3.1600 \(\int \frac{1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

e^3/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*(b*d - a*e)*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + e/(3*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - e^2/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (e^4*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
e^4*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.405376, antiderivative size = 253, 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 \[ \frac{e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

e^3/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*(b*d - a*e)*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + e/(3*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - e^2/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (e^4*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
e^4*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 53.3957, size = 231, normalized size = 0.91 \[ - \frac{e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{e^{3}}{\left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e^{2} \left (2 a + 2 b x\right )}{4 \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{e}{3 \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 2 b x}{8 \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*d)**5) +
e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(a*e - b*d)**5) +
e**3/((a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + e**2*(2*a + 2*b*x)/(4*
(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + e/(3*(a*e - b*d)**2*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)) + (2*a + 2*b*x)/(8*(a*e - b*d)*(a**2 + 2*a*b*x +
b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.226926, size = 163, normalized size = 0.64 \[ \frac{-(b d-a e) \left (-25 a^3 e^3+a^2 b e^2 (23 d-52 e x)+a b^2 e \left (-13 d^2+20 d e x-42 e^2 x^2\right )+b^3 \left (3 d^3-4 d^2 e x+6 d e^2 x^2-12 e^3 x^3\right )\right )-12 e^4 (a+b x)^4 \log (d+e x)+12 e^4 (a+b x)^4 \log (a+b x)}{12 (a+b x)^3 \sqrt{(a+b x)^2} (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*d - a*e)*(-25*a^3*e^3 + a^2*b*e^2*(23*d - 52*e*x) + a*b^2*e*(-13*d^2 + 20*
d*e*x - 42*e^2*x^2) + b^3*(3*d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 12*e^3*x^3))) + 12*
e^4*(a + b*x)^4*Log[a + b*x] - 12*e^4*(a + b*x)^4*Log[d + e*x])/(12*(b*d - a*e)^
5*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.028, size = 359, normalized size = 1.4 \[ -{\frac{ \left ( 12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}+48\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-48\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}+72\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-72\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-12\,{x}^{3}a{b}^{3}{e}^{4}+12\,{x}^{3}{b}^{4}d{e}^{3}+48\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-48\,\ln \left ( ex+d \right ) x{a}^{3}b{e}^{4}-42\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+48\,{x}^{2}a{b}^{3}d{e}^{3}-6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-12\,\ln \left ( ex+d \right ){a}^{4}{e}^{4}-52\,x{a}^{3}b{e}^{4}+72\,x{a}^{2}{b}^{2}d{e}^{3}-24\,xa{b}^{3}{d}^{2}{e}^{2}+4\,x{b}^{4}{d}^{3}e-25\,{a}^{4}{e}^{4}+48\,{a}^{3}bd{e}^{3}-36\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+16\,a{b}^{3}{d}^{3}e-3\,{b}^{4}{d}^{4} \right ) \left ( bx+a \right ) }{12\, \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(12*ln(b*x+a)*x^4*b^4*e^4-12*ln(e*x+d)*x^4*b^4*e^4+48*ln(b*x+a)*x^3*a*b^3*
e^4-48*ln(e*x+d)*x^3*a*b^3*e^4+72*ln(b*x+a)*x^2*a^2*b^2*e^4-72*ln(e*x+d)*x^2*a^2
*b^2*e^4-12*x^3*a*b^3*e^4+12*x^3*b^4*d*e^3+48*ln(b*x+a)*x*a^3*b*e^4-48*ln(e*x+d)
*x*a^3*b*e^4-42*x^2*a^2*b^2*e^4+48*x^2*a*b^3*d*e^3-6*x^2*b^4*d^2*e^2+12*ln(b*x+a
)*a^4*e^4-12*ln(e*x+d)*a^4*e^4-52*x*a^3*b*e^4+72*x*a^2*b^2*d*e^3-24*x*a*b^3*d^2*
e^2+4*x*b^4*d^3*e-25*a^4*e^4+48*a^3*b*d*e^3-36*a^2*b^2*d^2*e^2+16*a*b^3*d^3*e-3*
b^4*d^4)*(b*x+a)/(a*e-b*d)^5/((b*x+a)^2)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224984, size = 887, normalized size = 3.51 \[ -\frac{3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (a^{4} b^{5} d^{5} - 5 \, a^{5} b^{4} d^{4} e + 10 \, a^{6} b^{3} d^{3} e^{2} - 10 \, a^{7} b^{2} d^{2} e^{3} + 5 \, a^{8} b d e^{4} - a^{9} e^{5} +{\left (b^{9} d^{5} - 5 \, a b^{8} d^{4} e + 10 \, a^{2} b^{7} d^{3} e^{2} - 10 \, a^{3} b^{6} d^{2} e^{3} + 5 \, a^{4} b^{5} d e^{4} - a^{5} b^{4} e^{5}\right )} x^{4} + 4 \,{\left (a b^{8} d^{5} - 5 \, a^{2} b^{7} d^{4} e + 10 \, a^{3} b^{6} d^{3} e^{2} - 10 \, a^{4} b^{5} d^{2} e^{3} + 5 \, a^{5} b^{4} d e^{4} - a^{6} b^{3} e^{5}\right )} x^{3} + 6 \,{\left (a^{2} b^{7} d^{5} - 5 \, a^{3} b^{6} d^{4} e + 10 \, a^{4} b^{5} d^{3} e^{2} - 10 \, a^{5} b^{4} d^{2} e^{3} + 5 \, a^{6} b^{3} d e^{4} - a^{7} b^{2} e^{5}\right )} x^{2} + 4 \,{\left (a^{3} b^{6} d^{5} - 5 \, a^{4} b^{5} d^{4} e + 10 \, a^{5} b^{4} d^{3} e^{2} - 10 \, a^{6} b^{3} d^{2} e^{3} + 5 \, a^{7} b^{2} d e^{4} - a^{8} b e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*d^4 - 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 - 48*a^3*b*d*e^3 + 25*a^4
*e^4 - 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 - 8*a*b^3*d*e^3 + 7*a^2*b
^2*e^4)*x^2 - 4*(b^4*d^3*e - 6*a*b^3*d^2*e^2 + 18*a^2*b^2*d*e^3 - 13*a^3*b*e^4)*
x - 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*
e^4)*log(b*x + a) + 12*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^
3*b*e^4*x + a^4*e^4)*log(e*x + d))/(a^4*b^5*d^5 - 5*a^5*b^4*d^4*e + 10*a^6*b^3*d
^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b*d*e^4 - a^9*e^5 + (b^9*d^5 - 5*a*b^8*d^4*e
+ 10*a^2*b^7*d^3*e^2 - 10*a^3*b^6*d^2*e^3 + 5*a^4*b^5*d*e^4 - a^5*b^4*e^5)*x^4
+ 4*(a*b^8*d^5 - 5*a^2*b^7*d^4*e + 10*a^3*b^6*d^3*e^2 - 10*a^4*b^5*d^2*e^3 + 5*a
^5*b^4*d*e^4 - a^6*b^3*e^5)*x^3 + 6*(a^2*b^7*d^5 - 5*a^3*b^6*d^4*e + 10*a^4*b^5*
d^3*e^2 - 10*a^5*b^4*d^2*e^3 + 5*a^6*b^3*d*e^4 - a^7*b^2*e^5)*x^2 + 4*(a^3*b^6*d
^5 - 5*a^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^2 - 10*a^6*b^3*d^2*e^3 + 5*a^7*b^2*d*e^4
- a^8*b*e^5)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((d + e*x)*((a + b*x)**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.655701, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

sage0*x