3.339 \(\int \frac{1}{x (d+e x) \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{e (a e+c d x)}{a d \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}+\frac{1}{a d \sqrt{a+c x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.325103, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{e (a e+c d x)}{a d \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}+\frac{1}{a d \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.3311, size = 124, normalized size = 0.84 \[ \frac{e^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{e \left (a e + c d x\right )}{a d \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{1}{a d \sqrt{a + c x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d*(a*e**2 +
c*d**2)**(3/2)) - e*(a*e + c*d*x)/(a*d*sqrt(a + c*x**2)*(a*e**2 + c*d**2)) + 1/(
a*d*sqrt(a + c*x**2)) - atanh(sqrt(a + c*x**2)/sqrt(a))/(a**(3/2)*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.461422, size = 167, normalized size = 1.14 \[ -\frac{\log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2} d}+\frac{\log (x)}{a^{3/2} d}+\frac{c d-c e x}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{d \left (a e^2+c d^2\right )^{3/2}}-\frac{e^3 \log (d+e x)}{d \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 318, normalized size = 2.2 \[{\frac{1}{ad}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{1}{d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{e}^{2}}{d \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{cex}{ \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{2}}{d \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.543231, 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/((c*x^2 + a)^(3/2)*(e*x + d)*x),x, algorithm="fricas")

[Out]

[-1/2*(2*(c*d*e*x - c*d^2)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(a) - (a*c*e^
3*x^2 + a^2*e^3)*sqrt(a)*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)) - (a*c*d^2 + a^2*e^2 + (c^2*d^
2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2
+ a)*a)/x^2))/((a^2*c*d^3 + a^3*d*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(c*d
^2 + a*e^2)*sqrt(a)), -1/2*(2*(c*d*e*x - c*d^2)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2
+ a)*sqrt(a) + 2*(a*c*e^3*x^2 + a^2*e^3)*sqrt(a)*arctan(sqrt(-c*d^2 - a*e^2)*(c*
d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) - (a*c*d^2 + a^2*e^2 + (c^2*d^2 +
a*c*e^2)*x^2)*sqrt(-c*d^2 - a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 +
a)*a)/x^2))/((a^2*c*d^3 + a^3*d*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(-c*d^2
- a*e^2)*sqrt(a)), -1/2*(2*(c*d*e*x - c*d^2)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a
)*sqrt(-a) + 2*(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2)
*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a*c*e^3*x^2 + a^2*e^3)*sqrt(-a)*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)))/((a^2*c*d^3 + a^3*d*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(c*d
^2 + a*e^2)*sqrt(-a)), -((c*d*e*x - c*d^2)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*
sqrt(-a) + (a*c*e^3*x^2 + a^2*e^3)*sqrt(-a)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x -
a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) + (a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e
^2)*x^2)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)))/((a^2*c*d^3 + a^
3*d*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(-c*d^2 - a*e^2)*sqrt(-a))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError