3.1964 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3 (a+b x) (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.222969, antiderivative size = 146, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.7281, size = 73, normalized size = 0.5 \[ \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 \left (d + e x\right )^{4} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

b*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*(d + e*x)**3*(a*e - b*d)**2) - (a**2 +
2*a*b*x + b**2*x**2)**(3/2)/(4*(d + e*x)**4*(a*e - b*d))

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Mathematica [A]  time = 0.0593258, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(3*a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^
2*x^2)))/(12*e^3*(a + b*x)*(d + e*x)^4)

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Maple [A]  time = 0.01, size = 78, normalized size = 0.5 \[ -{\frac{6\,{x}^{2}{b}^{2}{e}^{2}+8\,xab{e}^{2}+4\,x{b}^{2}de+3\,{a}^{2}{e}^{2}+2\,abde+{b}^{2}{d}^{2}}{12\,{e}^{3} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288085, size = 132, normalized size = 0.9 \[ -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.76381, size = 104, normalized size = 0.71 \[ - \frac{3 a^{2} e^{2} + 2 a b d e + b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (8 a b e^{2} + 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

-(3*a**2*e**2 + 2*a*b*d*e + b**2*d**2 + 6*b**2*e**2*x**2 + x*(8*a*b*e**2 + 4*b**
2*d*e))/(12*d**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*d*e**6*x**3 + 12
*e**7*x**4)

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GIAC/XCAS [A]  time = 0.286706, size = 130, normalized size = 0.89 \[ -\frac{{\left (6 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 8 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a b d e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(6*b^2*x^2*e^2*sign(b*x + a) + 4*b^2*d*x*e*sign(b*x + a) + b^2*d^2*sign(b*
x + a) + 8*a*b*x*e^2*sign(b*x + a) + 2*a*b*d*e*sign(b*x + a) + 3*a^2*e^2*sign(b*
x + a))*e^(-3)/(x*e + d)^4