3.1973 \(\int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=125 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{3 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3} \]

[Out]

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

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Rubi [A]  time = 0.326745, antiderivative size = 125, 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 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{3 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 31.0694, size = 100, normalized size = 0.8 \[ \frac{\left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{7 b} - \frac{\left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{21 b^{2}} + \frac{\left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{105 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(7*b) - (d + e*x)*(a*e - b*d)*(
a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(21*b**2) + (a*e - b*d)**2*(a**2 + 2*a*b*x +
b**2*x**2)**(5/2)/(105*b**3)

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Mathematica [A]  time = 0.105959, size = 157, normalized size = 1.26 \[ \frac{x \sqrt{(a+b x)^2} \left (35 a^4 \left (3 d^2+3 d e x+e^2 x^2\right )+35 a^3 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+21 a^2 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+7 a b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )}{105 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(35*a^4*(3*d^2 + 3*d*e*x + e^2*x^2) + 35*a^3*b*x*(6*d^2 + 8
*d*e*x + 3*e^2*x^2) + 21*a^2*b^2*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 7*a*b^3*x
^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + b^4*x^4*(21*d^2 + 35*d*e*x + 15*e^2*x^2)))
/(105*(a + b*x))

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Maple [B]  time = 0.011, size = 189, normalized size = 1.5 \[{\frac{x \left ( 15\,{e}^{2}{b}^{4}{x}^{6}+70\,{x}^{5}{e}^{2}a{b}^{3}+35\,{x}^{5}de{b}^{4}+126\,{x}^{4}{e}^{2}{a}^{2}{b}^{2}+168\,{x}^{4}a{b}^{3}de+21\,{x}^{4}{b}^{4}{d}^{2}+105\,{a}^{3}b{e}^{2}{x}^{3}+315\,{a}^{2}{b}^{2}de{x}^{3}+105\,a{b}^{3}{d}^{2}{x}^{3}+35\,{x}^{2}{e}^{2}{a}^{4}+280\,{x}^{2}de{a}^{3}b+210\,{x}^{2}{d}^{2}{a}^{2}{b}^{2}+105\,{a}^{4}dex+210\,{a}^{3}b{d}^{2}x+105\,{d}^{2}{a}^{4} \right ) }{105\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*x*(15*b^4*e^2*x^6+70*a*b^3*e^2*x^5+35*b^4*d*e*x^5+126*a^2*b^2*e^2*x^4+168*
a*b^3*d*e*x^4+21*b^4*d^2*x^4+105*a^3*b*e^2*x^3+315*a^2*b^2*d*e*x^3+105*a*b^3*d^2
*x^3+35*a^4*e^2*x^2+280*a^3*b*d*e*x^2+210*a^2*b^2*d^2*x^2+105*a^4*d*e*x+210*a^3*
b*d^2*x+105*a^4*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)*(b*x + a)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276897, size = 211, normalized size = 1.69 \[ \frac{1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac{1}{3} \,{\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} +{\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} +{\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28122, size = 351, normalized size = 2.81 \[ \frac{1}{7} \, b^{4} x^{7} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{4} d x^{6} e{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, b^{4} d^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, a b^{3} x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{8}{5} \, a b^{3} d x^{5} e{\rm sign}\left (b x + a\right ) + a b^{3} d^{2} x^{4}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, a^{2} b^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{2} d x^{4} e{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{2} x^{3}{\rm sign}\left (b x + a\right ) + a^{3} b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{8}{3} \, a^{3} b d x^{3} e{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{4} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + a^{4} d x^{2} e{\rm sign}\left (b x + a\right ) + a^{4} d^{2} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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