3.325 \(\int \frac{\sqrt{a+c x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=274 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2} d}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{e^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^5}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{\sqrt{a+c x^2}}{4 d x^4} \]

[Out]

-Sqrt[a + c*x^2]/(4*d*x^4) - (c*Sqrt[a + c*x^2])/(8*a*d*x^2) - (e^2*Sqrt[a + c*x
^2])/(2*d^3*x^2) + (e^3*Sqrt[a + c*x^2])/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^
2*x^3) + (e^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqr
t[a + c*x^2])])/d^5 + (c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(3/2)*d) - (c*
e^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d^3) - (Sqrt[a]*e^4*ArcTanh[Sqr
t[a + c*x^2]/Sqrt[a]])/d^5

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Rubi [A]  time = 0.701576, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2} d}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{e^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^5}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{\sqrt{a+c x^2}}{4 d x^4} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + c*x^2]/(4*d*x^4) - (c*Sqrt[a + c*x^2])/(8*a*d*x^2) - (e^2*Sqrt[a + c*x
^2])/(2*d^3*x^2) + (e^3*Sqrt[a + c*x^2])/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^
2*x^3) + (e^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqr
t[a + c*x^2])])/d^5 + (c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(3/2)*d) - (c*
e^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d^3) - (Sqrt[a]*e^4*ArcTanh[Sqr
t[a + c*x^2]/Sqrt[a]])/d^5

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Rubi in Sympy [A]  time = 73.7002, size = 243, normalized size = 0.89 \[ - \frac{\sqrt{a} e^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{d^{5}} - \frac{\sqrt{a + c x^{2}}}{4 d x^{4}} - \frac{e^{2} \sqrt{a + c x^{2}}}{2 d^{3} x^{2}} + \frac{e^{3} \sqrt{a + c x^{2}}}{d^{4} x} + \frac{e^{3} \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{5}} - \frac{c \sqrt{a + c x^{2}}}{8 a d x^{2}} + \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 a d^{2} x^{3}} - \frac{c e^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 \sqrt{a} d^{3}} + \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*e**4*atanh(sqrt(a + c*x**2)/sqrt(a))/d**5 - sqrt(a + c*x**2)/(4*d*x**4)
- e**2*sqrt(a + c*x**2)/(2*d**3*x**2) + e**3*sqrt(a + c*x**2)/(d**4*x) + e**3*s
qrt(a*e**2 + c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)
))/d**5 - c*sqrt(a + c*x**2)/(8*a*d*x**2) + e*(a + c*x**2)**(3/2)/(3*a*d**2*x**3
) - c*e**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(2*sqrt(a)*d**3) + c**2*atanh(sqrt(a
+ c*x**2)/sqrt(a))/(8*a**(3/2)*d)

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Mathematica [A]  time = 0.644674, size = 248, normalized size = 0.91 \[ \frac{\frac{3 \left (-8 a^2 e^4-4 a c d^2 e^2+c^2 d^4\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2}}+\frac{3 \log (x) \left (8 a^2 e^4+4 a c d^2 e^2-c^2 d^4\right )}{a^{3/2}}+24 e^3 \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-24 e^3 \sqrt{a e^2+c d^2} \log (d+e x)+\frac{d \sqrt{a+c x^2} \left (a \left (-6 d^3+8 d^2 e x-12 d e^2 x^2+24 e^3 x^3\right )+c d^2 x^2 (8 e x-3 d)\right )}{a x^4}}{24 d^5} \]

Antiderivative was successfully verified.

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

[Out]

((d*Sqrt[a + c*x^2]*(c*d^2*x^2*(-3*d + 8*e*x) + a*(-6*d^3 + 8*d^2*e*x - 12*d*e^2
*x^2 + 24*e^3*x^3)))/(a*x^4) + (3*(-(c^2*d^4) + 4*a*c*d^2*e^2 + 8*a^2*e^4)*Log[x
])/a^(3/2) - 24*e^3*Sqrt[c*d^2 + a*e^2]*Log[d + e*x] + (3*(c^2*d^4 - 4*a*c*d^2*e
^2 - 8*a^2*e^4)*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/a^(3/2) + 24*e^3*Sqrt[c*d^2 +
a*e^2]*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(24*d^5)

