3.2040 \(\int \frac{a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=260 \[ -\frac{e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{4 b e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{4 b e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{3 b e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{b e}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

(-3*b*e^2)/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b/(3*(b*d - a*e)^2*(a
+ b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*e)/((b*d - a*e)^3*(a + b*x)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (4*b*e^3*(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 = 0.457124, antiderivative size = 260, 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 \[ -\frac{e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{4 b e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{4 b e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{3 b e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{b e}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*e^2)/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b/(3*(b*d - a*e)^2*(a
+ b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*e)/((b*d - a*e)^3*(a + b*x)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/((b*d - a*e)^4*(d + e*x)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (4*b*e^3*(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 = 64.5224, size = 253, normalized size = 0.97 \[ \frac{4 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} - \frac{4 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} - \frac{4 e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{5}} + \frac{2 e^{2}}{\left (d + e x\right ) \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e \left (2 a + 2 b x\right )}{3 \left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{1}{3 \left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*d)**5
) - 4*b*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(a*e - b*d
)**5) - 4*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*d)**5) + 2*e
**2/((d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + e*(2*a + 2*b*x
)/(3*(d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + 1/(3*(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 = 0.195316, size = 144, normalized size = 0.55 \[ \frac{\frac{3 e^3 (a+b x)^3 (a e-b d)}{d+e x}+12 b e^3 (a+b x)^3 \log (d+e x)-9 b e^2 (a+b x)^2 (b d-a e)+3 b e (a+b x) (b d-a e)^2-b (b d-a e)^3-12 b e^3 (a+b x)^3 \log (a+b x)}{3 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

(-(b*(b*d - a*e)^3) + 3*b*e*(b*d - a*e)^2*(a + b*x) - 9*b*e^2*(b*d - a*e)*(a + b
*x)^2 + (3*e^3*(-(b*d) + a*e)*(a + b*x)^3)/(d + e*x) - 12*b*e^3*(a + b*x)^3*Log[
a + b*x] + 12*b*e^3*(a + b*x)^3*Log[d + e*x])/(3*(b*d - a*e)^5*((a + b*x)^2)^(3/
2))

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Maple [B]  time = 0.033, size = 483, normalized size = 1.9 \[{\frac{ \left ( 24\,{x}^{2}a{b}^{3}d{e}^{3}+36\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-36\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}-12\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}+36\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+12\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-12\,\ln \left ( ex+d \right ) x{a}^{3}b{e}^{4}-36\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-36\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}-10\,{a}^{3}bd{e}^{3}+12\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}-12\,{x}^{3}a{b}^{3}{e}^{4}+12\,{x}^{3}{b}^{4}d{e}^{3}-30\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-22\,x{a}^{3}b{e}^{4}-2\,x{b}^{4}{d}^{3}e-12\,\ln \left ( ex+d \right ){a}^{3}bd{e}^{3}+6\,x{a}^{2}{b}^{2}d{e}^{3}+18\,xa{b}^{3}{d}^{2}{e}^{2}-3\,{a}^{4}{e}^{4}-36\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+36\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}+36\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}+18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+{b}^{4}{d}^{4}-6\,a{b}^{3}{d}^{3}e \right ) \left ( bx+a \right ) ^{2}}{ \left ( 3\,ex+3\,d \right ) \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(24*x^2*a*b^3*d*e^3+36*ln(b*x+a)*x^3*a*b^3*e^4-36*ln(e*x+d)*x^3*a*b^3*e^4-12
*ln(e*x+d)*x^3*b^4*d*e^3+12*ln(b*x+a)*x^3*b^4*d*e^3+36*ln(b*x+a)*x^2*a^2*b^2*e^4
+12*ln(b*x+a)*x*a^3*b*e^4-12*ln(e*x+d)*x*a^3*b*e^4-36*ln(e*x+d)*x*a^2*b^2*d*e^3-
36*ln(e*x+d)*x^2*a*b^3*d*e^3-10*a^3*b*d*e^3+12*ln(b*x+a)*a^3*b*d*e^3-12*x^3*a*b^
3*e^4+12*x^3*b^4*d*e^3-30*x^2*a^2*b^2*e^4+6*x^2*b^4*d^2*e^2-22*x*a^3*b*e^4-2*x*b
^4*d^3*e-12*ln(e*x+d)*a^3*b*d*e^3+6*x*a^2*b^2*d*e^3+18*x*a*b^3*d^2*e^2-3*a^4*e^4
-36*ln(e*x+d)*x^2*a^2*b^2*e^4+36*ln(b*x+a)*x*a^2*b^2*d*e^3+12*ln(b*x+a)*x^4*b^4*
e^4-12*ln(e*x+d)*x^4*b^4*e^4+36*ln(b*x+a)*x^2*a*b^3*d*e^3+18*a^2*b^2*d^2*e^2+b^4
*d^4-6*a*b^3*d^3*e)*(b*x+a)^2/(e*x+d)/(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((b*x + a)/((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 = 0.305456, size = 1014, normalized size = 3.9 \[ -\frac{b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} +{\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} +{\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} +{\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b^4*d^4 - 6*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 - 3*a^4*e^4
+ 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 - 5*a^2*b^2*e^
4)*x^2 - 2*(b^4*d^3*e - 9*a*b^3*d^2*e^2 - 3*a^2*b^2*d*e^3 + 11*a^3*b*e^4)*x + 12
*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a
^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(b*x + a) - 12*(b^4*e^4*x^
4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*
x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(e*x + d))/(a^3*b^5*d^6 - 5*a^4*b^4*d^
5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b
^8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4
*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^
3*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(
a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*
d*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2
- 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*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((b*x+a)/(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. \[ \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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