3.1996 \(\int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=78 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^2} \]

[Out]

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

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Rubi [A]  time = 0.155041, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.2296, size = 60, normalized size = 0.77 \[ \frac{\left (d + e x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{8 b} - \frac{\left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{56 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(8*b) - (a*e - b*d)*(a**2 + 2*a*b*
x + b**2*x**2)**(7/2)/(56*b**2)

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Mathematica [A]  time = 0.0919996, size = 140, normalized size = 1.79 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right )}{56 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(28*a^6*(2*d + e*x) + 56*a^5*b*x*(3*d + 2*e*x) + 70*a^4*b^2
*x^2*(4*d + 3*e*x) + 56*a^3*b^3*x^3*(5*d + 4*e*x) + 28*a^2*b^4*x^4*(6*d + 5*e*x)
+ 8*a*b^5*x^5*(7*d + 6*e*x) + b^6*x^6*(8*d + 7*e*x)))/(56*(a + b*x))

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Maple [B]  time = 0.008, size = 162, normalized size = 2.1 \[{\frac{x \left ( 7\,e{b}^{6}{x}^{7}+48\,{x}^{6}ea{b}^{5}+8\,{x}^{6}d{b}^{6}+140\,{x}^{5}e{a}^{2}{b}^{4}+56\,{x}^{5}da{b}^{5}+224\,{a}^{3}{b}^{3}e{x}^{4}+168\,{a}^{2}{b}^{4}d{x}^{4}+210\,{x}^{3}e{b}^{2}{a}^{4}+280\,{x}^{3}d{a}^{3}{b}^{3}+112\,{a}^{5}be{x}^{2}+280\,{a}^{4}{b}^{2}d{x}^{2}+28\,xe{a}^{6}+168\,xd{a}^{5}b+56\,d{a}^{6} \right ) }{56\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/56*x*(7*b^6*e*x^7+48*a*b^5*e*x^6+8*b^6*d*x^6+140*a^2*b^4*e*x^5+56*a*b^5*d*x^5+
224*a^3*b^3*e*x^4+168*a^2*b^4*d*x^4+210*a^4*b^2*e*x^3+280*a^3*b^3*d*x^3+112*a^5*
b*e*x^2+280*a^4*b^2*d*x^2+28*a^6*e*x+168*a^5*b*d*x+56*a^6*d)*((b*x+a)^2)^(5/2)/(
b*x+a)^5

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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)^(5/2)*(b*x + a)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295275, size = 192, normalized size = 2.46 \[ \frac{1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac{1}{7} \,{\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} +{\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} +{\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.286473, size = 319, normalized size = 4.09 \[ \frac{1}{8} \, b^{6} x^{8} e{\rm sign}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d x^{7}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, a b^{5} x^{7} e{\rm sign}\left (b x + a\right ) + a b^{5} d x^{6}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{4} x^{6} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d x^{5}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} b^{3} x^{5} e{\rm sign}\left (b x + a\right ) + 5 \, a^{3} b^{3} d x^{4}{\rm sign}\left (b x + a\right ) + \frac{15}{4} \, a^{4} b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + 2 \, a^{5} b x^{3} e{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{6} x^{2} e{\rm sign}\left (b x + a\right ) + a^{6} d x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d),x, algorithm="giac")

[Out]

1/8*b^6*x^8*e*sign(b*x + a) + 1/7*b^6*d*x^7*sign(b*x + a) + 6/7*a*b^5*x^7*e*sign
(b*x + a) + a*b^5*d*x^6*sign(b*x + a) + 5/2*a^2*b^4*x^6*e*sign(b*x + a) + 3*a^2*
b^4*d*x^5*sign(b*x + a) + 4*a^3*b^3*x^5*e*sign(b*x + a) + 5*a^3*b^3*d*x^4*sign(b
*x + a) + 15/4*a^4*b^2*x^4*e*sign(b*x + a) + 5*a^4*b^2*d*x^3*sign(b*x + a) + 2*a
^5*b*x^3*e*sign(b*x + a) + 3*a^5*b*d*x^2*sign(b*x + a) + 1/2*a^6*x^2*e*sign(b*x
+ a) + a^6*d*x*sign(b*x + a)