3.1601 \(\int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=307 \[ \frac{e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac{5 b e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{5 b e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{4 b e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{3 b e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.540045, antiderivative size = 307, 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 \[ \frac{e^4 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac{5 b e^4 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{5 b e^4 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{4 b e^3}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{3 b e^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.908, size = 304, normalized size = 0.99 \[ \frac{5 b e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{5 b e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{5 e^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{6}} + \frac{5 e^{3}}{2 \left (d + e x\right ) \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 e^{2} \left (2 a + 2 b x\right )}{12 \left (d + e x\right ) \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5 e}{12 \left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 2 b x}{8 \left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*d)**6
) - 5*b*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(a*e - b*d
)**6) - 5*e**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*d)**6) + 5*e
**3/(2*(d + e*x)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + 5*e**2*(2*a
+ 2*b*x)/(12*(d + e*x)*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + 5*e
/(12*(d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + (2*a + 2*b*
x)/(8*(d + e*x)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.214735, size = 167, normalized size = 0.54 \[ \frac{\frac{12 e^4 (a+b x)^3 (b d-a e)}{d+e x}-60 b e^4 (a+b x)^3 \log (d+e x)+48 b e^3 (a+b x)^2 (b d-a e)-18 b e^2 (a+b x) (b d-a e)^2-\frac{3 b (b d-a e)^4}{a+b x}+8 b e (b d-a e)^3+60 b e^4 (a+b x)^3 \log (a+b x)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]

Antiderivative was successfully verified.

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

[Out]

(8*b*e*(b*d - a*e)^3 - (3*b*(b*d - a*e)^4)/(a + b*x) - 18*b*e^2*(b*d - a*e)^2*(a
+ b*x) + 48*b*e^3*(b*d - a*e)*(a + b*x)^2 + (12*e^4*(b*d - a*e)*(a + b*x)^3)/(d
+ e*x) + 60*b*e^4*(a + b*x)^3*Log[a + b*x] - 60*b*e^4*(a + b*x)^3*Log[d + e*x])
/(12*(b*d - a*e)^6*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.036, size = 651, normalized size = 2.1 \[{\frac{ \left ( -240\,\ln \left ( ex+d \right ){x}^{4}a{b}^{4}{e}^{5}-60\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}-360\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{3}{e}^{5}-60\,\ln \left ( ex+d \right ) x{a}^{4}b{e}^{5}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{4}-60\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-12\,{a}^{5}{e}^{5}-3\,{b}^{5}{d}^{5}-360\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{4}-240\,\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{4}-60\,\ln \left ( ex+d \right ){x}^{5}{b}^{5}{e}^{5}-240\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{2}d{e}^{4}+240\,\ln \left ( bx+a \right ){x}^{3}a{b}^{4}d{e}^{4}-60\,{x}^{4}a{b}^{4}{e}^{5}+60\,{x}^{4}{b}^{5}d{e}^{4}-210\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+30\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-260\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-10\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-125\,x{a}^{4}b{e}^{5}+5\,x{b}^{5}{d}^{4}e+240\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}{e}^{5}+360\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}{e}^{5}+60\,\ln \left ( bx+a \right ) x{a}^{4}b{e}^{5}+60\,\ln \left ( bx+a \right ){a}^{4}bd{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{4}{b}^{5}d{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{5}{b}^{5}{e}^{5}-65\,{a}^{4}bd{e}^{4}+120\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+20\,a{b}^{4}{d}^{4}e+180\,{x}^{3}a{b}^{4}d{e}^{4}+150\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+120\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-60\,\ln \left ( ex+d \right ){a}^{4}bd{e}^{4}-20\,x{a}^{3}{b}^{2}d{e}^{4}+180\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}d{e}^{4}-240\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{2}{e}^{5}-40\,xa{b}^{4}{d}^{3}{e}^{2} \right ) \left ( bx+a \right ) }{ \left ( 12\,ex+12\,d \right ) \left ( ae-bd \right ) ^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(-240*ln(e*x+d)*x^4*a*b^4*e^5-60*ln(e*x+d)*x^4*b^5*d*e^4-360*ln(e*x+d)*x^3*
a^2*b^3*e^5-60*ln(e*x+d)*x*a^4*b*e^5+360*ln(b*x+a)*x^2*a^2*b^3*d*e^4-60*a^2*b^3*
d^3*e^2-12*a^5*e^5-3*b^5*d^5-360*ln(e*x+d)*x^2*a^2*b^3*d*e^4-240*ln(e*x+d)*x^3*a
*b^4*d*e^4-60*ln(e*x+d)*x^5*b^5*e^5-240*ln(e*x+d)*x*a^3*b^2*d*e^4+240*ln(b*x+a)*
x^3*a*b^4*d*e^4-60*x^4*a*b^4*e^5+60*x^4*b^5*d*e^4-210*x^3*a^2*b^3*e^5+30*x^3*b^5
*d^2*e^3-260*x^2*a^3*b^2*e^5-10*x^2*b^5*d^3*e^2-125*x*a^4*b*e^5+5*x*b^5*d^4*e+24
0*ln(b*x+a)*x^4*a*b^4*e^5+360*ln(b*x+a)*x^3*a^2*b^3*e^5+240*ln(b*x+a)*x^2*a^3*b^
2*e^5+60*ln(b*x+a)*x*a^4*b*e^5+60*ln(b*x+a)*a^4*b*d*e^4+60*ln(b*x+a)*x^4*b^5*d*e
^4+60*ln(b*x+a)*x^5*b^5*e^5-65*a^4*b*d*e^4+120*a^3*b^2*d^2*e^3+20*a*b^4*d^4*e+18
0*x^3*a*b^4*d*e^4+150*x^2*a^2*b^3*d*e^4+120*x^2*a*b^4*d^2*e^3-60*ln(e*x+d)*a^4*b
*d*e^4-20*x*a^3*b^2*d*e^4+180*x*a^2*b^3*d^2*e^3+240*ln(b*x+a)*x*a^3*b^2*d*e^4-24
0*ln(e*x+d)*x^2*a^3*b^2*e^5-40*x*a*b^4*d^3*e^2)*(b*x+a)/(e*x+d)/(a*e-b*d)^6/((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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23707, size = 1462, normalized size = 4.76 \[ \text{result too large to display} \]

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)^2),x, algorithm="fricas")

[Out]

-1/12*(3*b^5*d^5 - 20*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 6
5*a^4*b*d*e^4 + 12*a^5*e^5 - 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 - 30*(b^5*d^2*e^3 +
6*a*b^4*d*e^4 - 7*a^2*b^3*e^5)*x^3 + 10*(b^5*d^3*e^2 - 12*a*b^4*d^2*e^3 - 15*a^2
*b^3*d*e^4 + 26*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 8*a*b^4*d^3*e^2 + 36*a^2*b^3*d
^2*e^3 - 4*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^
5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^
3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*x + a) + 6
0*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^4
+ 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*e^
4 + a^4*b*e^5)*x)*log(e*x + d))/(a^4*b^6*d^7 - 6*a^5*b^5*d^6*e + 15*a^6*b^4*d^5*
e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (
b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*
b^6*d^2*e^5 - 6*a^5*b^5*d*e^6 + a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9
*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 + 54*a^5*b^5*d^2*e^5
- 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*
a^3*b^7*d^5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 -
16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 + 2*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33
*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5*a^6*b^4*d^3*e^4 + 12*a^7*b^3*d^2*e^5 -
9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5
*b^5*d^5*e^2 - 65*a^6*b^4*d^4*e^3 + 40*a^7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a
^9*b*d*e^6 + a^10*e^7)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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