3.1562 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{6 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (e*(a^2 + 2*a*
b*x + b^2*x^2)^(7/2))/(7*b^2)

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Rubi [A]  time = 0.0752155, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{6 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (e*(a^2 + 2*a*
b*x + b^2*x^2)^(7/2))/(7*b^2)

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Rubi in Sympy [A]  time = 9.38976, size = 66, normalized size = 0.96 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{7 b^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(7*b**2) - (2*a + 2*b*x)*(a*e - b*d)*(a**2
+ 2*a*b*x + b**2*x**2)**(5/2)/(12*b**2)

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Mathematica [A]  time = 0.0813301, size = 121, normalized size = 1.75 \[ \frac{x \sqrt{(a+b x)^2} \left (21 a^5 (2 d+e x)+35 a^4 b x (3 d+2 e x)+35 a^3 b^2 x^2 (4 d+3 e x)+21 a^2 b^3 x^3 (5 d+4 e x)+7 a b^4 x^4 (6 d+5 e x)+b^5 x^5 (7 d+6 e x)\right )}{42 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(21*a^5*(2*d + e*x) + 35*a^4*b*x*(3*d + 2*e*x) + 35*a^3*b^2
*x^2*(4*d + 3*e*x) + 21*a^2*b^3*x^3*(5*d + 4*e*x) + 7*a*b^4*x^4*(6*d + 5*e*x) +
b^5*x^5*(7*d + 6*e*x)))/(42*(a + b*x))

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Maple [B]  time = 0.007, size = 138, normalized size = 2. \[{\frac{x \left ( 6\,e{b}^{5}{x}^{6}+35\,{x}^{5}ea{b}^{4}+7\,{x}^{5}d{b}^{5}+84\,{a}^{2}{b}^{3}e{x}^{4}+42\,a{b}^{4}d{x}^{4}+105\,{x}^{3}e{a}^{3}{b}^{2}+105\,{x}^{3}d{a}^{2}{b}^{3}+70\,{x}^{2}{a}^{4}be+140\,{x}^{2}{a}^{3}{b}^{2}d+21\,xe{a}^{5}+105\,xd{a}^{4}b+42\,d{a}^{5} \right ) }{42\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*x*(6*b^5*e*x^6+35*a*b^4*e*x^5+7*b^5*d*x^5+84*a^2*b^3*e*x^4+42*a*b^4*d*x^4+1
05*a^3*b^2*e*x^3+105*a^2*b^3*d*x^3+70*a^4*b*e*x^2+140*a^3*b^2*d*x^2+21*a^5*e*x+1
05*a^4*b*d*x+42*a^5*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)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204772, size = 155, normalized size = 2.25 \[ \frac{1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac{1}{6} \,{\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} +{\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21338, size = 269, normalized size = 3.9 \[ \frac{1}{7} \, b^{5} x^{7} e{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, b^{5} d x^{6}{\rm sign}\left (b x + a\right ) + \frac{5}{6} \, a b^{4} x^{6} e{\rm sign}\left (b x + a\right ) + a b^{4} d x^{5}{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b^{3} x^{5} e{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} d x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{3} b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, a^{4} b x^{3} e{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{5} x^{2} e{\rm sign}\left (b x + a\right ) + a^{5} d x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^5*x^7*e*sign(b*x + a) + 1/6*b^5*d*x^6*sign(b*x + a) + 5/6*a*b^4*x^6*e*sign
(b*x + a) + a*b^4*d*x^5*sign(b*x + a) + 2*a^2*b^3*x^5*e*sign(b*x + a) + 5/2*a^2*
b^3*d*x^4*sign(b*x + a) + 5/2*a^3*b^2*x^4*e*sign(b*x + a) + 10/3*a^3*b^2*d*x^3*s
ign(b*x + a) + 5/3*a^4*b*x^3*e*sign(b*x + a) + 5/2*a^4*b*d*x^2*sign(b*x + a) + 1
/2*a^5*x^2*e*sign(b*x + a) + a^5*d*x*sign(b*x + a)