3.1661 \(\int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=213 \[ \frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}-\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)} \]

[Out]

-Sqrt[d + e*x]/(5*(b*d - a*e)*(a + b*x)^5) + (9*e*Sqrt[d + e*x])/(40*(b*d - a*e)
^2*(a + b*x)^4) - (21*e^2*Sqrt[d + e*x])/(80*(b*d - a*e)^3*(a + b*x)^3) + (21*e^
3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)^2) - (63*e^4*Sqrt[d + e*x])/(128*(b
*d - a*e)^5*(a + b*x)) + (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(128*Sqrt[b]*(b*d - a*e)^(11/2))

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Rubi [A]  time = 0.369686, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{63 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{11/2}}-\frac{63 e^4 \sqrt{d+e x}}{128 (a+b x) (b d-a e)^5}+\frac{21 e^3 \sqrt{d+e x}}{64 (a+b x)^2 (b d-a e)^4}-\frac{21 e^2 \sqrt{d+e x}}{80 (a+b x)^3 (b d-a e)^3}+\frac{9 e \sqrt{d+e x}}{40 (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x}}{5 (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

-Sqrt[d + e*x]/(5*(b*d - a*e)*(a + b*x)^5) + (9*e*Sqrt[d + e*x])/(40*(b*d - a*e)
^2*(a + b*x)^4) - (21*e^2*Sqrt[d + e*x])/(80*(b*d - a*e)^3*(a + b*x)^3) + (21*e^
3*Sqrt[d + e*x])/(64*(b*d - a*e)^4*(a + b*x)^2) - (63*e^4*Sqrt[d + e*x])/(128*(b
*d - a*e)^5*(a + b*x)) + (63*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(128*Sqrt[b]*(b*d - a*e)^(11/2))

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Rubi in Sympy [A]  time = 109.514, size = 189, normalized size = 0.89 \[ \frac{63 e^{4} \sqrt{d + e x}}{128 \left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{21 e^{3} \sqrt{d + e x}}{64 \left (a + b x\right )^{2} \left (a e - b d\right )^{4}} + \frac{21 e^{2} \sqrt{d + e x}}{80 \left (a + b x\right )^{3} \left (a e - b d\right )^{3}} + \frac{9 e \sqrt{d + e x}}{40 \left (a + b x\right )^{4} \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x}}{5 \left (a + b x\right )^{5} \left (a e - b d\right )} + \frac{63 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 \sqrt{b} \left (a e - b d\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

63*e**4*sqrt(d + e*x)/(128*(a + b*x)*(a*e - b*d)**5) + 21*e**3*sqrt(d + e*x)/(64
*(a + b*x)**2*(a*e - b*d)**4) + 21*e**2*sqrt(d + e*x)/(80*(a + b*x)**3*(a*e - b*
d)**3) + 9*e*sqrt(d + e*x)/(40*(a + b*x)**4*(a*e - b*d)**2) + sqrt(d + e*x)/(5*(
a + b*x)**5*(a*e - b*d)) + 63*e**5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(
128*sqrt(b)*(a*e - b*d)**(11/2))

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Mathematica [A]  time = 0.78894, size = 168, normalized size = 0.79 \[ \frac{1}{640} \left (\frac{315 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{11/2}}-\frac{\sqrt{d+e x} \left (210 e^3 (a+b x)^3 (a e-b d)+168 e^2 (a+b x)^2 (b d-a e)^2+144 e (a+b x) (a e-b d)^3+128 (b d-a e)^4+315 e^4 (a+b x)^4\right )}{(a+b x)^5 (b d-a e)^5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-((Sqrt[d + e*x]*(128*(b*d - a*e)^4 + 144*e*(-(b*d) + a*e)^3*(a + b*x) + 168*e^
2*(b*d - a*e)^2*(a + b*x)^2 + 210*e^3*(-(b*d) + a*e)*(a + b*x)^3 + 315*e^4*(a +
b*x)^4))/((b*d - a*e)^5*(a + b*x)^5)) + (315*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])
/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(11/2)))/640

