3.1715 \(\int \frac{\sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{5 e^3 \sqrt{d+e x}}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{5 e^2 \sqrt{d+e x}}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e \sqrt{d+e x}}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

(-5*e^3*Sqrt[d + e*x])/(64*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt
[d + e*x]/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(2
4*b*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^2*Sqrt[d + e*x
])/(96*b*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^4*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(7/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.472202, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 e^3 \sqrt{d+e x}}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{5 e^2 \sqrt{d+e x}}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{e \sqrt{d+e x}}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{\sqrt{d+e x}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*e^3*Sqrt[d + e*x])/(64*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt
[d + e*x]/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(2
4*b*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^2*Sqrt[d + e*x
])/(96*b*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^4*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(7/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.581864, size = 169, normalized size = 0.6 \[ \frac{(a+b x)^5 \left (\frac{5 e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (10 e^2 (a+b x)^2 (a e-b d)+8 e (a+b x) (b d-a e)^2+48 (b d-a e)^3+15 e^3 (a+b x)^3\right )}{3 b (a+b x)^4 (b d-a e)^3}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^5*(-(Sqrt[d + e*x]*(48*(b*d - a*e)^3 + 8*e*(b*d - a*e)^2*(a + b*x) +
10*e^2*(-(b*d) + a*e)*(a + b*x)^2 + 15*e^3*(a + b*x)^3))/(3*b*(b*d - a*e)^3*(a +
b*x)^4) + (5*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*(b*
d - a*e)^(7/2))))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.026, size = 500, normalized size = 1.8 \[{\frac{bx+a}{192\,b \left ( ae-bd \right ) \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( 15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{4}{b}^{4}{e}^{4}+60\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{3}a{b}^{3}{e}^{4}+15\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{7/2}{b}^{3}+90\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+55\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}a{b}^{2}e-55\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{5/2}{b}^{3}d+60\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{3}b{e}^{4}+73\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{a}^{2}b{e}^{2}-146\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}a{b}^{2}de+73\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{3}{d}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{4}{e}^{4}-15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{3}{e}^{3}+45\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}bd{e}^{2}-45\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}a{b}^{2}{d}^{2}e+15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(b*x+a)*(15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^4*e^4+60*arc
tan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^3*e^4+15*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(7/2)*b^3+90*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*e^4+5
5*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-55*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)
*b^3*d+60*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b*e^4+73*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-146*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e
+73*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+15*arctan((e*x+d)^(1/2)*b/(b*(a*e-
b*d))^(1/2))*a^4*e^4-15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+45*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2-45*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^
2*e+15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)/(b*(a*e-b*d))^(1/2)/b/(a*e-b*d
)/(a^2*e^2-2*a*b*d*e+b^2*d^2)/((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(sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225585, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(2*(15*b^3*e^3*x^3 + 48*b^3*d^3 - 136*a*b^2*d^2*e + 118*a^2*b*d*e^2 - 15
*a^3*e^3 - 5*(2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 + (8*b^3*d^2*e - 36*a*b^2*d*e^2 +
73*a^2*b*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(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((sqrt(b^2*d - a*b*e)*(
b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^4*b^4*d^
3 - 3*a^5*b^3*d^2*e + 3*a^6*b^2*d*e^2 - a^7*b*e^3 + (b^8*d^3 - 3*a*b^7*d^2*e + 3
*a^2*b^6*d*e^2 - a^3*b^5*e^3)*x^4 + 4*(a*b^7*d^3 - 3*a^2*b^6*d^2*e + 3*a^3*b^5*d
*e^2 - a^4*b^4*e^3)*x^3 + 6*(a^2*b^6*d^3 - 3*a^3*b^5*d^2*e + 3*a^4*b^4*d*e^2 - a
^5*b^3*e^3)*x^2 + 4*(a^3*b^5*d^3 - 3*a^4*b^4*d^2*e + 3*a^5*b^3*d*e^2 - a^6*b^2*e
^3)*x)*sqrt(b^2*d - a*b*e)), -1/192*((15*b^3*e^3*x^3 + 48*b^3*d^3 - 136*a*b^2*d^
2*e + 118*a^2*b*d*e^2 - 15*a^3*e^3 - 5*(2*b^3*d*e^2 - 11*a*b^2*e^3)*x^2 + (8*b^3
*d^2*e - 36*a*b^2*d*e^2 + 73*a^2*b*e^3)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) -
15*(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)
*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^4*b^4*d^3 - 3*a^
5*b^3*d^2*e + 3*a^6*b^2*d*e^2 - a^7*b*e^3 + (b^8*d^3 - 3*a*b^7*d^2*e + 3*a^2*b^6
*d*e^2 - a^3*b^5*e^3)*x^4 + 4*(a*b^7*d^3 - 3*a^2*b^6*d^2*e + 3*a^3*b^5*d*e^2 - a
^4*b^4*e^3)*x^3 + 6*(a^2*b^6*d^3 - 3*a^3*b^5*d^2*e + 3*a^4*b^4*d*e^2 - a^5*b^3*e
^3)*x^2 + 4*(a^3*b^5*d^3 - 3*a^4*b^4*d^2*e + 3*a^5*b^3*d*e^2 - a^6*b^2*e^3)*x)*s
qrt(-b^2*d + a*b*e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.243211, size = 671, normalized size = 2.4 \[ \frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{4}}{64 \,{\left (b^{4} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{3} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b^{2} d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} b e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} e^{4} - 55 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{4} + 73 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt{x e + d} b^{3} d^{3} e^{4} + 55 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{5} - 146 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{5} - 45 \, \sqrt{x e + d} a b^{2} d^{2} e^{5} + 73 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{6} + 45 \, \sqrt{x e + d} a^{2} b d e^{6} - 15 \, \sqrt{x e + d} a^{3} e^{7}}{192 \,{\left (b^{4} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{3} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b^{2} d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} b e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^3*sign(-(x*e + d)*
b*e + b*d*e - a*e^2) - 3*a*b^3*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3*a^
2*b^2*d*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*b*e^3*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) + 1/192*(15*(x*e + d)^(7/2)*b^3*e^4 - 5
5*(x*e + d)^(5/2)*b^3*d*e^4 + 73*(x*e + d)^(3/2)*b^3*d^2*e^4 + 15*sqrt(x*e + d)*
b^3*d^3*e^4 + 55*(x*e + d)^(5/2)*a*b^2*e^5 - 146*(x*e + d)^(3/2)*a*b^2*d*e^5 - 4
5*sqrt(x*e + d)*a*b^2*d^2*e^5 + 73*(x*e + d)^(3/2)*a^2*b*e^6 + 45*sqrt(x*e + d)*
a^2*b*d*e^6 - 15*sqrt(x*e + d)*a^3*e^7)/((b^4*d^3*sign(-(x*e + d)*b*e + b*d*e -
a*e^2) - 3*a*b^3*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3*a^2*b^2*d*e^2*si
gn(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*b*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e
^2))*((x*e + d)*b - b*d + a*e)^4)