3.1641 \(\int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=124 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]

[Out]

(-5*e)/(3*(b*d - a*e)^2*(d + e*x)^(3/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(3
/2)) - (5*b*e)/((b*d - a*e)^3*Sqrt[d + e*x]) + (5*b^(3/2)*e*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

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Rubi [A]  time = 0.214962, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*e)/(3*(b*d - a*e)^2*(d + e*x)^(3/2)) - 1/((b*d - a*e)*(a + b*x)*(d + e*x)^(3
/2)) - (5*b*e)/((b*d - a*e)^3*Sqrt[d + e*x]) + (5*b^(3/2)*e*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

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Rubi in Sympy [A]  time = 51.3248, size = 109, normalized size = 0.88 \[ \frac{5 b^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{7}{2}}} + \frac{5 b e}{\sqrt{d + e x} \left (a e - b d\right )^{3}} - \frac{5 e}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(3/2)*e*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(a*e - b*d)**(7/2) + 5*
b*e/(sqrt(d + e*x)*(a*e - b*d)**3) - 5*e/(3*(d + e*x)**(3/2)*(a*e - b*d)**2) + 1
/((a + b*x)*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.275198, size = 125, normalized size = 1.01 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}+\sqrt{d+e x} \left (-\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{4 b e}{(d+e x) (b d-a e)^3}-\frac{2 e}{3 (d+e x)^2 (b d-a e)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*(-(b^2/((b*d - a*e)^3*(a + b*x))) - (2*e)/(3*(b*d - a*e)^2*(d + e*
x)^2) - (4*b*e)/((b*d - a*e)^3*(d + e*x))) + (5*b^(3/2)*e*ArcTanh[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

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Maple [A]  time = 0.027, size = 125, normalized size = 1. \[ -{\frac{2\,e}{3\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{be}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) }\sqrt{ex+d}}+5\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*e/(a*e-b*d)^2/(e*x+d)^(3/2)+4*e/(a*e-b*d)^3*b/(e*x+d)^(1/2)+e*b^2/(a*e-b*d)
^3*(e*x+d)^(1/2)/(b*e*x+a*e)+5*e*b^2/(a*e-b*d)^3/(b*(a*e-b*d))^(1/2)*arctan((e*x
+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22775, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 28 \, a b d e - 4 \, a^{2} e^{2} + 15 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 20 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{6 \,{\left (a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(30*b^2*e^2*x^2 + 6*b^2*d^2 + 28*a*b*d*e - 4*a^2*e^2 + 15*(b^2*e^2*x^2 + a
*b*d*e + (b^2*d*e + a*b*e^2)*x)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2
*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 20*(2
*b^2*d*e + a*b*e^2)*x)/((a*b^3*d^4 - 3*a^2*b^2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e
^3 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x^2 + (b^4*d^4
- 2*a*b^3*d^3*e + 2*a^3*b*d*e^3 - a^4*e^4)*x)*sqrt(e*x + d)), -1/3*(15*b^2*e^2*x
^2 + 3*b^2*d^2 + 14*a*b*d*e - 2*a^2*e^2 - 15*(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e +
a*b*e^2)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d
- a*e))/(sqrt(e*x + d)*b)) + 10*(2*b^2*d*e + a*b*e^2)*x)/((a*b^3*d^4 - 3*a^2*b^
2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e^3 + (b^4*d^3*e - 3*a*b^3*d^2*e^2 + 3*a^2*b^2
*d*e^3 - a^3*b*e^4)*x^2 + (b^4*d^4 - 2*a*b^3*d^3*e + 2*a^3*b*d*e^3 - a^4*e^4)*x)
*sqrt(e*x + d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215269, size = 302, normalized size = 2.44 \[ -\frac{5 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{\sqrt{x e + d} b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} b e + b d e - a e^{2}\right )}}{3 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e/((b^3*d^3 - 3*a*b^2*d^2*e
+ 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) - sqrt(x*e + d)*b^2*e/((b^3*d^3
- 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)*b - b*d + a*e)) - 2/3*(6*
(x*e + d)*b*e + b*d*e - a*e^2)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e
^3)*(x*e + d)^(3/2))