3.1965 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^3 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^5} \]

[Out]

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

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Rubi [A]  time = 0.224175, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^3 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.9242, size = 114, normalized size = 0.78 \[ - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{30 \left (d + e x\right )^{3} \left (a e - b d\right )^{3}} + \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{10 \left (d + e x\right )^{4} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 \left (d + e x\right )^{5} \left (a e - b d\right )} \]

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)**6,x)

[Out]

-b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(30*(d + e*x)**3*(a*e - b*d)**3) + b*(
a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(10*(d + e*x)**4*(a*e - b*d)**2) - (a**2 + 2*
a*b*x + b**2*x**2)**(3/2)/(5*(d + e*x)**5*(a*e - b*d))

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Mathematica [A]  time = 0.0699505, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{30 e^3 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) + b^2*(d^2 + 5*d*e*x + 10*e
^2*x^2)))/(30*e^3*(a + b*x)*(d + e*x)^5)

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Maple [A]  time = 0.011, size = 78, normalized size = 0.5 \[ -{\frac{10\,{x}^{2}{b}^{2}{e}^{2}+15\,xab{e}^{2}+5\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+3\,abde+{b}^{2}{d}^{2}}{30\,{e}^{3} \left ( ex+d \right ) ^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30/e^3*(10*b^2*e^2*x^2+15*a*b*e^2*x+5*b^2*d*e*x+6*a^2*e^2+3*a*b*d*e+b^2*d^2)*
((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)*(b*x + a)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282353, size = 147, normalized size = 1.01 \[ -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*e^2*x^2 + b^2*d^2 + 3*a*b*d*e + 6*a^2*e^2 + 5*(b^2*d*e + 3*a*b*e^2
)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^
5*e^3)

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Sympy [A]  time = 4.4895, size = 116, normalized size = 0.79 \[ - \frac{6 a^{2} e^{2} + 3 a b d e + b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} + 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(6*a**2*e**2 + 3*a*b*d*e + b**2*d**2 + 10*b**2*e**2*x**2 + x*(15*a*b*e**2 + 5*b
**2*d*e))/(30*d**5*e**3 + 150*d**4*e**4*x + 300*d**3*e**5*x**2 + 300*d**2*e**6*x
**3 + 150*d*e**7*x**4 + 30*e**8*x**5)

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GIAC/XCAS [A]  time = 0.28631, size = 130, normalized size = 0.89 \[ -\frac{{\left (10 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 15 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b d e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{30 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*x^2*e^2*sign(b*x + a) + 5*b^2*d*x*e*sign(b*x + a) + b^2*d^2*sign(b
*x + a) + 15*a*b*x*e^2*sign(b*x + a) + 3*a*b*d*e*sign(b*x + a) + 6*a^2*e^2*sign(
b*x + a))*e^(-3)/(x*e + d)^5