3.1542 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=92 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.125515, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2 (a+b x) (d+e x)^5}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.8144, size = 73, normalized size = 0.79 \[ - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 e \left (d + e x\right )^{5}} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{20 e^{2} \left (a + b x\right ) \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**6,x)

[Out]

-sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*e*(d + e*x)**5) + (a*e - b*d)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(20*e**2*(a + b*x)*(d + e*x)**5)

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Mathematica [A]  time = 0.0428105, size = 45, normalized size = 0.49 \[ -\frac{\sqrt{(a+b x)^2} (4 a e+b (d+5 e x))}{20 e^2 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(4*a*e + b*(d + 5*e*x)))/(20*e^2*(a + b*x)*(d + e*x)^5)

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Maple [A]  time = 0.005, size = 42, normalized size = 0.5 \[ -{\frac{5\,bex+4\,ae+bd}{20\,{e}^{2} \left ( ex+d \right ) ^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/(e*x+d)^6,x)

[Out]

-1/20/e^2*(5*b*e*x+4*a*e+b*d)*((b*x+a)^2)^(1/2)/(e*x+d)^5/(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)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.205681, size = 97, normalized size = 1.05 \[ -\frac{5 \, b e x + b d + 4 \, a e}{20 \,{\left (e^{7} x^{5} + 5 \, d e^{6} x^{4} + 10 \, d^{2} e^{5} x^{3} + 10 \, d^{3} e^{4} x^{2} + 5 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*e*x + b*d + 4*a*e)/(e^7*x^5 + 5*d*e^6*x^4 + 10*d^2*e^5*x^3 + 10*d^3*e
^4*x^2 + 5*d^4*e^3*x + d^5*e^2)

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Sympy [A]  time = 2.75656, size = 76, normalized size = 0.83 \[ - \frac{4 a e + b d + 5 b e x}{20 d^{5} e^{2} + 100 d^{4} e^{3} x + 200 d^{3} e^{4} x^{2} + 200 d^{2} e^{5} x^{3} + 100 d e^{6} x^{4} + 20 e^{7} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/(e*x+d)**6,x)

[Out]

-(4*a*e + b*d + 5*b*e*x)/(20*d**5*e**2 + 100*d**4*e**3*x + 200*d**3*e**4*x**2 +
200*d**2*e**5*x**3 + 100*d*e**6*x**4 + 20*e**7*x**5)

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GIAC/XCAS [A]  time = 0.21116, size = 61, normalized size = 0.66 \[ -\frac{{\left (5 \, b x e{\rm sign}\left (b x + a\right ) + b d{\rm sign}\left (b x + a\right ) + 4 \, a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{20 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^6,x, algorithm="giac")

[Out]

-1/20*(5*b*x*e*sign(b*x + a) + b*d*sign(b*x + a) + 4*a*e*sign(b*x + a))*e^(-2)/(
x*e + d)^5