^5*e*sign(b*x + a) + 1/4*b^3*d^2*x^4*
sign(b*x + a) + 3/5*a*b^2*x^5*e^2*sign(b*x + a) + 3/2*a*b^2*d*x^4*e*sign(b*x + a
) + a*b^2*d^2*x^3*sign(b*x + a) + 3/4*a^2*b*x^4*e^2*sign(b*x + a) + 2*a^2*b*d*x^
3*e*sign(b*x + a) + 3/2*a^2*b*d^2*x^2*sign(b*x + a) + 1/3*a^3*x^3*e^2*sign(b*x +
a) + a^3*d*x^2*e*sign(b*x + a) + a^3*d^2*x*sign(b*x + a)