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Maple [B]  time = 0.02, size = 703, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/d/a/x^4*(c*x^2+a)^(3/2)+1/8/d*c/a^2/x^2*(c*x^2+a)^(3/2)+1/8/d*c^2/a^(3/2)*l
n((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-1/8/d*c^2/a^2*(c*x^2+a)^(1/2)-1/d^5*e^4*a^(
1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+1/d^5*e^4*(c*x^2+a)^(1/2)-1/2/d^3*e^2
/a/x^2*(c*x^2+a)^(3/2)-1/2/d^3*e^2*c/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/
x)+1/2/d^3*e^2*c/a*(c*x^2+a)^(1/2)-1/d^5*e^4*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2
+c*d^2)/e^2)^(1/2)+1/d^4*e^3*c^(1/2)*ln((-c*d/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c-
2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))+1/d^5*e^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln
((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2
*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a+1/d^3*e^2/((a*e^2+c*d^2)/e^2
)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+
d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*c+1/d^4*e^3/a/x*(c*x
^2+a)^(3/2)-1/d^4*e^3*c/a*x*(c*x^2+a)^(1/2)-1/d^4*e^3*c^(1/2)*ln(x*c^(1/2)+(c*x^
2+a)^(1/2))+1/3*e*(c*x^2+a)^(3/2)/a/d^2/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}}{{\left (e x + d\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.419208, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(24*sqrt(c*d^2 + a*e^2)*a^(3/2)*e^3*x^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2
*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^
2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*x^4
*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) + 2*(8*a*d^3*e*x - 6*a*
d^4 + 8*(c*d^3*e + 3*a*d*e^3)*x^3 - 3*(c*d^4 + 4*a*d^2*e^2)*x^2)*sqrt(c*x^2 + a)
*sqrt(a))/(a^(3/2)*d^5*x^4), -1/48*(48*sqrt(-c*d^2 - a*e^2)*a^(3/2)*e^3*x^4*arct
an((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) + 3*(c^2*d^4 - 4*a*c*d^
2*e^2 - 8*a^2*e^4)*x^4*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) -
2*(8*a*d^3*e*x - 6*a*d^4 + 8*(c*d^3*e + 3*a*d*e^3)*x^3 - 3*(c*d^4 + 4*a*d^2*e^2
)*x^2)*sqrt(c*x^2 + a)*sqrt(a))/(a^(3/2)*d^5*x^4), 1/24*(12*sqrt(c*d^2 + a*e^2)*
sqrt(-a)*a*e^3*x^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2
)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x
+ d^2)) + 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*x^4*arctan(sqrt(-a)/sqrt(c*x^2
+ a)) + (8*a*d^3*e*x - 6*a*d^4 + 8*(c*d^3*e + 3*a*d*e^3)*x^3 - 3*(c*d^4 + 4*a*d
^2*e^2)*x^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*d^5*x^4), -1/24*(24*sqrt(-c*d
^2 - a*e^2)*sqrt(-a)*a*e^3*x^4*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c
*x^2 + a))) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*x^4*arctan(sqrt(-a)/sqrt(c
*x^2 + a)) - (8*a*d^3*e*x - 6*a*d^4 + 8*(c*d^3*e + 3*a*d*e^3)*x^3 - 3*(c*d^4 + 4
*a*d^2*e^2)*x^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*d^5*x^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + c*x**2)/(x**5*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.288165, size = 805, normalized size = 2.94 \[ -\frac{2 \,{\left (c d^{2} e^{3} + a e^{5}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{\sqrt{-c d^{2} - a e^{2}} d^{5}} - \frac{{\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a d^{5}} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} c^{2} d^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} a c^{\frac{3}{2}} d^{2} e + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a c^{2} d^{3} + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} a c d e^{2} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a^{2} c^{\frac{3}{2}} d^{2} e + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} c^{2} d^{3} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a^{2} c d e^{2} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} a^{2} \sqrt{c} e^{3} - 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} c^{\frac{3}{2}} d^{2} e + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} c^{2} d^{3} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{3} c d e^{2} + 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a^{3} \sqrt{c} e^{3} + 8 \, a^{4} c^{\frac{3}{2}} d^{2} e + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{4} c d e^{2} - 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{4} \sqrt{c} e^{3} + 24 \, a^{5} \sqrt{c} e^{3}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4} a d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d^2*e^3 + a*e^5)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqr
t(-c*d^2 - a*e^2))/(sqrt(-c*d^2 - a*e^2)*d^5) - 1/4*(c^2*d^4 - 4*a*c*d^2*e^2 - 8
*a^2*e^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*d^5) + 1/1
2*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^2*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^
6*a*c^(3/2)*d^2*e + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d^3 + 12*(sqrt(c)*x
- sqrt(c*x^2 + a))^7*a*c*d*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)
*d^2*e + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d^3 - 12*(sqrt(c)*x - sqrt(c
*x^2 + a))^5*a^2*c*d*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*sqrt(c)*e^3 -
8*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^3*c^2*d^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c*d*e^2 + 72*(sqrt(c)
*x - sqrt(c*x^2 + a))^4*a^3*sqrt(c)*e^3 + 8*a^4*c^(3/2)*d^2*e + 12*(sqrt(c)*x -
sqrt(c*x^2 + a))*a^4*c*d*e^2 - 72*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*sqrt(c)*e^
3 + 24*a^5*sqrt(c)*e^3)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^4*a*d^4)