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

Optimal. Leaf size=194 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{e^2 (a e+c d x)}{a d^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}} \]

[Out]

-(e/(a*d^2*Sqrt[a + c*x^2])) - 1/(a*d*x*Sqrt[a + c*x^2]) - (2*c*x)/(a^2*d*Sqrt[a
+ c*x^2]) + (e^2*(a*e + c*d*x))/(a*d^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (e^4*
ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^2*(c*d^2 + a*e^
2)^(3/2)) + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2)*d^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.401405, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^2}-\frac{2 c x}{a^2 d \sqrt{a+c x^2}}+\frac{e^2 (a e+c d x)}{a d^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{e}{a d^2 \sqrt{a+c x^2}}-\frac{1}{a d x \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(e/(a*d^2*Sqrt[a + c*x^2])) - 1/(a*d*x*Sqrt[a + c*x^2]) - (2*c*x)/(a^2*d*Sqrt[a
+ c*x^2]) + (e^2*(a*e + c*d*x))/(a*d^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (e^4*
ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^2*(c*d^2 + a*e^
2)^(3/2)) + (e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2)*d^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.8053, size = 172, normalized size = 0.89 \[ - \frac{e^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{2} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{1}{a d x \sqrt{a + c x^{2}}} + \frac{e^{2} \left (a e + c d x\right )}{a d^{2} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{e}{a d^{2} \sqrt{a + c x^{2}}} - \frac{2 c x}{a^{2} d \sqrt{a + c x^{2}}} + \frac{e \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**4*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**2*(a*e**
2 + c*d**2)**(3/2)) - 1/(a*d*x*sqrt(a + c*x**2)) + e**2*(a*e + c*d*x)/(a*d**2*sq
rt(a + c*x**2)*(a*e**2 + c*d**2)) - e/(a*d**2*sqrt(a + c*x**2)) - 2*c*x/(a**2*d*
sqrt(a + c*x**2)) + e*atanh(sqrt(a + c*x**2)/sqrt(a))/(a**(3/2)*d**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.589887, size = 188, normalized size = 0.97 \[ \frac{e \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2} d^2}-\frac{e \log (x)}{a^{3/2} d^2}-\frac{\sqrt{a+c x^2} \left (\frac{c (a e+c d x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{1}{d x}\right )}{a^2}-\frac{e^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}+\frac{e^4 \log (d+e x)}{d^2 \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[a + c*x^2]*(1/(d*x) + (c*(a*e + c*d*x))/((c*d^2 + a*e^2)*(a + c*x^2))))/
a^2) - (e*Log[x])/(a^(3/2)*d^2) + (e^4*Log[d + e*x])/(d^2*(c*d^2 + a*e^2)^(3/2))
+ (e*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(a^(3/2)*d^2) - (e^4*Log[a*e - c*d*x + S
qrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(d^2*(c*d^2 + a*e^2)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 363, normalized size = 1.9 \[ -{\frac{1}{adx}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-2\,{\frac{cx}{{a}^{2}d\sqrt{c{x}^{2}+a}}}+{\frac{{e}^{3}}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{2}cx}{d \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{e}^{3}}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{e}{a{d}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{e}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.534645, 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^2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(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^2),x, algorithm="giac")

[Out]

Exception raised: TypeError