3.2041 \(\int \frac{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=323 \[ -\frac{4 b e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac{10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{6 b^2 e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b^2 e}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

(-6*b^2*e^2)/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(3*(b*d - a*e)^
3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e)/(2*(b*d - a*e)^4*(a + b
*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/(2*(b*d - a*e)^4*(d + e*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x))/((b*d - a*e)^5*(d + e*x)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10*b^2*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e
)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (10*b^2*e^3*(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 = 0.576238, antiderivative size = 323, 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 \[ -\frac{4 b e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}-\frac{e^3 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^4}-\frac{10 b^2 e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac{10 b^2 e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac{6 b^2 e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{3 b^2 e}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-6*b^2*e^2)/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(3*(b*d - a*e)^
3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e)/(2*(b*d - a*e)^4*(a + b
*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/(2*(b*d - a*e)^4*(d + e*x)^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x))/((b*d - a*e)^5*(d + e*x)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10*b^2*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e
)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (10*b^2*e^3*(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 = 85.9442, size = 318, normalized size = 0.98 \[ - \frac{10 b^{2} e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} + \frac{10 b^{2} e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{6}} + \frac{10 b e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{6}} - \frac{5 e^{3} \left (2 a + 2 b x\right )}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{10 e^{2}}{3 \left (d + e x\right )^{2} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{5 e \left (2 a + 2 b x\right )}{12 \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{1}{3 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

-10*b**2*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/((a + b*x)*(a*e - b*
d)**6) + 10*b**2*e**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/((a + b*x)*(
a*e - b*d)**6) + 10*b*e**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/((d + e*x)*(a*e - b*
d)**6) - 5*e**3*(2*a + 2*b*x)/(2*(d + e*x)**2*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)) + 10*e**2/(3*(d + e*x)**2*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b*
*2*x**2)) + 5*e*(2*a + 2*b*x)/(12*(d + e*x)**2*(a*e - b*d)**2*(a**2 + 2*a*b*x +
b**2*x**2)**(3/2)) + 1/(3*(d + e*x)**2*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)*
*(3/2))

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Mathematica [A]  time = 0.228132, size = 184, normalized size = 0.57 \[ \frac{60 b^2 e^3 (a+b x)^3 \log (d+e x)-36 b^2 e^2 (a+b x)^2 (b d-a e)+9 b^2 e (a+b x) (b d-a e)^2-2 b^2 (b d-a e)^3-60 b^2 e^3 (a+b x)^3 \log (a+b x)-\frac{3 e^3 (a+b x)^3 (b d-a e)^2}{(d+e x)^2}-\frac{24 b e^3 (a+b x)^3 (b d-a e)}{d+e x}}{6 \left ((a+b x)^2\right )^{3/2} (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-2*b^2*(b*d - a*e)^3 + 9*b^2*e*(b*d - a*e)^2*(a + b*x) - 36*b^2*e^2*(b*d - a*e)
*(a + b*x)^2 - (3*e^3*(b*d - a*e)^2*(a + b*x)^3)/(d + e*x)^2 - (24*b*e^3*(b*d -
a*e)*(a + b*x)^3)/(d + e*x) - 60*b^2*e^3*(a + b*x)^3*Log[a + b*x] + 60*b^2*e^3*(
a + b*x)^3*Log[d + e*x])/(6*(b*d - a*e)^6*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.038, size = 753, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/6*(60*x^3*a*b^4*d*e^4-120*x^2*a^2*b^3*d*e^4+60*x*a*b^4*d^3*e^2+180*ln(b*x+a)*
x^3*a^2*b^3*e^5-60*ln(e*x+d)*x^5*b^5*e^5+90*x^3*b^5*d^2*e^3+60*ln(b*x+a)*x^5*b^5
*e^5-360*ln(e*x+d)*x^3*a*b^4*d*e^4-120*ln(e*x+d)*x*a^3*b^2*d*e^4-180*ln(e*x+d)*x
*a^2*b^3*d^2*e^3-180*ln(e*x+d)*x^2*a*b^4*d^2*e^3+60*a^2*b^3*d^3*e^2+60*ln(b*x+a)
*x^2*a^3*b^2*e^5+210*x^2*a*b^4*d^2*e^3-160*x*a^3*b^2*d*e^4+120*x*a^2*b^3*d^2*e^3
+60*ln(b*x+a)*x^3*b^5*d^2*e^3+3*a^5*e^5+2*b^5*d^5-60*ln(e*x+d)*a^3*b^2*d^2*e^3+1
20*ln(b*x+a)*x^4*b^5*d*e^4-60*ln(e*x+d)*x^2*a^3*b^2*e^5-180*ln(e*x+d)*x^3*a^2*b^
3*e^5-60*ln(e*x+d)*x^3*b^5*d^2*e^3-180*ln(e*x+d)*x^4*a*b^4*e^5-120*ln(e*x+d)*x^4
*b^5*d*e^4+180*ln(b*x+a)*x^4*a*b^4*e^5+60*ln(b*x+a)*a^3*b^2*d^2*e^3-30*a^4*b*d*e
^4-20*a^3*b^2*d^2*e^3-15*a*b^4*d^4*e+360*ln(b*x+a)*x^2*a^2*b^3*d*e^4+120*ln(b*x+
a)*x*a^3*b^2*d*e^4+180*ln(b*x+a)*x^2*a*b^4*d^2*e^3+180*ln(b*x+a)*x*a^2*b^3*d^2*e
^3-360*ln(e*x+d)*x^2*a^2*b^3*d*e^4+60*x^4*b^5*d*e^4-150*x^3*a^2*b^3*e^5-110*x^2*
a^3*b^2*e^5+20*x^2*b^5*d^3*e^2-15*x*a^4*b*e^5-5*x*b^5*d^4*e-60*x^4*a*b^4*e^5+360
*ln(b*x+a)*x^3*a*b^4*d*e^4)*(b*x+a)^2/(e*x+d)^2/(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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31599, size = 1554, normalized size = 4.81 \[ \text{result too large to display} \]

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)^3),x, algorithm="fricas")

[Out]

-1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*
a^4*b*d*e^4 + 3*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5*d^2*e^3 + 2
*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a*b^4*d^2*e^3 - 12*a^
2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3
*d^2*e^3 + 32*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3
+ (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^
5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^2*b^3*d^2*
e^3 + 2*a^3*b^2*d*e^4)*x)*log(b*x + a) - 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*
b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3
+ (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^2*b^3*d^2*e^3 +
2*a^3*b^2*d*e^4)*x)*log(e*x + d))/(a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^
6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6
+ (b^9*d^6*e^2 - 6*a*b^8*d^5*e^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*
a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^9*d^7*e - 9*a*b^8*d^
6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4
*d^2*e^6 - 16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2
+ 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2
*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7*d^7*e + 1
0*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5
- 18*a^7*b^2*d^2*e^6 + a^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^
4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6*b^3*d^4*e^4 + 12*a^7*b^2*d^3*e^5 - 9*
a^8*b*d^2*e^6 + 2*a^9*d*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((b*x+a)/(e*x+d)**3/(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. \[ \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{3}}\,{d x} \]

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)^3),x, algorithm="giac")

[Out]

integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^3), x)