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Maple [A]  time = 0.016, size = 211, normalized size = 1. \[{\frac{{e}^{5}}{ \left ( 5\,ae-5\,bd \right ) \left ( bex+ae \right ) ^{5}}\sqrt{ex+d}}+{\frac{9\,{e}^{5}}{40\, \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) ^{4}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{80\, \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{21\,{e}^{5}}{64\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{63\,{e}^{5}}{128\, \left ( ae-bd \right ) ^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/5*e^5*(e*x+d)^(1/2)/(a*e-b*d)/(b*e*x+a*e)^5+9/40*e^5/(a*e-b*d)^2*(e*x+d)^(1/2)
/(b*e*x+a*e)^4+21/80*e^5/(a*e-b*d)^3*(e*x+d)^(1/2)/(b*e*x+a*e)^3+21/64*e^5/(a*e-
b*d)^4*(e*x+d)^(1/2)/(b*e*x+a*e)^2+63/128*e^5/(a*e-b*d)^5*(e*x+d)^(1/2)/(b*e*x+a
*e)+63/128*e^5/(a*e-b*d)^5/(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)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232973, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-1/1280*(2*(315*b^4*e^4*x^4 + 128*b^4*d^4 - 656*a*b^3*d^3*e + 1368*a^2*b^2*d^2*
e^2 - 1490*a^3*b*d*e^3 + 965*a^4*e^4 - 210*(b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 42*(4
*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 64*a^2*b^2*e^4)*x^2 - 6*(24*b^4*d^3*e - 128*a*b^
3*d^2*e^2 + 289*a^2*b^2*d*e^3 - 395*a^3*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x +
d) + 315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x
^2 + 5*a^4*b*e^5*x + a^5*e^5)*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^5*b^5*d^5 - 5*a^6*b^4*d^4*e + 10
*a^7*b^3*d^3*e^2 - 10*a^8*b^2*d^2*e^3 + 5*a^9*b*d*e^4 - a^10*e^5 + (b^10*d^5 - 5
*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b
^5*e^5)*x^5 + 5*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d
^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^4 + 10*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e
+ 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^3 +
10*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5
*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x^2 + 5*(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^
4*d^3*e^2 - 10*a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e^4 - a^9*b*e^5)*x)*sqrt(b^2*d - a*
b*e)), -1/640*((315*b^4*e^4*x^4 + 128*b^4*d^4 - 656*a*b^3*d^3*e + 1368*a^2*b^2*d
^2*e^2 - 1490*a^3*b*d*e^3 + 965*a^4*e^4 - 210*(b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 42
*(4*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 64*a^2*b^2*e^4)*x^2 - 6*(24*b^4*d^3*e - 128*a
*b^3*d^2*e^2 + 289*a^2*b^2*d*e^3 - 395*a^3*b*e^4)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e
*x + d) - 315*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e
^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt
(e*x + d))))/((a^5*b^5*d^5 - 5*a^6*b^4*d^4*e + 10*a^7*b^3*d^3*e^2 - 10*a^8*b^2*d
^2*e^3 + 5*a^9*b*d*e^4 - a^10*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e
^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5)*x^5 + 5*(a*b^9*d^5 - 5*
a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*
b^4*e^5)*x^4 + 10*(a^2*b^8*d^5 - 5*a^3*b^7*d^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b
^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3*e^5)*x^3 + 10*(a^3*b^7*d^5 - 5*a^4*b^6*d^
4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6*b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x
^2 + 5*(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10*a^7*b^3*d^2*e^3
+ 5*a^8*b^2*d*e^4 - a^9*b*e^5)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{6} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Integral(1/((a + b*x)**6*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.217374, size = 613, normalized size = 2.88 \[ -\frac{63 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{315 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} + 965 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 1470 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 5376 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} - 3860 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 2688 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 7110 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} + 5790 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} - 3860 \, \sqrt{x e + d} a^{3} b d e^{8} + 965 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

-63/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d^5 - 5*a*b^4*d^4
*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(-b^
2*d + a*b*e)) - 1/640*(315*(x*e + d)^(9/2)*b^4*e^5 - 1470*(x*e + d)^(7/2)*b^4*d*
e^5 + 2688*(x*e + d)^(5/2)*b^4*d^2*e^5 - 2370*(x*e + d)^(3/2)*b^4*d^3*e^5 + 965*
sqrt(x*e + d)*b^4*d^4*e^5 + 1470*(x*e + d)^(7/2)*a*b^3*e^6 - 5376*(x*e + d)^(5/2
)*a*b^3*d*e^6 + 7110*(x*e + d)^(3/2)*a*b^3*d^2*e^6 - 3860*sqrt(x*e + d)*a*b^3*d^
3*e^6 + 2688*(x*e + d)^(5/2)*a^2*b^2*e^7 - 7110*(x*e + d)^(3/2)*a^2*b^2*d*e^7 +
5790*sqrt(x*e + d)*a^2*b^2*d^2*e^7 + 2370*(x*e + d)^(3/2)*a^3*b*e^8 - 3860*sqrt(
x*e + d)*a^3*b*d*e^8 + 965*sqrt(x*e + d)*a^4*e^9)/((b^5*d^5 - 5*a*b^4*d^4*e + 10
*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*((x*e + d)*b -
b*d + a*e)^5)