3.1591 \(\int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{3 b e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{3 b e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.318402, antiderivative size = 217, 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 \[ \frac{e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{3 b e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{3 b e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 46.3085, size = 214, normalized size = 0.99 \[ \frac{3 b e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{3 b e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{3 e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{4}} + \frac{3 e}{2 \left (d + e x\right ) \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 a + 2 b x}{4 \left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.183662, size = 141, normalized size = 0.65 \[ \frac{-(b d-a e) \left (-2 a^2 e^2-a b e (5 d+9 e x)+b^2 \left (d^2-3 d e x-6 e^2 x^2\right )\right )+6 b e^2 (a+b x)^2 (d+e x) \log (a+b x)-6 b e^2 (a+b x)^2 (d+e x) \log (d+e x)}{2 (a+b x) \sqrt{(a+b x)^2} (d+e x) (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

(-((b*d - a*e)*(-2*a^2*e^2 - a*b*e*(5*d + 9*e*x) + b^2*(d^2 - 3*d*e*x - 6*e^2*x^
2))) + 6*b*e^2*(a + b*x)^2*(d + e*x)*Log[a + b*x] - 6*b*e^2*(a + b*x)^2*(d + e*x
)*Log[d + e*x])/(2*(b*d - a*e)^4*(a + b*x)*Sqrt[(a + b*x)^2]*(d + e*x))

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Maple [B]  time = 0.03, size = 330, normalized size = 1.5 \[{\frac{ \left ( 6\,\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}{e}^{3}+6\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}-12\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}-6\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ) x{a}^{2}b{e}^{3}+12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}-6\,\ln \left ( ex+d \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}-6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}-6\,\ln \left ( ex+d \right ){a}^{2}bd{e}^{2}-9\,x{a}^{2}b{e}^{3}+6\,xa{b}^{2}d{e}^{2}+3\,x{b}^{3}{d}^{2}e-2\,{a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+6\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) }{ \left ( 2\,ex+2\,d \right ) \left ( ae-bd \right ) ^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/2*(6*ln(b*x+a)*x^3*b^3*e^3-6*ln(e*x+d)*x^3*b^3*e^3+12*ln(b*x+a)*x^2*a*b^2*e^3+
6*ln(b*x+a)*x^2*b^3*d*e^2-12*ln(e*x+d)*x^2*a*b^2*e^3-6*ln(e*x+d)*x^2*b^3*d*e^2+6
*ln(b*x+a)*x*a^2*b*e^3+12*ln(b*x+a)*x*a*b^2*d*e^2-6*ln(e*x+d)*x*a^2*b*e^3-12*ln(
e*x+d)*x*a*b^2*d*e^2-6*x^2*a*b^2*e^3+6*x^2*b^3*d*e^2+6*ln(b*x+a)*a^2*b*d*e^2-6*l
n(e*x+d)*a^2*b*d*e^2-9*x*a^2*b*e^3+6*x*a*b^2*d*e^2+3*x*b^3*d^2*e-2*a^3*e^3-3*a^2
*b*d*e^2+6*a*b^2*d^2*e-b^3*d^3)*(b*x+a)/(e*x+d)/(a*e-b*d)^4/((b*x+a)^2)^(3/2)

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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(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220812, size = 667, normalized size = 3.07 \[ -\frac{b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} +{\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} +{\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} +{\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral(1/((d + e*x)**2*((a + b*x)**2)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^2), x)