3.333 \(\int \frac{1}{x^3 (d+e x) \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=168 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d}-\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^3}+\frac{e \sqrt{a+c x^2}}{a d^2 x}+\frac{e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{2 a d x^2} \]

[Out]

-Sqrt[a + c*x^2]/(2*a*d*x^2) + (e*Sqrt[a + c*x^2])/(a*d^2*x) + (e^3*ArcTanh[(a*e
- c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*Sqrt[c*d^2 + a*e^2]) + (c
*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d) - (e^2*ArcTanh[Sqrt[a + c*x^2]/
Sqrt[a]])/(Sqrt[a]*d^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.356062, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d}-\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^3}+\frac{e \sqrt{a+c x^2}}{a d^2 x}+\frac{e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \sqrt{a e^2+c d^2}}-\frac{\sqrt{a+c x^2}}{2 a d x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + c*x^2]/(2*a*d*x^2) + (e*Sqrt[a + c*x^2])/(a*d^2*x) + (e^3*ArcTanh[(a*e
- c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*Sqrt[c*d^2 + a*e^2]) + (c
*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(3/2)*d) - (e^2*ArcTanh[Sqrt[a + c*x^2]/
Sqrt[a]])/(Sqrt[a]*d^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.5291, size = 146, normalized size = 0.87 \[ \frac{e^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{3} \sqrt{a e^{2} + c d^{2}}} - \frac{\sqrt{a + c x^{2}}}{2 a d x^{2}} + \frac{e \sqrt{a + c x^{2}}}{a d^{2} x} - \frac{e^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d^{3}} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**3*sqrt(a*
e**2 + c*d**2)) - sqrt(a + c*x**2)/(2*a*d*x**2) + e*sqrt(a + c*x**2)/(a*d**2*x)
- e**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(sqrt(a)*d**3) + c*atanh(sqrt(a + c*x**2)
/sqrt(a))/(2*a**(3/2)*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.404555, size = 177, normalized size = 1.05 \[ \frac{\frac{\left (c d^2-2 a e^2\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2}}+\frac{\log (x) \left (2 a e^2-c d^2\right )}{a^{3/2}}+\frac{2 e^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{2 e^3 \log (d+e x)}{\sqrt{a e^2+c d^2}}+\frac{d \sqrt{a+c x^2} (2 e x-d)}{a x^2}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

((d*(-d + 2*e*x)*Sqrt[a + c*x^2])/(a*x^2) + ((-(c*d^2) + 2*a*e^2)*Log[x])/a^(3/2
) - (2*e^3*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] + ((c*d^2 - 2*a*e^2)*Log[a + Sqrt[a
]*Sqrt[a + c*x^2]])/a^(3/2) + (2*e^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[
a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(2*d^3)

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 236, normalized size = 1.4 \[ -{\frac{1}{2\,ad{x}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{e}^{2}}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{{e}^{2}}{{d}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{a{d}^{2}x}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(c*x^2+a)^(1/2)/a/d/x^2+1/2/d*c/a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/
x)-1/d^3*e^2/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+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))+e*(c*x^2+a)^(
1/2)/a/d^2/x

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(c*x^2 + a)*(e*x + d)*x^3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.357967, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*a^(3/2)*e^3*x^2*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)*sqrt(c*d^2 + a*e^2) - 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2
)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - sqrt(c*d^2 + a*e^2)*(c*d^2 -
2*a*e^2)*x^2*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) + 2*sqrt(c*
d^2 + a*e^2)*(2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(c*d^2 + a*e^2)*a^(3/
2)*d^3*x^2), -1/4*(4*a^(3/2)*e^3*x^2*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/(
(c*d^2 + a*e^2)*sqrt(c*x^2 + a))) + (c*d^2 - 2*a*e^2)*sqrt(-c*d^2 - a*e^2)*x^2*l
og(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*sqrt(-c*d^2 - a*e^2)*
(2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(-c*d^2 - a*e^2)*a^(3/2)*d^3*x^2),
1/2*(sqrt(-a)*a*e^3*x^2*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)*sqrt(c*d^2 + a*e^2) - 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^
2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + sqrt(c*d^2 + a*e^2)*(c*d^2 -
2*a*e^2)*x^2*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + sqrt(c*d^2 + a*e^2)*(2*d*e*x -
d^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(c*d^2 + a*e^2)*sqrt(-a)*a*d^3*x^2), -1/2*(2
*sqrt(-a)*a*e^3*x^2*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*s
qrt(c*x^2 + a))) - (c*d^2 - 2*a*e^2)*sqrt(-c*d^2 - a*e^2)*x^2*arctan(sqrt(-a)/sq
rt(c*x^2 + a)) - sqrt(-c*d^2 - a*e^2)*(2*d*e*x - d^2)*sqrt(c*x^2 + a)*sqrt(-a))/
(sqrt(-c*d^2 - a*e^2)*sqrt(-a)*a*d^3*x^2)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + c x^{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(a + c*x**2)*(d + e*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286229, size = 323, normalized size = 1.92 \[ -c^{\frac{3}{2}}{\left (\frac{2 \, \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{3}}{\sqrt{-c d^{2} - a e^{2}} c^{\frac{3}{2}} d^{3}} + \frac{{\left (c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a c^{\frac{3}{2}} d^{3}} - \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} \sqrt{c} d - 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a e +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a \sqrt{c} d + 2 \, a^{2} e}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a c d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c^(3/2)*(2*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 -
a*e^2))*e^3/(sqrt(-c*d^2 - a*e^2)*c^(3/2)*d^3) + (c*d^2 - 2*a*e^2)*arctan(-(sqrt
(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*c^(3/2)*d^3) - ((sqrt(c)*x - sqrt
(c*x^2 + a))^3*sqrt(c)*d - 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*e + (sqrt(c)*x -
sqrt(c*x^2 + a))*a*sqrt(c)*d + 2*a^2*e)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^2
*a*c*d^2))